1.2 Discrepancies and metrics

Let ${\mathcal{M}}$ be a compact metric measure space with a fixed metric $\unicode[STIX]{x1D703}$ and a finite Borel measure  $\unicode[STIX]{x1D707}$ , normalized by

(1.10) $$\begin{eqnarray}\operatorname{diam}({\mathcal{M}},\unicode[STIX]{x1D703})=\unicode[STIX]{x1D70B},\qquad \unicode[STIX]{x1D707}({\mathcal{M}})=1,\end{eqnarray}$$

where $\operatorname{diam}({\mathcal{E}},\unicode[STIX]{x1D70C})=\sup \{\unicode[STIX]{x1D70C}(x_{1},x_{2}):x_{1},x_{2}\in {\mathcal{E}}\}$ denotes the diameter of a subset ${\mathcal{E}}\subseteq {\mathcal{M}}$ with respect to a metric  $\unicode[STIX]{x1D70C}$ .

We write $B(r,y)=\{x:\unicode[STIX]{x1D703}(x,y)<r\}$ for the ball of radius $r\in {\mathcal{I}}$ centered at $y\in {\mathcal{M}}$ and of volume $v(r,y)=\unicode[STIX]{x1D707}(B(r,y))$ ; here ${\mathcal{I}}=\{r=\unicode[STIX]{x1D703}(x_{1},x_{2}):x_{1},x_{2}\in {\mathcal{M}}\}$ is the set of all possible radii. If ${\mathcal{M}}$ is connected, then ${\mathcal{I}}=[0,\unicode[STIX]{x1D70B}]$ . We can conveniently write $B(y,r)=\emptyset$ and $v(y,r)=0$ if $r\leqslant 0$ and $B(y,r)={\mathcal{M}}$ and $v(y,r)=1$ if $r\geqslant \unicode[STIX]{x1D70B}$ .

Let ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ be a subset (distribution) of $N$ points, not necessarily different. The local discrepancy of ${\mathcal{D}}_{N}$ in the ball $B(r,y)$ is defined by

(1.11) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}[B(r,y),{\mathcal{D}}_{N}]=\#\{B(r,y)\cap {\mathcal{D}}_{N}\}-Nv(r,y)=\mathop{\sum }_{x\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D6EC}(B(r,y),x),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(B(r,y),x)=\unicode[STIX]{x1D712}(B(r,y),x)-v(r,y)\end{eqnarray}$$

and $\unicode[STIX]{x1D712}({\mathcal{E}},x)$ denotes the characteristic function of a subset ${\mathcal{E}}\subset {\mathcal{M}}$ . The quadratic discrepancy is defined by

(1.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]=\unicode[STIX]{x1D706}[{\mathcal{M}},\unicode[STIX]{x1D709},{\mathcal{D}}_{N}] & = & \displaystyle \iint _{{\mathcal{M}}\times {\mathcal{I}}}\unicode[STIX]{x1D6EC}[B(r,y),{\mathcal{D}}_{N}]^{2}\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D709}(r)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{x_{1},x_{2}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D709},x_{1},x_{2}),\end{eqnarray}$$

where $\unicode[STIX]{x1D709}$ is a measure on the set of radii ${\mathcal{I}}$ and

(1.13) $$\begin{eqnarray}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D709},y_{1},y_{2})=\int _{{\mathcal{I}}}\unicode[STIX]{x1D706}_{r}(y_{1},y_{2})\,d\unicode[STIX]{x1D709}(r),\end{eqnarray}$$

with

(1.14) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{r}(y_{1},y_{2})=\int _{{\mathcal{M}}}\unicode[STIX]{x1D6EC}(B(r,y),y_{1})\unicode[STIX]{x1D6EC}(B(r,y),y_{2})\,d\unicode[STIX]{x1D707}(y).\end{eqnarray}$$

The quantity $\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]^{1/2}$ is known as the $L_{2}$ -discrepancy. In the present paper it is more convenient to deal with the quadratic discrepancies (1.12).

We introduce the extremal characteristic

(1.15) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{N}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D706}_{N}({\mathcal{M}},\unicode[STIX]{x1D709})=\inf _{{\mathcal{D}}_{N}}\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}],\end{eqnarray}$$

where the infimum is taken over all $N$ -point subsets ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ .

It is worth noting that for the sphere $S^{d}$ the quadratic discrepancies (1.1) and (1.12) are related by

(1.16) $$\begin{eqnarray}\unicode[STIX]{x1D706}^{\text{cap}}[{\mathcal{D}}_{N}]=\unicode[STIX]{x1D706}[S^{d},\unicode[STIX]{x1D709}^{\natural },{\mathcal{D}}_{N}]\end{eqnarray}$$

with the specific measure $d\unicode[STIX]{x1D709}^{\natural }(r)=\sin r\,dr$ .

For an arbitrary metric $\unicode[STIX]{x1D70C}$ on ${\mathcal{M}}$ we define the sum of pairwise distances

(1.17) $$\begin{eqnarray}\unicode[STIX]{x1D70C}[{\mathcal{D}}_{N}]=\mathop{\sum }_{x_{1},x_{2}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D70C}(x_{1},x_{2}),\end{eqnarray}$$

and introduce the extremal characteristic

(1.18) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{N}=\unicode[STIX]{x1D70C}_{N}({\mathcal{M}})=\sup _{{\mathcal{D}}_{N}}\unicode[STIX]{x1D70C}[{\mathcal{D}}_{N}],\end{eqnarray}$$

where the supremum is taken over all $N$ -point subsets ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ . We write $\langle \unicode[STIX]{x1D70C}\rangle$ for the average value of a metric  $\unicode[STIX]{x1D70C}$ ,

(1.19) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70C}\rangle =\iint _{{\mathcal{M}}\times {\mathcal{M}}}\unicode[STIX]{x1D70C}(y_{1},y_{2})\,d\unicode[STIX]{x1D707}(y_{1})\,d\unicode[STIX]{x1D707}(y_{2}).\end{eqnarray}$$

The symmetric difference metric $\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})$ on the space ${\mathcal{M}}$ is defined by

(1.20) $$\begin{eqnarray}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},y_{1},y_{2})=\int _{{\mathcal{I}}}\unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}(y_{1},y_{2})\,d\unicode[STIX]{x1D709}(r),\end{eqnarray}$$

where

(1.21) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}(y_{1},y_{2})=\frac{1}{2}\int _{{\mathcal{M}}}\unicode[STIX]{x1D712}(B(r,y_{1})\unicode[STIX]{x1D6E5}B(r,y_{2}),y)\,d\unicode[STIX]{x1D707}(y)=\frac{1}{2}\unicode[STIX]{x1D707}(B(r,y_{1})\unicode[STIX]{x1D6E5}B(r,y_{2})),\end{eqnarray}$$

and

$$\begin{eqnarray}B(r,y_{1})\unicode[STIX]{x1D6E5}B(r,y_{2})=B(r,y_{1})\cup B(r,y_{2})\setminus B(r,y_{1})\cap B(r,y_{2})\end{eqnarray}$$

is the symmetric difference of the balls $B(r,y_{1})$ and $B(r,y_{2})$ . We have

(1.22) $$\begin{eqnarray}\unicode[STIX]{x1D712}(B(r,y_{1})\unicode[STIX]{x1D6E5}B(r,y_{2}),y)=|\unicode[STIX]{x1D712}(B(r,y_{1}),y)-\unicode[STIX]{x1D712}(B(r,y_{2}),y)|.\end{eqnarray}$$

Therefore,

(1.23) $$\begin{eqnarray}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},y_{1},y_{2})=\frac{1}{2}\iint _{{\mathcal{M}}\times {\mathcal{I}}}|\unicode[STIX]{x1D712}(B(r,y_{1}),y)-\unicode[STIX]{x1D712}(B(r,y_{2}),y)|\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D709}(r)\end{eqnarray}$$

is the $L_{1}$ -distance between the characteristic functions of the balls $B(r,y_{1})$ and $B(r,y_{2})$ . On the other hand, we have

(1.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(B(r,y_{1})\unicode[STIX]{x1D6E5}B(r,y_{2}),y) & = & \displaystyle \unicode[STIX]{x1D712}(B(r,y_{1}),y)+\unicode[STIX]{x1D712}(B(r,y_{2}),y)\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D712}(B(r,y_{1}),y)\unicode[STIX]{x1D712}(B(r,y_{2}),y).\end{eqnarray}$$

Substituting (1.24) into (1.21), we obtain

(1.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},y_{1},y_{2}) & = & \displaystyle \frac{1}{2}\iint _{{\mathcal{M}}\times {\mathcal{I}}}\big(\unicode[STIX]{x1D712}(B(r,y_{1}),y)+\unicode[STIX]{x1D712}(B(r,y_{2}),y)\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D712}(B(r,y_{1}),y)\unicode[STIX]{x1D712}(B(r,y_{2}),y)\big)\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D709}(r).\end{eqnarray}$$

The symmetry of the metric $\unicode[STIX]{x1D703}$ implies the relation

(1.26) $$\begin{eqnarray}\unicode[STIX]{x1D712}(B(r,y),x)=\unicode[STIX]{x1D712}(B(r,x),y)=\unicode[STIX]{x1D712}(r-\unicode[STIX]{x1D703}(x,y))=\unicode[STIX]{x1D712}_{r}(\unicode[STIX]{x1D703}(x,y)),\end{eqnarray}$$

where $\unicode[STIX]{x1D712}(z)$ , $z\in \mathbb{R}$ , is the characteristic function of the half-axis $(0,\infty )$ , and $\unicode[STIX]{x1D712}_{r}(\cdot )$ is the characteristic function of the interval $[0,r)$ , $0\leqslant r\leqslant \unicode[STIX]{x1D70B}$ . Using (1.26) in (1.25), we obtain the following formula for the average values (1.19) of the metric (1.21):

(1.27) $$\begin{eqnarray}\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle =\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}({\mathcal{M}},\unicode[STIX]{x1D709})\rangle =\int _{{\mathcal{I}}}\langle \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}\rangle \,d\unicode[STIX]{x1D709}(r),\end{eqnarray}$$

where

(1.28) $$\begin{eqnarray}\langle \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}\rangle =\langle \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}({\mathcal{M}})\rangle =\int _{{\mathcal{M}}}(v(r,y)-v(r,y)^{2})\,d\unicode[STIX]{x1D707}(y)=\int _{{\mathcal{M}}}v(r,y)v^{\prime }(r,y)\,d\unicode[STIX]{x1D707}(y);\end{eqnarray}$$

here $v^{\prime }(r,y)=1-v(r,y)=\unicode[STIX]{x1D707}(B^{\prime }(r,y))$ , and $B^{\prime }(r,y)={\mathcal{M}}\setminus B(r,y)$ denotes the complement of $B(r,y)$ .

A metric measure space ${\mathcal{M}}$ is called distance-invariant if the volume of any ball $v(r)=v(r,y)$ is independent of $y\in {\mathcal{M}}$ ; see [Reference Levenshtein, Pless and Huffman24, p. 504]. For such spaces the above formulas for the discrepancies and the symmetric difference metrics can be essentially simplified. A simple calculation leads to

(1.29) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D706}_{r}(y_{1},y_{2})=\unicode[STIX]{x1D707}(B(r,y_{1})\cap B(r,y_{2}))-v(r)^{2},\\ \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}(y_{1},y_{2})=v(r)-\unicode[STIX]{x1D707}(B(r,y_{1})\cap B(r,y_{2})),\\ \langle \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}\rangle =v(r)-v(r)^{2}=v(r)v^{\prime }(r).\end{array}\right\}\end{eqnarray}$$

Integrating these relations with a measure $\unicode[STIX]{x1D709}$ , one can obtain the corresponding formulas for the quantities (1.13), (1.20), and (1.27). Comparing the formulas (1.29), we obtain the identities

(1.30) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D706}_{r}(y_{1},y_{2})+\unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}(y_{1},y_{2})=\langle \unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}}\rangle ,\\ \unicode[STIX]{x1D706}(\unicode[STIX]{x1D709},y_{1},y_{2})+\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},y_{1},y_{2})=\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle .\end{array}\right\}\end{eqnarray}$$

In line with the definitions (1.17) and (1.18), we put

(1.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]=\mathop{\sum }_{x_{1},x_{2}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},x_{1},x_{2}),\end{eqnarray}$$

(1.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D703}_{N}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D703}_{N}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709},{\mathcal{M}})=\sup _{{\mathcal{D}}_{N}}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}],\end{eqnarray}$$

where the supremum is taken over all $N$ -point subsets ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ .

Comparing the identities (1.30) with the definitions (1.12) and (1.31), we obtain the $L_{1}$ -invariance principle

(1.33) $$\begin{eqnarray}\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]+\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]=\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2},\end{eqnarray}$$

for an arbitrary $N$ -point subset ${\mathcal{D}}_{N}\in {\mathcal{M}}$ . Our terminology for $L_{q}$ -invariance principles will be explained below in the comments on the relation (2.9). From (1.15), (1.32) and (1.33), we also obtain the identity

(1.34) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{N}({\mathcal{M}},\unicode[STIX]{x1D709})+\unicode[STIX]{x1D703}_{N}^{\unicode[STIX]{x1D6E5}}({\mathcal{M}},\unicode[STIX]{x1D709})=\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}({\mathcal{M}},\unicode[STIX]{x1D709})\rangle N^{2}.\end{eqnarray}$$

The typical examples of distance-invariant spaces are homogeneous spaces ${\mathcal{M}}=G/K$ , where $G$ is a compact group, $K\subset G$ is a closed subgroup, while $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D707}$ are the $G$ -invariant metric and measure on  ${\mathcal{M}}$ . In this case, the quantities (1.13), (1.14) and (1.20), (1.21) are also $G$ -invariant.

2.1 Two-point homogeneous spaces

Recall the definition and necessary facts on two-point homogeneous spaces; for more details, we refer to [Reference Besse6, Reference Helgason21–Reference Koornwinder, Askey, Koornwinder and Schempp23, Reference Wolf33, Reference Wolf34]. Let $G=G({\mathcal{M}})$ be the group of isometries of a metric space ${\mathcal{M}}$ with a metric $\unicode[STIX]{x1D703}$ , that is, $\unicode[STIX]{x1D703}(gx_{1},gx_{2})=\unicode[STIX]{x1D703}(x_{1},x_{2})$ for all $x_{1}$ , $x_{2}\in {\mathcal{M}}$ and $g\in G$ . The space ${\mathcal{M}}$ is called two-point homogeneous if for any two pairs of points $x_{1}$ , $x_{2}$ and $y_{1}$ , $y_{2}$ with $\unicode[STIX]{x1D703}(x_{1},x_{2})=\unicode[STIX]{x1D703}(y_{1},y_{2})$ there exists an isometry $g\in G$ , such that $y_{1}=gx_{1}$ , $y_{2}=gx_{2}$ . In this case, the group $G$ is obviously transitive on ${\mathcal{M}}$ and ${\mathcal{M}}=G/K$ is a homogeneous space, where the subgroup $K\subset G$ is the stabilizer of a point $x_{0}\in {\mathcal{M}}$ . Furthermore, the homogeneous space ${\mathcal{M}}$ is symmetric, that is, for any two points $y_{1}$ , $y_{2}\in {\mathcal{M}}$ there exists an isometry $g\in G$ , such that $gy_{1}=y_{2}$ , $gy_{2}=y_{1}$ .

We consider compact connected two-point homogeneous spaces ${\mathcal{M}}=G/K$ . It is known that for such spaces $G$ and $K\subset G$ are Lie groups and ${\mathcal{M}}=G/K$ are Riemannian symmetric spaces of rank one. This means that all flat totally geodesic submanifolds in ${\mathcal{M}}$ are one-dimensional and coincide with geodesics. This also means that all $G$ -invariant differential operators on ${\mathcal{M}}$ are polynomials of the Laplace–Beltrami operator on ${\mathcal{M}}$ . All such spaces are classified completely; see, for example, [Reference Wolf33, §8.12]. They are as follows:

1. (i) the $d$ -dimensional Euclidean spheres $S^{d}=\text{SO}(d+1)/\text{SO}(d)\times \{1\}$ , $d\geqslant 2$ , and $S^{1}=O(2)/O(1)\times \{1\}$ ;

2. (ii) the real projective spaces $\mathbb{R}P^{n}=O(n+1)/O(n)\times O(1)$ ;

3. (iii) the complex projective spaces $\mathbb{C}P^{n}=U(n+1)/U(n)\times U(1)$ ;

4. (iv) the quaternionic projective spaces $\mathbb{H}P^{n}=\text{Sp}(n+1)/\text{Sp}(n)\times \text{Sp}(1)$ ;

5. (v) the octonionic projective plane $\mathbb{O}P^{2}=F_{4}/\operatorname{Spin}(9)$ .

Here we use the standard notation from the theory of Lie groups; in particular, $F_{4}$ is one of the exceptional Lie groups in Cartan’s classification.

As a compact Riemannian manifold the projective space $\mathbb{F}P^{n}$ has dimension

(2.1) $$\begin{eqnarray}d=\dim _{\mathbb{R}}\mathbb{F}P^{n}=nd_{0},\qquad d_{0}=\dim _{\mathbb{R}}\mathbb{F},\end{eqnarray}$$

where $d_{0}=1,2,4,8$ for $\mathbb{F}=\mathbb{R}$ , $\mathbb{C}$ , $\mathbb{H}$ , $\mathbb{O}$ , correspondingly.

For the spheres $S^{d}$ we put $d_{0}=d$ by definition. Projective spaces of dimension $d_{0}$ ( $n=1$ ) are isomorphic to the spheres $S^{d_{0}}$ : $\mathbb{R}P^{1}\approx S^{1}$ , $\mathbb{C}P^{1}\approx S^{2}$ , $\mathbb{H}P^{1}\approx S^{4}$ , $\mathbb{O}P^{1}\approx S^{8}$ . We can conveniently agree that $d>d_{0}$ ( $n\geqslant 2$ ) for projective spaces, while the equality $d=d_{0}$ holds only for spheres. Under this convention, the dimensions $d=nd_{0}$ and $d_{0}$ define uniquely (up to isomorphism) the corresponding two-point homogeneous space which we denote by $Q=Q(d,d_{0})$ .

We consider $Q(d,d_{0})$ as a metric measure space with a metric $\unicode[STIX]{x1D703}$ and a measure $\unicode[STIX]{x1D707}$ proportional to the invariant Riemannian distance and measure on $Q(d,d_{0})$ . The coefficients of proportionality are defined to satisfy (1.10). In what follows we always assume that $n=2$ if $\mathbb{F}=\mathbb{O}$ , since projective spaces $\mathbb{O}P^{n}$ do not exist for $n>2$ . Any space $Q(d,d_{0})$ is distance-invariant and the volume of balls is given by

(2.2) $$\begin{eqnarray}v(r)=\unicode[STIX]{x1D705}(d,d_{0})\int _{0}^{r}\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\,du,\quad r\in [0,\unicode[STIX]{x1D70B}],\end{eqnarray}$$

where $\unicode[STIX]{x1D705}(d,d_{0})=B(d/2,d_{0}/2)^{-1}$ ; $B(a,b)=\unicode[STIX]{x1D6E4}(a)\unicode[STIX]{x1D6E4}(b)/\unicode[STIX]{x1D6E4}(a+b)$ and $\unicode[STIX]{x1D6E4}(a)$ are the beta and gamma functions (see [Reference Gangolli19, pp. 177–178], [Reference Helgason22, pp. 165–168], [Reference Levenshtein, Pless and Huffman24, pp. 508–510]).

The following extension of Stolarsky’s invariance principle (1.4) to all two-point homogeneous spaces $Q(d,d_{0})$ is given in [Reference Skriganov29]. The chordal metric on the spaces $Q(d,d_{0})$ can be defined by

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(x_{1},x_{2})=\sin {\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}(x_{1},x_{2}),\quad x_{1},x_{2}\in Q(d,d_{0}).\end{eqnarray}$$

Notice that expression (2.3) defines a metric because the function $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D703})=\sin \unicode[STIX]{x1D703}/2$ , $0\leqslant \unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x1D70B}$ , is concave, increasing and $\unicode[STIX]{x1D711}(0)=0$ , which implies the triangle inequality; see, for example, [Reference Deza and Laurent17, Lemma 9.0.2]. For the sphere  $S^{d}$ , we have

(2.4) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\cos \unicode[STIX]{x1D703}(x_{1},x_{2})=(x_{1},x_{2}),x_{1},x_{2}\in S^{d},\\ \unicode[STIX]{x1D70F}(x_{1},x_{2})=\sin {\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}(x_{1},x_{2})={\textstyle \frac{1}{2}}\Vert x_{1}-x_{2}\Vert ,\end{array}\right\}\end{eqnarray}$$

where $(\cdot ,\cdot )$ is the inner product and $\Vert \cdot \Vert$ is the Euclidean norm in $\mathbb{R}^{d+1}$ .

Each projective space $\mathbb{F}P^{n}$ can be canonically embedded into the unit sphere

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6F1}:Q(d,d_{0})\ni x\rightarrow \unicode[STIX]{x1D6F1}(x)\in S^{m-1}\subset \mathbb{R}^{m},\quad m={\textstyle \frac{1}{2}}(n+1)(d+2),\end{eqnarray}$$

such that

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(x_{1},x_{2})=\frac{1}{\sqrt{2}}\Vert \unicode[STIX]{x1D6F1}(x_{1})-\unicode[STIX]{x1D6F1}(x_{2})\Vert ,\quad x_{1},x_{2}\in \mathbb{F}P^{n},\end{eqnarray}$$

where $\Vert \cdot \Vert$ is the Euclidean norm in $\mathbb{R}^{m}$ . Hence, the metric $\unicode[STIX]{x1D70F}(x_{1},x_{2})$ coincides with the Euclidean length of a segment joining the points $\unicode[STIX]{x1D6F1}(x_{1})$ and $\unicode[STIX]{x1D6F1}(x_{2})$ on the unit sphere and normalized by $\operatorname{diam}(Q(d,d_{0}),\unicode[STIX]{x1D70F})=1$ . The space $Q(d,d_{0})$ canonically embedded by (2.5) is a smooth connected compact affine algebraic submanifold in  $\mathbb{R}^{m}$ . An explicit description of the embedding (2.5) in the terms of the present paper can be found in [Reference Skriganov29].

The chordal metric $\unicode[STIX]{x1D70F}$ on the complex projective space $\mathbb{C}P^{n}$ is known as the Fubini–Study metric. In connection with special point configurations in two-point homogeneous spaces the chordal metric on projective spaces has been discussed in the papers [Reference Cohn and Kumar14, Reference Cohn, Kumar and Minton15]; see also the paper [Reference Conway, Hardin and Sloane16], where the chordal metric has been defined for Grassmannian manifolds.

A careful analysis of the embedding (2.5) shows that for any space $Q=Q(d,d_{0})$ the chordal metric (2.3) and the symmetric difference metric (1.20) are related by

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(x_{1},x_{2})=\unicode[STIX]{x1D6FE}(Q)\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709}^{\natural },x_{1},x_{2}),\quad x_{1},x_{2}\in Q,\end{eqnarray}$$

where $d\unicode[STIX]{x1D709}^{\natural }(r)=\sin r\,dr$ , $r\in {\mathcal{I}}$ , and

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(Q(d,d_{0}))=\frac{\sqrt{\unicode[STIX]{x1D70B}}}{4}(d+d_{0})\frac{\unicode[STIX]{x1D6E4}(d_{0}/2)}{\unicode[STIX]{x1D6E4}((d_{0}+1)/2)}=\frac{d+d_{0}}{2d_{0}}\unicode[STIX]{x1D6FE}(S^{d_{0}}),\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}(S^{d_{0}})$ defined by (1.5). Comparing (2.7) with (1.33) and (1.34), we obtain the $L_{2}$ -invariance principle

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(Q)\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709}^{\natural },{\mathcal{D}}_{N}]+\unicode[STIX]{x1D70F}[{\mathcal{D}}_{N}]=\langle \unicode[STIX]{x1D70F}\rangle N^{2},\end{eqnarray}$$

for an arbitrary $N$ -point subset ${\mathcal{D}}_{N}\subset Q$ , and

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(Q)\unicode[STIX]{x1D706}_{N}(\unicode[STIX]{x1D709}^{\natural })+\unicode[STIX]{x1D70F}_{N}(Q)=\langle \unicode[STIX]{x1D70F}\rangle N^{2}.\end{eqnarray}$$

For the spheres $S^{d}$ relation (2.9) coincides with (1.4), while for the spaces $Q(d,d_{0})$ with $d>d_{0}$ this relation can be thought of as an extension of Stolarsky’s invariance principle to projective spaces.

Recall that a metric space ${\mathcal{M}}$ with a metric $\unicode[STIX]{x1D70C}$ is called isometrically $L_{q}$ -embeddable ( $q=1,2$ ), if there exists a map $\unicode[STIX]{x1D711}:{\mathcal{M}}\ni x\rightarrow \unicode[STIX]{x1D711}(x)\in L_{q}$ , such that $\unicode[STIX]{x1D70C}(x_{1},x_{2})=\Vert \unicode[STIX]{x1D711}(x_{1})-\unicode[STIX]{x1D711}(x_{2})\Vert _{L_{q}}$ for all $x_{1}$ , $x_{2}\in {\mathcal{M}}$ . Notice that $L_{2}$ -embeddability is stronger and implies $L_{1}$ -embeddability; see [Reference Deza and Laurent17, Theorem 6.3.1]. A compact metric space ${\mathcal{M}}$ is isometrically $L_{1}$ -embeddable with respect to a symmetric difference metric $\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})$ ; see (1.23). At the same time, a two-point homogeneous space $Q(d,d_{0})$ is isometrically $L_{2}$ -embeddable with respect to the chordal metric  $\unicode[STIX]{x1D70F}$ ; see (2.4) and (2.6). This explains our terminology of $L_{1}$ - and $L_{2}$ -invariance principles. It should be mentioned that the above discussion raises the question whether there are other measures $\unicode[STIX]{x1D709}\neq \unicode[STIX]{x1D709}^{\natural }$ for which the spaces $Q(d,d_{0})$ with the metrics $\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})$ are also $L_{2}$ -embeddable.

2.2 Bounds for discrepancies and sums of distances

We introduce the following classes of measures on the set of radii. The class $W(a,b)$ , $a\geqslant b\geqslant 1$ , consists of (non-negative) measures on ${\mathcal{I}}=[0,\unicode[STIX]{x1D70B}]$ such that

(2.11) $$\begin{eqnarray}w(a,b,\unicode[STIX]{x1D709})=\int _{0}^{\unicode[STIX]{x1D70B}}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{a-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{b-1}\,d\unicode[STIX]{x1D709}(r)<\infty .\end{eqnarray}$$

The subclass $AW(a,b)\subset W(a,b)$ consists of absolutely continuous measures. In other words, $d\unicode[STIX]{x1D709}(r)=\unicode[STIX]{x1D702}(r)\,dr$ , where $\unicode[STIX]{x1D702}$ is a locally integrable function on the interval $(0,1)$ and

(2.12) $$\begin{eqnarray}w(a,b,\unicode[STIX]{x1D709})=\int _{0}^{\unicode[STIX]{x1D70B}}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{a-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{b-1}\unicode[STIX]{x1D702}(r)\,dr<\infty .\end{eqnarray}$$

Notice that the functions $\unicode[STIX]{x1D702}$ can have rather large singularities at $r=0$ and $r=\unicode[STIX]{x1D70B}$ .

Theorem 2.1. For a space $Q=Q(d,d_{0})$ and a measure $\unicode[STIX]{x1D709}\in W(d,d_{0})$ , we have the bounds

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{N}(Q,\unicode[STIX]{x1D709})<CN^{1-1/d}\end{eqnarray}$$

and

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{N}^{\unicode[STIX]{x1D6E5}}(Q,\unicode[STIX]{x1D709})>\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2}-CN^{1-1/d}.\end{eqnarray}$$

The proof of Theorem 2.1 is given in §3.

The following theorem shows that the bounds (2.13) and (2.14) have the best possible order if the measure $\unicode[STIX]{x1D709}$ is absolutely continuous.

Theorem 2.2. For a space $Q=Q(d,d_{0})$ and a measure $\unicode[STIX]{x1D709}\in AW(d,d_{0})$ with $w(d,d_{0},\unicode[STIX]{x1D709})>0$ , we have the two-sided bounds

(2.15) $$\begin{eqnarray}\begin{array}{@{}c@{}}cN^{1-1/d}<\unicode[STIX]{x1D706}_{N}(Q,\unicode[STIX]{x1D709})<CN^{1-1/d}\end{array}\end{eqnarray}$$

and

(2.16) $$\begin{eqnarray}\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2}-cN^{1-1/d}>\unicode[STIX]{x1D703}_{N}^{\unicode[STIX]{x1D6E5}}(Q,\unicode[STIX]{x1D709})>\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2}-CN^{1-1/d}.\end{eqnarray}$$

In particular, for the chordal metric on $Q$ , we have

(2.17) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70F}\rangle N^{2}-cN^{1-1/d}>\unicode[STIX]{x1D70F}_{N}(Q)>\langle \unicode[STIX]{x1D70F}\rangle N^{2}-CN^{1-1/d}.\end{eqnarray}$$

The bounds (2.17) can be thought of as an extension of the bounds (1.6) and (1.9) to projective spaces. At the same time, the bounds (2.15) and (2.16) with an arbitrary measure $\unicode[STIX]{x1D709}\in AW(d,d_{0})$ are new even in the case of spheres.

The proof of Theorem 2.2 is given in §4. It is clear that the right bounds in (2.15) and (2.16) follow immediately from Theorem 2.1. In §4 we prove the left bound in (2.15). By the invariance principle (1.34), this immediately implies the left bound in (2.16). Finally, the bounds (2.17) follow from (2.16) and the invariance principle (2.10).

The following example shows that Theorem 2.2 fails if $\unicode[STIX]{x1D709}\in W(d,d_{0})$ is not absolutely continuous. It is known that for spheres the geodesic distance is a symmetric difference metric

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D703}(y_{1},y_{2})=\unicode[STIX]{x1D70B}\{1-2\unicode[STIX]{x1D707}(B(\unicode[STIX]{x1D70B}/2,y_{1})\cap B(\unicode[STIX]{x1D70B}/2,y_{2}))\},\end{eqnarray}$$

where the balls $B(\unicode[STIX]{x1D70B}/2,y)\subset S^{d}$ are hemispheres; see [Reference Deza and Laurent17, §6.4]. In this case, the invariance principle (1.34) takes the form

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{N}(S^{d})+2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D706}_{N}(S^{d},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70B}/2})=\langle \unicode[STIX]{x1D703}\rangle N^{2},\end{eqnarray}$$

where $\langle \unicode[STIX]{x1D703}\rangle =\unicode[STIX]{x1D70B}/2$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70B}/2}$ is the Dirac mass at the point $r=\unicode[STIX]{x1D70B}/2$ .

On the other hand, one can easily check that $\unicode[STIX]{x1D706}_{N}(S^{d},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70B}/2})=0$ for even $N$ and $\unicode[STIX]{x1D706}_{N}(S^{d},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70B}/2})\leqslant \unicode[STIX]{x1D70B}/2$ for odd  $N$ . Hence, the bounds (2.15) and (2.16) fail for the singular measure $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70B}/2}$ . This example is considered in more detail in [Reference Bilyk, Dai and Matzke8, Reference Skriganov28].

It can be proved that the statements of Theorem 2.2 always hold for any measure $\unicode[STIX]{x1D709}\in W(d,d_{0})$ such that the Fourier integrals

(2.20) $$\begin{eqnarray}\int _{c_{0}}^{\unicode[STIX]{x1D70B}-c_{0}}\exp (\text{i}ur)\,d\unicode[STIX]{x1D709}(r)\rightarrow 0\quad \text{as}~u\rightarrow \infty ,\end{eqnarray}$$

for each arbitrarily small $c_{0}>0$ ; see the proof of Lemma 4.2 below. Theorem 2.2 is stated above for measures $\unicode[STIX]{x1D709}\in AW(d.d_{0})$ , since in this case the condition (2.20) obviously holds by the Riemann–Lebesgue lemma.

One further comment should be made to mention relations with the theory of random fields on manifolds. Recall that the Lévi–Schoenberg kernel for a metric $\unicode[STIX]{x1D70C}$ on a space ${\mathcal{M}}$ is defined by

(2.21) $$\begin{eqnarray}k(\unicode[STIX]{x1D70C},y_{1},y_{2})={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70C}(y_{1},y_{0})+\unicode[STIX]{x1D70C}(y_{2},y_{0})-\unicode[STIX]{x1D70C}(y_{1},y_{2})),\end{eqnarray}$$

where $y_{0}\in {\mathcal{M}}$ is a fixed point; see [Reference Gangolli19]. The metric $\unicode[STIX]{x1D70C}$ can be recovered from the kernel $k$ by $\unicode[STIX]{x1D70C}(y_{1},y_{2})=k(\unicode[STIX]{x1D70C},y_{1},y_{1})+k(\unicode[STIX]{x1D70C},y_{2},y_{2})-2k(\unicode[STIX]{x1D70C},y_{1},y_{2})$ .

If the kernel (2.21) is positive definite (i.e. $\sum _{1\leqslant i,j\leqslant N}\bar{c}_{i}c_{j}k(\unicode[STIX]{x1D70C},y_{i},y_{j})\geqslant 0$ for any points $y_{1},\ldots ,y_{N}\in {\mathcal{M}}$ and complex numbers $c_{1},\ldots ,c_{N}$ ), then it can be thought of as the covariance of a Gaussian process (a random field) on  ${\mathcal{M}}$ . The standard methods of probability theory enable one to construct such a random field as a mapping $W:{\mathcal{M}}\ni y\rightarrow W(y)=W(y,\unicode[STIX]{x1D714})\in L_{2}(\unicode[STIX]{x1D6FA},d\unicode[STIX]{x1D714})$ , with $W(y_{0})=0$ , $\mathbb{E}W(y_{1})=0$ , $\mathbb{E}W(y_{1})W(y_{2})=k(\unicode[STIX]{x1D70C},y_{1},y_{2})$ , and $\mathbb{E}(W(y_{1})-W(y_{2}))^{2}=\unicode[STIX]{x1D70C}(y_{1},y_{2})$ . Here $L_{2}(\unicode[STIX]{x1D6FA},d\unicode[STIX]{x1D714})$ is the Hilbert space of real-valued square-integrable random variables on a probability space $\unicode[STIX]{x1D6FA}$ with a probability measure $d\unicode[STIX]{x1D714}$ , and $\mathbb{E}$ denotes the expectation on $L_{2}(\unicode[STIX]{x1D6FA},d\unicode[STIX]{x1D714})$ . Furthermore, if ${\mathcal{M}}$ is a Riemannian manifold and $\unicode[STIX]{x1D70C}$ is Hölder continuous with respect to the geodesic distance  $\unicode[STIX]{x1D703}$ (i.e.  $\unicode[STIX]{x1D70C}(y_{1},y_{2})\lesssim \unicode[STIX]{x1D703}(y_{1},y_{2})^{\unicode[STIX]{x1D6FD}}$ , $\unicode[STIX]{x1D6FD}>0$ ), then trajectories of the random field $W(y,\unicode[STIX]{x1D714})$ are continuous functions of $y\in {\mathcal{M}}$ for almost all $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$ . For more details; see, for example, [Reference Gangolli19, Reference Malyarenko25].

For the symmetric difference metrics (1.20) and (1.21) the definition (2.21) takes the form

(2.22) $$\begin{eqnarray}k(\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709}),y_{1},y_{2})=\int _{{\mathcal{I}}}k(\unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}},y_{1},y_{2})\,d\unicode[STIX]{x1D709}(r),\end{eqnarray}$$

where

(2.23) $$\begin{eqnarray}k(\unicode[STIX]{x1D703}_{r}^{\unicode[STIX]{x1D6E5}},y_{1},y_{2})=\frac{1}{2}\int _{{\mathcal{M}}}F_{r}(y_{1},y)F_{r}(y_{2},y)\,d\unicode[STIX]{x1D707}(y),\end{eqnarray}$$

and $F_{r}(x,y)=\unicode[STIX]{x1D712}(B(r,x),y)-\unicode[STIX]{x1D712}(B(r,y_{0}),y)$ . Therefore, the Lévi–Schoenberg kernels are positive definite for all symmetric difference metrics. In particular, in view of (2.18), for the geodesic metric on  $S^{d}$ , the formula (2.23) can be written as

(2.24) $$\begin{eqnarray}k(\unicode[STIX]{x1D703},y_{1},y_{2})=\unicode[STIX]{x1D70B}\int _{S^{d}}F_{\unicode[STIX]{x1D70B}/2}(y_{1},y)F_{\unicode[STIX]{x1D70B}/2}(y_{2},y)\,d\unicode[STIX]{x1D707}(y).\end{eqnarray}$$

Therefore, the kernel $\unicode[STIX]{x1D703}(y_{1},y_{0})+\unicode[STIX]{x1D703}(y_{2},y_{0})-\unicode[STIX]{x1D703}(y_{1},y_{2})$ is positive definite. This is the well-known theorem of Lévy; see [Reference Gangolli19]. Originally, its proof was obtained in terms of “white noise” integrals for random fields on  $S^{d}$ . A direct proof was given in [Reference Gangolli19, §4] in terms of an expansion of the metric $\unicode[STIX]{x1D703}$ in Gegenbauer polynomials. The above proof is likely to be the simplest.

Notice that in contrast to the spheres $S^{d}$ , the geodesic metrics $\unicode[STIX]{x1D703}$ on the projective spaces $\mathbb{R}P^{d}$ , $\mathbb{C}P^{n}$ , $\mathbb{H}P^{n}$ , and $\mathbb{O}P^{2}$ are not symmetric difference metrics, and analogs of Lévy’s theorem are not true for projective spaces. This follows from [Reference Gangolli19, §4, pp. 225–226]. At the same time, the Lévi–Schoenberg kernel $k(\unicode[STIX]{x1D70F},y_{1},y_{2})$ for the chordal metric $\unicode[STIX]{x1D70F}$ is positive definite for all spaces $Q(d,d_{0})$ ; see (2.7).

Hence the kernels (2.22) and (2.23) are covariances of random fields on the spaces $Q(d,d_{0})$ . Notice that trajectories of such fields are continuous almost surely. This follows from Lemma 3.1(ii) given below. A comprehensive analysis of the random fields with the covariances (2.22) and (2.23) is beyond the scope of the present paper.

2.3 Applications to $t$ -designs

Consider a smooth compact connected manifold ${\mathcal{M}}$ in $\mathbb{R}^{m}$ of dimension $d$ equipped with a smooth Riemannian structure, and the geodesic distance $\unicode[STIX]{x1D703}$ and the Riemanniam measure $\unicode[STIX]{x1D707}$ normalized by (1.10). An $N$ -point subset ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ is called a $t$ -design if the exact quadrature formula

(2.25) $$\begin{eqnarray}\mathop{\sum }_{x\in {\mathcal{D}}_{N}}F(x)=N\int _{{\mathcal{M}}}F(y)\,d\unicode[STIX]{x1D707}(y)\end{eqnarray}$$

holds for all polynomials $F(x)$ , $x\in \mathbb{R}^{m}$ , of total degree not exceeding  $t$ .

It is known (see, for example, [Reference Etayo, Marzo and Ortega-Cerdà18]) that any $N$ -point $t$ -design ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ satisfies the bound $N\gtrsim t^{d}$ . An $N$ -point $t$ -design ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ is called optimal if

(2.26) $$\begin{eqnarray}c_{+}t^{d}\geqslant N\geqslant c_{-}t^{d}\end{eqnarray}$$

with some positive constants $c_{+}$ and $c_{-}$ independent of $N$ and  $t$ . Actually, in this definition we deal with sequences of $N$ -point $t$ -designs ${\mathcal{D}}_{N}$ as $N\rightarrow \infty$ .

The above definition of $t$ -design is applicable to any space $Q(d,d_{0})$ , since it can be realized as a Riemannian submanifold in $\mathbb{R}^{m}$ by the embedding (2.5). However, for the spaces $Q(d,d_{0})$ , one can give an equivalent definition in invariant terms; see [Reference Banai, Deza, Frankl and Rosenberg3, Reference Levenshtein, Pless and Huffman24]. An $N$ -point subset ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ is a $t$ -design if the exact quadrature formula

(2.27) $$\begin{eqnarray}\mathop{\sum }_{x_{1},x_{2}\in {\mathcal{D}}_{N}}f(\cos \unicode[STIX]{x1D703}(x_{1},x_{2}))=N^{2}\iint _{Q\times Q}f(\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))\,d\unicode[STIX]{x1D707}(y_{1})\,d\unicode[STIX]{x1D707}(y_{2})\end{eqnarray}$$

holds for all polynomials $f(z)$ , $z\in \mathbb{C}$ , of degree not exceeding  $t$ . The formula (2.27) is also equivalent to the quadrature formula

(2.28) $$\begin{eqnarray}\mathop{\sum }_{x\in {\mathcal{D}}_{N}}f(\cos \unicode[STIX]{x1D703}(x,y))=N\int _{Q}f(\cos \unicode[STIX]{x1D703}(x,y))\,d\unicode[STIX]{x1D707}(x),\end{eqnarray}$$

for all $y\in Q$ . Another equivalent definition of $t$ -designs can be given in terms of spherical functions on the spaces $Q(d,d_{0})$ ; see (4.32) below.

For any $N$ -point subset ${\mathcal{D}}_{N}\subset {\mathcal{M}}$ we put

(2.29) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r]=\int _{Q}(\#\{B(r,y)\cap {\mathcal{D}}_{N}\})^{2}\,d\unicode[STIX]{x1D707}(y),\\ \displaystyle \unicode[STIX]{x1D708}_{\infty }[{\mathcal{D}}_{N},r]=\Big(\max _{y\in Q}\#\{B(r,y)\cap {\mathcal{D}}_{N}\}\Big)^{2},\end{array}\right\}\end{eqnarray}$$

and $\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r]=\unicode[STIX]{x1D708}_{\infty }[{\mathcal{D}}_{N},r]=N^{2}$ if $r\geqslant \unicode[STIX]{x1D70B}$ . Obviously, $\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r]\leqslant$ $\unicode[STIX]{x1D708}_{\infty }[{\mathcal{D}}_{N},r]$ .

Our result on $t$ -designs can be stated as follows.

Inequality (2.31) follows immediately from (2.30) and definition (2.26). The proof of the bound (2.30) is given in §4. The proof relies on the theory of spherical functions on $Q(d,d_{0})$ .

If the factor $\unicode[STIX]{x1D708}_{1}$ in (2.31) is bounded from above by a constant independent of  $N$ , then the order of the bound (2.31) is best possible. Two simple sufficient conditions for this are given below in Lemma 2.1. We introduce some definitions. For an arbitrary $N$ -point subset ${\mathcal{D}}_{N}\subset Q=Q(d,d_{0})$ , we put

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}[{\mathcal{D}}_{N}]={\textstyle \frac{1}{2}}\min \{\unicode[STIX]{x1D703}(x_{1},x_{2}):x_{1},x_{2}\in {\mathcal{D}}_{N},x_{1}\neq x_{2}\}.\end{eqnarray}$$

The balls $B(\unicode[STIX]{x1D6FF},x)$ , $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}[{\mathcal{D}}_{N}]$ , $x\in {\mathcal{D}}_{N}$ , do not overlap. Therefore, $\unicode[STIX]{x1D6FF}\lesssim N^{-1/d}$ , since the volume $v(r)\simeq r^{d}$ . An $N$ -point subset ${\mathcal{D}}_{N}\subset Q$ is called well separated if

(2.33) $$\begin{eqnarray}C_{1}N^{-1/d}\geqslant \unicode[STIX]{x1D6FF}[{\mathcal{D}}_{N}]\geqslant c_{1}N^{-1/d}\end{eqnarray}$$

with constants $C_{1}\geqslant c_{1}>0$ independent of $N$ .

Consider an equal-measure partition ${\mathcal{P}}_{N}=\{P_{i}\}_{1}^{N}$ of  $Q$ ,

$$\begin{eqnarray}Q=\mathop{\bigcup }_{i=1}^{N}P_{i},\qquad \unicode[STIX]{x1D707}(P_{i}\cap P_{j})=0,\qquad i\neq j,\,\unicode[STIX]{x1D707}(P_{i})=1/N,\end{eqnarray}$$

and put $\operatorname{Diam}({\mathcal{P}}_{N},\unicode[STIX]{x1D703})=\max _{1\leqslant i\leqslant N}\operatorname{diam}(P_{i},\unicode[STIX]{x1D703})$ . We say that an equal-measure partition ${\mathcal{P}}_{N}$ is of small diameter if

(2.34) $$\begin{eqnarray}\operatorname{Diam}({\mathcal{P}}_{N},\unicode[STIX]{x1D703})\leqslant C_{2}N^{-1/d}\end{eqnarray}$$

with a constant $C_{2}>0$ independent of $N$ . Constructions of equal-measure partitions of small diameter are known for a large class of compact metric measure spaces, including all compact Riemannian manifolds; see [Reference Gigante and Leopardi20].

We say that an $N$ -point subset ${\mathcal{D}}_{N}=\{x_{i}\}_{1}^{N}\in Q$ is subordinated to a partition ${\mathcal{P}}_{N}=\{P_{i}\}_{1}^{N}$ of $Q$ if $x_{i}\in P_{i}$ , $i=1,\ldots ,N$ .

Lemma 2.1. Suppose that an $N$ -point subset ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ satisfies one of the following conditions:

1. (i) ${\mathcal{D}}_{N}$ is well separated;

2. (ii) ${\mathcal{D}}_{N}$ is subordinated to an equal-measure partition of small diameter.

Then, for any constant $c>0$ , there exists a constant $C=C(c)$ independent of $N$ such that

(2.35) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{\infty }[{\mathcal{D}}_{N},cN^{-1/d}]\leqslant C.\end{eqnarray}$$

Comparing Theorem 2.3 with Lemma 2.1, and taking into account the left bounds of Theorem 2.2, we arrive at the following result.

Corollary 2.1. Let $\unicode[STIX]{x1D709}\in AW(d,d_{0})$ with $w(d,d_{0},\unicode[STIX]{x1D709})>0$ and an $N$ -point subset ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ be an optimal $t$ -design. Suppose that ${\mathcal{D}}_{N}$ satisfies one of the conditions (i) or (ii) of Lemma 2.1. Then, for all sufficiently large $N$ , we have the two-sided bounds

(2.36) $$\begin{eqnarray}\displaystyle \displaystyle \langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2}-cN^{1-1/d} & {>} & \displaystyle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]>\langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D709})\rangle N^{2}-CN^{1-1/d},\qquad\end{eqnarray}$$

(2.37) $$\begin{eqnarray}\displaystyle cN^{1-1/d} & {<} & \displaystyle \unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]<CN^{1-1/d}.\qquad\end{eqnarray}$$

In particular, for the chordal metric $\unicode[STIX]{x1D70F}$ on $Q(d,d_{0})$ , we have

(2.38) $$\begin{eqnarray}\langle r\rangle N^{2}-cN^{1-1/d}>\unicode[STIX]{x1D70F}[{\mathcal{D}}_{N}]>\langle r\rangle N^{2}-CN^{1-1/d}.\end{eqnarray}$$

The positive constants in (2.36)–(2.38) are independent of  $N$ .

The existence of optimal $t$ -designs was a long-standing open problem known as the Korevaar–Meyers conjecture. In the papers [Reference Bondarenko, Radchenko and Viazovska9, Reference Bondarenko, Radchenko and Viazovska10] by Bondarenko, Radchenko and Viazovska, the problem was solved for spherical $t$ -designs. In [Reference Bondarenko, Radchenko and Viazovska9] the existence of optimal $t$ -designs ${\mathcal{D}}_{N}\subset S^{d}$ was proved for all sufficiently large  $N$ . Such optimal $t$ -designs were constructed in [Reference Bondarenko, Radchenko and Viazovska9], as subsets subordinated to an equal-measure partition of small diameter. In the subsequent paper [Reference Bondarenko, Radchenko and Viazovska10], the authors proved that optimal spherical $t$ -designs can be constructed as well-separated subsets. Therefore, Corollary 2.1 is applicable for the spheres  $S^{d}$ .

Using optimal spherical $t$ -designs ${\mathcal{D}}_{N}\subset S^{d}$ , one can easily construct optimal $[t/2]$ -designs ${\mathcal{D}}_{N}^{\circ }$ on the real projective space $\mathbb{R}P^{d}=Q(d,1)$ , merely as a set of one-dimensional subspaces in $\mathbb{R}^{d}$ containing points $x\in {\mathcal{D}}_{N}$ . One can easily check that if ${\mathcal{D}}_{N}$ is well separated on $S^{d}$ , then ${\mathcal{D}}_{N}^{\circ }$ satisfies the bound (2.35) on $\mathbb{R}P^{d}$ . Therefore, the optimal $[t/2]$ -designs ${\mathcal{D}}_{N}^{\circ }\subset \mathbb{R}P^{d}$ satisfy the bounds (2.36)–(2.38).

The corresponding generalization to the projective spaces $\mathbb{C}P^{n}$ , $\mathbb{H}P^{n}$ , and $\mathbb{O}P^{2}$ is not so straightforward. Using the mapping (2.5), one can realize any space $Q(d,d_{0})$ as a smooth connected compact affine algebraic submanifold in  $\mathbb{R}^{m}$ . For such manifolds the existence of optimal $t$ -designs as subsets subordinated to an equal-measure partition of small diameter has been established in [Reference Etayo, Marzo and Ortega-Cerdà18]. Therefore, Corollary 2.1 is also applicable to the projective spaces $\mathbb{C}P^{n}$ , $\mathbb{H}P^{n}$ and $\mathbb{O}P^{2}$ .

4.1 Fourier analysis on two-point homogeneous spaces

The spaces $Q(d,d_{0})$ belong to a specific and very important class of commutative spaces. The general theory of commutative spaces can be found in [Reference Wolf34]; see also [Reference Helgason22, Reference Koornwinder, Askey, Koornwinder and Schempp23] and [Reference Vilenkin and Klimyk32, vol. III, Ch. 17]. In the case of compact groups the theory becomes significantly simpler. In this section we outline the necessary facts and formulas in the explicit form convenient for subsequent calculations.

Let $G$ be a compact group and $K\subset G$ a closed subgroup. Denote by $\unicode[STIX]{x1D707}_{G}$ and $\unicode[STIX]{x1D707}_{K}$ Haar measures on the groups $G$ and  $K$ ; correspondingly, $\unicode[STIX]{x1D707}_{G}(G)=\unicode[STIX]{x1D707}_{K}(K)=1$ . As before, $\unicode[STIX]{x1D707}$ denotes the invariant measure on the homogeneous space $Q=G/K$ , and $\unicode[STIX]{x1D707}_{G}=\unicode[STIX]{x1D707}_{K}\times \unicode[STIX]{x1D707}$ . $L_{q}(G)$ , $q=1,2$ , denote spaces of functions on $G$ integrable with the power $q$ with respect to the Haar measure. $L_{q}(G/K)$ and $L_{q}(K\setminus G/K)$ denote subspaces in $L_{q}(G)$ consisting of functions such that $f(gk)=f(g)$ , $k\in K$ , and, correspondingly, $f(k_{1}gk_{2})=f(g),k_{1},k_{2}\in K$ . Obviously, functions in these subspaces can be thought of as functions on $Q=G/K$ . The spaces $L_{1}(K\setminus G/K)\subset L_{1}(G/K)\subset L_{1}(G)$ are associative Banach algebras with the convolution product

(4.1) $$\begin{eqnarray}f_{1}\ast f_{2}(g)=\int _{G}f_{1}(gh^{-1})f_{2}(h)\,d\unicode[STIX]{x1D707}_{G}(h).\end{eqnarray}$$

If the algebra $L_{1}(K\setminus G/K)$ is commutative, the pair of groups $K\subset G$ is called a Gelfand pair and the corresponding homogeneous space $Q=G/K$ is called a commutative space; see [Reference Wolf34]. Riemannian symmetric spaces and two-point homogeneous spaces are two large classes of commutative spaces; see [Reference Helgason22, Reference Wolf34]. The spaces $Q(d,d_{0})$ belong to both of these classes.

Consider the unitary representation of a group $G$ in the space $L_{2}(G/K)$ ,

(4.2) $$\begin{eqnarray}T(g)f(h)=f(g^{-1}h),\quad f\in L_{2}(G/K),\quad g,h\in G,\end{eqnarray}$$

and its decomposition into the orthogonal sum, $T=\widehat{\bigoplus _{l\geqslant 0}}\,T_{l}$ , $L_{2}(G/K)=\widehat{\bigoplus _{l\geqslant 0}}\,V_{l}$ , of unitary irreducible representations $T_{l}$ in finite-dimensional spaces  $V_{l}$ . Let $m_{l}=\dim V_{l}$ , and $(\cdot ,\cdot )$ denote the inner product in  $V_{l}$ .

If $Q=G/K$ is a commutative space, then the irreducible representations $T_{l}$ are pairwise non-equivalent and each subspace $V_{l}$ contains a single $K$ -invariant unit vector $\mathbf{e}^{(l)}$ , that is, $T_{l}(k)\mathbf{e}^{(l)}=\mathbf{e}^{(l)}$ for all $k\in K$ . Fix an orthonormal basis $\mathbf{e}_{1},\ldots ,\mathbf{e}_{m_{l}}$ in the space $V_{l}$ , such that $\mathbf{e}_{1}=\mathbf{e}^{(l)}$ and define the matrix elements $t_{ij}^{(l)}(g)=(T_{l}(g)\mathbf{e}_{i},\mathbf{e}_{j})$ . Then we have

(4.3) $$\begin{eqnarray}t_{ij}^{(l)}(g_{1}g_{2})=\mathop{\sum }_{p=1}^{m_{l}}t_{ip}^{(l)}(g_{1})t_{pj}^{(l)}(g_{2})\quad \text{and}\quad t_{ij}^{(l)}(g^{-1})=\overline{t_{ji}^{(l)}(g)}.\end{eqnarray}$$

We also have the orthogonality relations

(4.4) $$\begin{eqnarray}\int _{G}t_{ij}^{(l)}(g)\overline{t_{ij}^{(l^{\prime })}(g)}\,d\unicode[STIX]{x1D707}_{G}(g)=m_{l}^{-1}\unicode[STIX]{x1D6FF}_{ll^{\prime }}\unicode[STIX]{x1D6FF}_{ii^{\prime }}\unicode[STIX]{x1D6FF}_{jj^{\prime }},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{ab}$ is Kronecker’s symbol. The sets of functions $\{m_{l}^{1/2}t_{1j}^{(l)}(g),j=1,\ldots ,m_{l},l\geqslant 0\}$ and $\{m_{l}^{1/2}t_{11}^{(l)}(g),l\geqslant 0\}$ are orthonormal bases in the spaces $L_{2}(G/K)$ and $L_{2}(K\setminus G/K)$ , respectively; see [Reference Vilenkin and Klimyk32, vol. I, §2.3] (notice that in [Reference Vilenkin and Klimyk32] the subgroup $K$ in a Gelfand pair $K\subset G$ is called massive).

The matrix elements $\unicode[STIX]{x1D711}_{l}(g)=t_{11}^{(l)}(g)$ are called zonal spherical functions or simply spherical functions (the matrix elements $t_{1j}^{(l)}(g)$ , $j=2,\ldots ,m_{l}$ are called associated spherical functions). The definition of $\unicode[STIX]{x1D711}_{l}(g)$ and formula (4.3) immediately imply that the spherical functions are continuous, $\unicode[STIX]{x1D711}_{l}(e)=1$ , where $e$ is the unit element in  $G$ , $|\unicode[STIX]{x1D711}_{l}(g)|\leqslant 1$ for all $g\in G$ , and

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{l}(g_{1}g_{2}^{-1})=\mathop{\sum }_{j=1}^{m_{l}}t_{1j}^{(l)}(g_{1})\overline{t_{1j}^{(l)}(g_{2})}\quad \text{and}\quad \unicode[STIX]{x1D711}_{l}(g)=\overline{\unicode[STIX]{x1D711}_{l}(g^{-1})}.\end{eqnarray}$$

It follows from (4.5) that all $\unicode[STIX]{x1D711}_{l}$ are positive definite:

(4.6) $$\begin{eqnarray}\mathop{\sum }_{1\leqslant i,j\leqslant N}\overline{c_{i}}c_{j}\unicode[STIX]{x1D711}_{l}(g_{i}^{-1}g_{j})\geqslant 0\end{eqnarray}$$

for any $g_{1},\ldots ,g_{N}\in G$ and any complex numbers $c_{1},\ldots ,c_{N}$ .

From (4.1), (4.4), and (4.5), we obtain the following relation for the convolution of two spherical functions:

(4.7) $$\begin{eqnarray}(\unicode[STIX]{x1D711}_{l}\ast \unicode[STIX]{x1D711}_{l^{\prime }})(g)=\unicode[STIX]{x1D6FF}_{ll^{\prime }}m_{l}^{-1}\unicode[STIX]{x1D711}_{l}(g).\end{eqnarray}$$

Putting $g=e$ in (4.7), we obtain the following formula for the dimensions  $m_{l}$ :

(4.8) $$\begin{eqnarray}m_{l}=\bigg(\int _{G}|\unicode[STIX]{x1D711}_{l}(g)|^{2}\,d\unicode[STIX]{x1D707}_{G}(g)\bigg)^{-1}.\end{eqnarray}$$

Functions $f\in L_{2}(K\setminus G/K)$ have the expansions

(4.9) $$\begin{eqnarray}f(g)\sim \mathop{\sum }_{l\geqslant 0}m_{l}c_{l}(f)\unicode[STIX]{x1D711}_{l}(g),\end{eqnarray}$$

where ${\sim}$ denotes $L_{2}$ -convergence, Fourier coefficients are given by

(4.10) $$\begin{eqnarray}c_{l}(f)=\int _{G}f(g)\overline{\unicode[STIX]{x1D711}_{l}(g)}\,d\unicode[STIX]{x1D707}_{G}(g),\end{eqnarray}$$

and Parseval’s equality has the form $\int _{G}|f(g)|^{2}\,d\unicode[STIX]{x1D707}_{G}(g)=\sum _{l\geqslant 0}m_{l}\,|c_{l}(f)|^{2}$ . Actually, this is the Peter–Weyl theorem written for the space $L_{2}(K\setminus G/K)$ ; see [Reference Vilenkin and Klimyk32, vol. I, Ch. 2].

Substituting expansion (4.9) for two functions $f_{1},f_{2}\in L_{2}(K\setminus G/K)$ into (4.1) and using the relation (4.7), we obtain

(4.11) $$\begin{eqnarray}f_{1}\ast f_{2}(g)=\mathop{\sum }_{l\geqslant 0}m_{l}c_{l}(f_{1})c_{l}(f_{2})\unicode[STIX]{x1D711}_{l}(g).\end{eqnarray}$$

Applying the Cauchy–Schwarz inequality to (4.11), we observe that the series (4.11) converges absolutely. Since the spherical functions $\unicode[STIX]{x1D711}_{l}$ are continuous and $|\unicode[STIX]{x1D711}_{l}(g)|\leqslant 1$ , we conclude that the convolution $f_{1}\ast f_{2}$ is a continuous function.

The facts listed above are true for all compact commutative spaces. Now we wish to specify these facts for two-point homogeneous spaces. Let $K\subset G$ be compact groups and $Q=G/K$ a two-point homogeneous space with a $G$ -invariant metric  $\unicode[STIX]{x1D703}$ . Suppose that $K$ is the stabilizer of a fixed point $y_{0}\in Q$ . It follows from the definition that the subgroup $K$ is transitive on each sphere $\unicode[STIX]{x1D6F4}(r,y_{0})=\{y:\unicode[STIX]{x1D703}(y,y_{0})=r\}\subset Q$ , $r\in {\mathcal{I}}$ . Thus, any function $f\in L_{q}(K\setminus G/K)$ , as a function on  $Q$ , is constant on each sphere $\unicode[STIX]{x1D6F4}(r,y_{0})$ , and we can write

(4.12) $$\begin{eqnarray}f(g)=F(\unicode[STIX]{x1D703}(gy_{0},y_{0}))\end{eqnarray}$$

with a function $F(r)$ , $r\in {\mathcal{I}}$ . In other words, the set of double cosets $K\setminus G/K$ is in one-to-one correspondence with the set of radii  ${\mathcal{I}}$ .

Using (4.12), the convolution (4.1) can be written in the form

(4.13) $$\begin{eqnarray}\displaystyle (f_{1}\ast f_{2})(g_{1}^{-1}g_{2}) & = & \displaystyle \int _{G}F_{1}(\unicode[STIX]{x1D703}(g_{1}y_{0},gy_{0}))F_{2}(\unicode[STIX]{x1D703}(gy_{0},g_{2}y_{0}))\,d\unicode[STIX]{x1D707}(g)\nonumber\\ \displaystyle & = & \displaystyle \int _{Q}F_{1}(\unicode[STIX]{x1D703}(y_{1},y))F_{2}(\unicode[STIX]{x1D703}(y,y_{2}))\,d\unicode[STIX]{x1D707}(y),\end{eqnarray}$$

where $y_{1}=g_{1}y_{0}$ , $y_{2}=g_{2}y_{0}$ . For a function of the form (4.12) we have

(4.14) $$\begin{eqnarray}\int _{G}|f(g)|^{2}\,d\unicode[STIX]{x1D707}_{G}(g)=\int _{Q}|F(\unicode[STIX]{x1D703}(y,y_{0}))|^{2}\,d\unicode[STIX]{x1D707}(y)=\int _{{\mathcal{I}}}|F(r)|^{2}\,dv(r),\end{eqnarray}$$

where the last integral is thought of as a Stieltjes integral with the non-decreasing function $v(r)=\unicode[STIX]{x1D707}(B(r,y_{0}))$ , $r\in {\mathcal{I}}$ . It follows from (4.12) and (4.14) that the mapping $f\rightarrow F$ is an isometry of the space $L_{2}(K\setminus G/K)$ onto the space $L_{2}({\mathcal{I}},v)$ of functions $F(r)$ , $r\in {\mathcal{I}}$ , with the norm $\Vert F\Vert =(\int _{{\mathcal{I}}}|F(r)|^{2}\,dv(r))^{1/2}$ .

Since the spherical functions $\unicode[STIX]{x1D711}_{l}\in L_{2}(K\setminus G/K)$ , they can be written in the form (4.12),

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{l}(g)=\unicode[STIX]{x1D6F7}_{l}(\unicode[STIX]{x1D703}(gy_{0},y_{0})),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}_{l}\in L_{2}({\mathcal{I}},v)$ , and putting $y_{1}=g_{1}y_{0}$ , $y_{2}=g_{2}y_{2}$ , $g_{1},g_{2}\in G$ , we can write

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{l}(g_{1}^{-1}g_{2})=\unicode[STIX]{x1D6F7}_{l}(\unicode[STIX]{x1D703}(g_{1}y_{0},g_{2}y_{0}))=\unicode[STIX]{x1D6F7}_{l}(\unicode[STIX]{x1D703}(y_{1},y_{2})).\end{eqnarray}$$

It follows from the properties of $\unicode[STIX]{x1D711}_{l}$ that $\unicode[STIX]{x1D6F7}_{l}$ are continuous and real-valued, $\unicode[STIX]{x1D6F7}_{l}(0)=1$ , $|\unicode[STIX]{x1D6F7}_{l}(r)|\leqslant 1$ , $r\in {\mathcal{I}}$ . The set of functions $\{m_{l}^{1/2}\unicode[STIX]{x1D6F7}_{l},l\geqslant 0\}$ is an orthonormal basis in the space $L_{2}({\mathcal{I}},v)$ , and the expansion (4.9) for $F\in L_{2}({\mathcal{I}},v)$ takes the form

(4.17) $$\begin{eqnarray}F(r)\sim \mathop{\sum }_{l\geqslant 0}m_{l}c_{l}(F)\unicode[STIX]{x1D6F7}_{l}(r)\end{eqnarray}$$

with the Fourier coefficients

(4.18) $$\begin{eqnarray}c_{l}(F)=\int _{{\mathcal{I}}}F(r)\unicode[STIX]{x1D6F7}_{l}(r)\,dv(r)\end{eqnarray}$$

and Parseval’s equality $\int _{{\mathcal{I}}}|F(r)|^{2}\,dv_{r}=\sum _{l\geqslant 0}m_{l}\,|c_{l}(F)|^{2}$ .

Comparing relations (4.11) and (4.13), we arrive at the formula

(4.19) $$\begin{eqnarray}\int _{Q}F_{1}(\unicode[STIX]{x1D703}(y_{1},y))F_{2}(\unicode[STIX]{x1D703}(y,y_{2}))\,d\unicode[STIX]{x1D707}(y)=\mathop{\sum }_{l\geqslant 0}m_{l}c_{l}(F_{1})c_{l}(F_{2})\unicode[STIX]{x1D6F7}_{l}(\unicode[STIX]{x1D703}(y_{1},y_{2})).\end{eqnarray}$$

For the spaces $Q=Q(d,d_{0})$ the matrix elements $t_{1j}^{(l)}(g)$ are eigenfunctions of the Laplace–Beltrami operator on $Q$ , and the spherical functions $\unicode[STIX]{x1D711}_{l}(g)=t_{11}^{(l)}(g)$ are eigenfunctions of the radial part of this operator and can be found explicitly; see [Reference Gangolli19, p. 178], [Reference Helgason22, Ch. V, Theorem 4.5], [Reference Koornwinder, Askey, Koornwinder and Schempp23], [Reference Levenshtein, Pless and Huffman24, pp. 514–512, 543–544], [Reference Wolf34, Theorem 11.4.21]. For the functions $\unicode[STIX]{x1D6F7}_{l}$ in (4.15), we have

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{l}(r)=\unicode[STIX]{x1D6F7}_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(r)=\frac{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos r)}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)},\quad r\in {\mathcal{I}}=[0,\unicode[STIX]{x1D70B}],\end{eqnarray}$$

where $P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)$ are the standard Jacobi polynomials of degree $l$ normalized by

(4.21) $$\begin{eqnarray}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)=\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FC}+l\\ l\end{array}\right)=\frac{(\unicode[STIX]{x1D6FC}+1)\cdots (\unicode[STIX]{x1D6FC}+l)}{l!}\simeq l^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

Jacobi polynomials can be defined by Rodrigues’ formula (see [Reference Szegö31, equation (4.3.1)]),

(4.22) $$\begin{eqnarray}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)=\frac{(-1)^{l}}{2^{l}l!}(1-z)^{-\unicode[STIX]{x1D6FC}}(1+z)^{-\unicode[STIX]{x1D6FD}}\frac{d^{l}}{dz^{l}}\{(1-z)^{l+\unicode[STIX]{x1D6FC}}(1+z)^{l+\unicode[STIX]{x1D6FD}}\}.\end{eqnarray}$$

The parameters $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ in (4.21) and dimensions  $d$ , $d_{0}$ in $Q(d,d_{0})$ are related by

(4.23) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}={\textstyle \frac{1}{2}}d-1,\qquad \unicode[STIX]{x1D6FD}={\textstyle \frac{1}{2}}d_{0}-1.\end{eqnarray}$$

In what follows, we assume that (4.23) always holds. Hence, $\unicode[STIX]{x1D6FC}\geqslant \unicode[STIX]{x1D6FD}\geqslant -1/2$ , and $|P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)|\leqslant P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1),z\in [-1,1]$ ; see [Reference Szegö31, Theorem 7.32.1].

We have the following orthogonality relations for Jacobi polynomials (see [Reference Szegö31, equation (4.3.3)]):

(4.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{0}^{\unicode[STIX]{x1D70B}}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos u)P_{l^{\prime }}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos u)\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\,du\nonumber\\ \displaystyle & & \displaystyle \quad =\big({\textstyle \frac{1}{2}}\big)^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+1}\int _{-1}^{1}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)P_{l^{\prime }}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)(1-z)^{\unicode[STIX]{x1D6FC}}(1+z)^{\unicode[STIX]{x1D6FD}}\,dz\nonumber\\ \displaystyle & & \displaystyle \quad =M_{l}^{-1}\unicode[STIX]{x1D6FF}_{ll^{\prime }},\end{eqnarray}$$

where $M_{0}=\unicode[STIX]{x1D705}(d,d_{0})$ and

(4.25) $$\begin{eqnarray}M_{l}=(2l+\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+1)\frac{\unicode[STIX]{x1D6E4}(l+1)\unicode[STIX]{x1D6E4}(l+\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+1)}{\unicode[STIX]{x1D6E4}(l+\unicode[STIX]{x1D6FC}+1)\unicode[STIX]{x1D6E4}(l+\unicode[STIX]{x1D6FD}+1)}\simeq l,\quad l\geqslant 1.\end{eqnarray}$$

Substituting (4.15) and (4.20) into (4.8) and using (4.24), we obtain the following explicit formula for the dimensions  $m_{l}$ :

(4.26) $$\begin{eqnarray}m_{l}=M_{l}\,B(d/2,d_{0}/2)\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FC}+l\\ l\end{array}\right)^{2}\simeq l^{d-1}.\end{eqnarray}$$

For functions $F\in L_{2}([0,\unicode[STIX]{x1D70B}],v_{r})$ the expansion (4.17) takes the form

(4.27) $$\begin{eqnarray}F(r)\sim \mathop{\sum }_{l\geqslant 0}M_{l}C_{l}(F)P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos r),\end{eqnarray}$$

with the Fourier–Jacobi coefficients

(4.28) $$\begin{eqnarray}C_{l}(F)=\int _{0}^{\unicode[STIX]{x1D70B}}F(u)P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos u)\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\,du,\end{eqnarray}$$

and Parseval’s equality $\int _{{\mathcal{I}}}|F(r)|^{2}\,dv(r)=\unicode[STIX]{x1D705}(d,d_{0})\sum _{l\geqslant 0}M_{l}\,|C_{l}(F)|^{2}$ . The Fourier–Jacobi coefficients (4.28) and Fourier coefficients (4.18) are related by

(4.29) $$\begin{eqnarray}c_{l}(F)=C_{l}(F)\frac{\unicode[STIX]{x1D705}(d,d_{0})}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)},\quad l\geqslant 0.\end{eqnarray}$$

Using the relations (4.20) and (4.29), we can write formula (4.19) in the form

(4.30) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{Q}F_{1}(\unicode[STIX]{x1D703}(y_{1},y))F_{2}(\unicode[STIX]{x1D703}(y,y_{2}))\,d\unicode[STIX]{x1D707}(y)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 0}M_{l}C_{l}(F_{1})C_{l}(F_{2})\frac{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)}.\end{eqnarray}$$

This formula will be used in the next section to obtain spherical function expansions for discrepancies and metrics.

The condition of positive definiteness (4.6) for the spherical functions (4.15) and (4.21) will be used in §4 in the following special form:

(4.31) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]=\mathop{\sum }_{x_{1},x_{2}\in {\mathcal{D}}_{N}}\frac{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos \unicode[STIX]{x1D703}(x_{1},x_{2}))}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)}\geqslant 0,\end{eqnarray}$$

where ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ is an arbitrary $N$ -point subset.

One can easily check (see also [Reference Banai, Deza, Frankl and Rosenberg3, Reference Levenshtein, Pless and Huffman24]) that conditions (2.27) and (2.28) in the definition of $t$ -designs ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ are equivalent to the equalities

(4.32) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]=0,\quad l=1,\ldots ,t,\end{eqnarray}$$

4.2 Spherical function expansions for discrepancies and metrics

In this section we obtain explicit spherical function expansions for the kernels (1.13) and (1.14) and the symmetric difference metrics (1.20) and (1.21) on the spaces $Q(d,d_{0})$ .

First of all, we recall the main facts on Jacobi polynomials $P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)$ , $z\in [-1,1]$ , $\unicode[STIX]{x1D6FC}\geqslant -1/2$ , $\unicode[STIX]{x1D6FD}\geqslant -1/2$ , as $l\rightarrow \infty$ . It is well known (see [Reference Szegö31]) that Jacobi polynomials behave extremely irregularly on the interval $z\in [-1,1]$ : inside the interval they oscillate and are of order $l^{-1/2}$ , while in neighborhoods of the end points $z=1$ and $z=-1$ they grow rapidly and are of order $l^{\unicode[STIX]{x1D6FC}}$ and $l^{\unicode[STIX]{x1D6FD}}$ , respectively. It is convenient to describe such behavior in terms of the functions

(4.33) $$\begin{eqnarray}J_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(r)=\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{\unicode[STIX]{x1D6FC}+1/2}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{\unicode[STIX]{x1D6FD}+1/2}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos r),\quad r\in [0,\unicode[STIX]{x1D70B}].\end{eqnarray}$$

(a) In the interval $r\in [c_{0}l^{-1},\unicode[STIX]{x1D70B}-c_{0}l^{-1}]$ , where $c_{0}>0$ is an arbitrary fixed constant, we have the asymptotic formula

(4.34) $$\begin{eqnarray}J_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(r)=(\unicode[STIX]{x1D70B}l)^{-1/2}\{\cos [(l+l_{0})r+r_{0}]+O((l\sin r)^{-1})\},\end{eqnarray}$$

where $l_{0}=(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+1)/2$ , $r_{0}=-\unicode[STIX]{x1D70B}(2\unicode[STIX]{x1D6FC}+1)/4$ ; see [Reference Szegö31, Theorem 8.21.3].

(b) In the intervals $r\in [0,c_{0}l^{-1}]$ and $r\in [\unicode[STIX]{x1D70B}-c_{0}l^{-1},\unicode[STIX]{x1D70B}]$ , we have the bound $J_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(r)=O(l^{-1/2})$ ; see [Reference Szegö31, Theorem 7.32.2]. This bound, together with (4.34), implies the bound

(4.35) $$\begin{eqnarray}|J_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(r)|<c(l+1)^{-1/2},\quad l\geqslant 0,\end{eqnarray}$$

which holds uniformly for all $r\in [0,\unicode[STIX]{x1D70B}]$ , with the constant $c$ depending only on $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ .

Consider the measure of the intersection of two balls $B(r,y_{1})$ and $B(r,y_{2})$ in the space $Q=Q(d,d_{0})$ ,

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{r}(y_{1},y_{2})=\unicode[STIX]{x1D707}(B(r,y_{1})\cap B(r,y_{2}))=\int _{Q}\unicode[STIX]{x1D712}_{r}(\unicode[STIX]{x1D703}(y_{1},y))\unicode[STIX]{x1D712}_{r}(\unicode[STIX]{x1D703}(y,y_{2}))\,d\unicode[STIX]{x1D707}(y),\end{eqnarray}$$

where $\unicode[STIX]{x1D712}_{r}(\cdot )$ is the characteristic function of the interval $[0,r]$ , $0\leqslant r\leqslant \unicode[STIX]{x1D70B}$ ; see (1.26).

Lemma 4.1. The kernel (4.36) has the spherical function expansion

(4.37) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{r}(y_{1},y_{2})=v(r)^{2}+\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}a_{l}(r)\frac{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)},\end{eqnarray}$$

where

(4.38) $$\begin{eqnarray}\displaystyle a_{l}(r) & = & \displaystyle \big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{2d}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{2d_{0}}\{P_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(\cos r)\}^{2}\nonumber\\ \displaystyle & = & \displaystyle \big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{d_{0}-1}\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r)\}^{2}.\end{eqnarray}$$

The coefficients in (4.37) satisfy

(4.39) $$\begin{eqnarray}M_{l}a_{l}(r)\leqslant c\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{d_{0}-1},\end{eqnarray}$$

with a constant depending only on $d$ and $d_{0}$ . Furthermore, we have the equality

(4.40) $$\begin{eqnarray}\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}a_{l}(r)=v(r)-v(r)^{2}=v(r)v^{\prime }(r).\end{eqnarray}$$

Proof. Applying the expansion (4.30) to the integral (4.36), we obtain

(4.41) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{r}(y_{1},y_{2})=\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 0}M_{l}\{C_{l}(\unicode[STIX]{x1D712}_{r})\}^{2}\frac{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))}{P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)},\end{eqnarray}$$

where $C_{l}(\unicode[STIX]{x1D712}_{r})$ are Fourier–Jacobi coefficients (4.28) of the characteristic function  $\unicode[STIX]{x1D712}_{r}$ ,

(4.42) $$\begin{eqnarray}\displaystyle C_{l}(\unicode[STIX]{x1D712}_{r}) & = & \displaystyle \int _{0}^{r}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos u)\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\,du\nonumber\\ \displaystyle & = & \displaystyle \big({\textstyle \frac{1}{2}}\big)^{(d-1)/2+(d_{0}-1)/2}\int _{\cos r}^{1}(1-z)^{\unicode[STIX]{x1D6FC}}(a+z)^{\unicode[STIX]{x1D6FD}}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)\,dz.\qquad\end{eqnarray}$$

In view of (2.2), we have $C_{0}(\unicode[STIX]{x1D712}_{r})=\unicode[STIX]{x1D705}(d,d_{0})^{-1}v(r)$ . Substituting Rodrigues’ formula (4.22) into (4.42), we obtain

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\cos r}^{1}(1-z)^{\unicode[STIX]{x1D6FC}}(1+z)^{\unicode[STIX]{x1D6FD}}P_{l}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)\,dz\nonumber\\ \displaystyle & & \displaystyle \quad =(2l)^{-1}(1-\cos r)^{\unicode[STIX]{x1D6FC}+1}(1+\cos r)^{\unicode[STIX]{x1D6FD}+1}P_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(\cos r)\nonumber\\ \displaystyle & & \displaystyle \quad =2^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+1}l^{-1}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{2\unicode[STIX]{x1D6FC}+2}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{2\unicode[STIX]{x1D6FD}+2}P_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(\cos r).\nonumber\end{eqnarray}$$

In view of (4.33) and (4.23), we have

(4.43) $$\begin{eqnarray}\displaystyle C_{l}(\unicode[STIX]{x1D712}_{r}) & = & \displaystyle l^{-1}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{d}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{d_{0}}P_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(\cos r)\nonumber\\ \displaystyle & = & \displaystyle l^{-1}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{(d-1)/2}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{(d_{0}-1)/2}J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r).\end{eqnarray}$$

Substituting (4.43) into (4.41), we obtain the formulas (4.37) and (4.38).

The bound (4.39) follows from (4.38), since $M_{l}\simeq l$ (see (4.25)), and $J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r)\lesssim l^{-1/2}$ (see (4.35)).

From (4.36), we obtain $\unicode[STIX]{x1D707}_{r}(y,y)=v(r)$ . Putting $y_{1}=y_{2}=y$ in (4.37), we obtain (4.40). In fact, formula (4.40) is Parseval’s equality for the expansion (4.27) of the characteristic function  $\unicode[STIX]{x1D712}_{r}$ .◻

The following theorem is an immediate corollary of Lemma 2.1.

It is worth noting the following. The chordal metric $\unicode[STIX]{x1D70F}$ is a symmetric difference metric (1.20) with the specific measure $\unicode[STIX]{x1D709}^{\natural }$ ; see (2.7). Therefore, it has the expansion (4.47). At the same time, the chordal metric can be written as

(4.49) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(y_{1},y_{2})=c(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\bigg[1-\frac{P_{1}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))}{P_{1}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)}\bigg]^{1/2},\end{eqnarray}$$

with $c(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(\unicode[STIX]{x1D6FC}+1/\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+2)^{1/2}=(d/d+d_{0})^{1/2}$ . Indeed, by Rodrigues’ formula (4.22), $P_{1}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)=\frac{1}{2}(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+2)z+\frac{1}{2}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})$ , and

(4.50) $$\begin{eqnarray}\frac{1}{2}(1-z)=\frac{\unicode[STIX]{x1D6FC}+1}{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}+2}\bigg[1-\frac{P_{1}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(z)}{P_{1}^{(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})}(1)}\bigg].\end{eqnarray}$$

On the other hand, by definition (2.3),

(4.51) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(y_{1},y_{2})=\sin {\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}(y_{1},y_{2})=\big[{\textstyle \frac{1}{2}}(1-\cos \unicode[STIX]{x1D703}(y_{1},y_{2}))\big]^{1/2}.\end{eqnarray}$$

Then formula (4.49) follows from (4.50) and (4.51).

Comparing Theorem 4.1(ii) with definitions (1.12) and (1.17), we arrive at the following theorem.

Theorem 4.2. If $\unicode[STIX]{x1D709}\in W(d,d_{0})$ , then for an arbitrary $N$ -point subset ${\mathcal{D}}_{N}\in Q(d,d_{0})$ , we have the expansions

(4.52) $$\begin{eqnarray}\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]=\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}A_{l}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]\end{eqnarray}$$

and

(4.53) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}] & = & \displaystyle \langle \unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D702})\rangle -\unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}A_{l}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}A_{l}(\unicode[STIX]{x1D709})[1-\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]],\end{eqnarray}$$

where the coefficients $\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}]\geqslant 0$ are defined by (4.31). The series (4.52) and (4.53) converge absolutely.

4.3 Bounds for Fourier–Jacobi coefficients

In this section we estimate the coefficients

(4.54) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}a_{l}(r)\ & =\ & \big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{d_{0}-1}\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r)\}^{2},\\ A_{l}(\unicode[STIX]{x1D712}_{r})\ & =\ & \displaystyle \int _{0}^{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D712}_{r}(u)a_{l}(u)\,du=\int _{0}^{r}a_{l}(u)\,du,\\ A_{l}\,(\unicode[STIX]{x1D709})\ & =\ & \displaystyle \int _{0}^{\unicode[STIX]{x1D70B}}a_{l}(u)\unicode[STIX]{x1D702}(u)\,du,\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D709}\in AW(d,d_{0})$ , $d\unicode[STIX]{x1D709}(r)=\unicode[STIX]{x1D702}(r)\,d(r)$ . In fact, we shall prove special weighted bounds for Jacobi polynomials.

Proof. The asymptotic formula (4.34) implies that

(4.57) $$\begin{eqnarray}\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r)\}^{2}=(\unicode[STIX]{x1D70B}l)^{-1}\big\{{\textstyle \frac{1}{2}}-{\textstyle \frac{1}{2}}\cos 2[(l+l_{0})r+r_{0}]+R_{l}(r)\big\},\end{eqnarray}$$

where the error term $R_{l}(r)$ satisfies

(4.58) $$\begin{eqnarray}R_{l}(r)=\left\{\begin{array}{@{}ll@{}}O(l^{-1}),\quad & \text{for}~0<c_{0}\leqslant r\leqslant \unicode[STIX]{x1D70B}-c_{0},\\ O((lr)^{-1}),\quad & \text{for}~l^{-1}\leqslant r\leqslant \unicode[STIX]{x1D70B}/2,\end{array}\right.\end{eqnarray}$$

with $0<c_{0}<\unicode[STIX]{x1D70B}/2$ arbitrarily fixed.

(i) Since $d\unicode[STIX]{x1D709}(r)=\unicode[STIX]{x1D702}(r)dr$ and $w(\unicode[STIX]{x1D709},d,d_{0})>0$ (see (2.12)), a sufficiently small constant $0<c_{0}<\unicode[STIX]{x1D70B}/2$ can be chosen to satisfy

(4.59) $$\begin{eqnarray}\int _{c_{0}}^{\unicode[STIX]{x1D70B}-c_{0}}\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\unicode[STIX]{x1D702}(u)\,du\geqslant \frac{1}{2}w(\unicode[STIX]{x1D709},d,d_{0})>0.\end{eqnarray}$$

Using (4.59) and (4.57) and the first bound in (4.58), we obtain

(4.60) $$\begin{eqnarray}\displaystyle A_{l}(\unicode[STIX]{x1D709}) & {\geqslant} & \displaystyle \int _{c_{0}}^{\unicode[STIX]{x1D70B}-c_{0}}\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(u)\}^{2}\unicode[STIX]{x1D702}(u)\,du\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D70B}l)^{-1}\big\{{\textstyle \frac{1}{4}}w(\unicode[STIX]{x1D709},d,d_{0})-I_{l}+O(l^{-1})\big\},\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle I_{l}=\frac{1}{2}\int _{c_{0}}^{\unicode[STIX]{x1D70B}-c_{0}}\big(\text{sin}{\textstyle \frac{1}{2}}u\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}u\big)^{d_{0}-1}\cos 2[(l+l_{0})u+r_{0}]\unicode[STIX]{x1D702}(u)\,du. & & \displaystyle \nonumber\end{eqnarray}$$

By the Riemann–Lebesgue lemma, $I_{l}\rightarrow 0$ as $l\rightarrow \infty$ , and from (4.60), we obtain the bound

(4.61) $$\begin{eqnarray}A_{l}(\unicode[STIX]{x1D709})>(8\unicode[STIX]{x1D70B}l)^{-1}w(\unicode[STIX]{x1D709},d,d_{0})\end{eqnarray}$$

for all sufficiently large $l>l_{1}$ , where $l_{1}$ depends only on $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ and  $\unicode[STIX]{x1D709}$ . Since $A_{l}(\unicode[STIX]{x1D709})>0$ for all $l\geqslant 1$ , we have the bound

(4.62) $$\begin{eqnarray}A_{l}(\unicode[STIX]{x1D709})>c_{1}l^{-1}\end{eqnarray}$$

for all $l\geqslant 1$ with a constant $c_{1}>0$ depending only on $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ and  $\unicode[STIX]{x1D709}$ .

On the other hand, the bound (4.35) implies

(4.63) $$\begin{eqnarray}r^{-d+1}a_{l}(r)=r^{-d+1}\big(\text{sin}{\textstyle \frac{1}{2}}r\big)^{d-1}\big(\text{cos}{\textstyle \frac{1}{2}}r\big)^{d_{0}-1}\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(r)\}^{2}\leqslant c_{2}l^{-1}.\end{eqnarray}$$

Comparing the bounds (4.62) and (4.63), we obtain the bound (4.55) with $c=c_{1}c_{2}^{-1}$ .

(ii) Let $0<r\leqslant \unicode[STIX]{x1D70B}/2$ and $lr\geqslant L$ , where $L\geqslant 1$ is a constant which will be fixed later. From definitions (4.54), we obtain

(4.64) $$\begin{eqnarray}\displaystyle r^{-d}A_{l}(\unicode[STIX]{x1D712}_{r}) & {\geqslant} & \displaystyle r^{-d}\int _{r/2}^{r}a_{l}(u)\,du\nonumber\\ \displaystyle & {\geqslant} & \displaystyle r^{-d}\bigg(\sin \frac{1}{4}r\bigg)^{d-1}\bigg(\cos \frac{1}{2}r\bigg)^{d_{0}-1}\int _{r/2}^{r}{\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(u)\}}^{2}\,du\nonumber\\ \displaystyle & {>} & \displaystyle c_{1}r^{-1}\int _{r/2}^{r}{\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}\}}^{2}\,du,\end{eqnarray}$$

where one can put $c_{1}=(1/8)^{d-1}(1/2)^{d_{0}-1}$ . Using the asymptotic formula (4.57) and the second bound in (4.58), we obtain

(4.65) $$\begin{eqnarray}\displaystyle & & \displaystyle r^{-1}\int _{r/2}^{r}{\{J_{l-1}^{(\unicode[STIX]{x1D6FC}+1,\unicode[STIX]{x1D6FD}+1)}(u)\}}^{2}\,du\nonumber\\ \displaystyle & & \displaystyle \quad =(\unicode[STIX]{x1D70B}l)^{-1}\bigg\{\frac{1}{4}-\frac{1}{2}r^{-1}\int _{r/2}^{r}\cos 2[(l+l_{0})u+r_{0}]\,du+O(L^{-1})\bigg\}.\qquad\end{eqnarray}$$

The integral on the right-hand side in (4.65) is of order $O((rl)^{-1})\lesssim O(L^{-1})$ . Substituting (4.65) into (4.64), we obtain

(4.66) $$\begin{eqnarray}r^{-d}A_{l}(\unicode[STIX]{x1D712}_{r})>c_{1}(4\unicode[STIX]{x1D70B}l)^{-1}\{1+O(L^{-1})\}.\end{eqnarray}$$

Now, in view of (4.66), we can fix a sufficiently large constant  $L$ , depending only on $\unicode[STIX]{x1D6FC}$ and  $\unicode[STIX]{x1D6FD}$ , to satisfy

(4.67) $$\begin{eqnarray}r^{-d}A_{l}(\unicode[STIX]{x1D712}_{r})>c_{1}(8\unicode[STIX]{x1D70B}l)^{-1}=c_{2}l^{-1}.\end{eqnarray}$$

On the other hand, the bound (4.35) implies

(4.68) $$\begin{eqnarray}A_{l}(\unicode[STIX]{x1D709})\leqslant C_{2}w(\unicode[STIX]{x1D709},d,d_{0})l^{-1}=C_{3}l^{-1}.\end{eqnarray}$$

Comparing (4.67) and (4.68), we obtain the bound (4.56) with $C=C_{3}c_{2}^{-1}$ .◻

4.4 Proof of Theorems 2.2 and 2.3

By Theorem 4.1, we have

(4.69) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}\unicode[STIX]{x1D706}_{r}[{\mathcal{D}}_{N}]\ & =\ & \displaystyle \unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}a_{l}(r)\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}],\\ \unicode[STIX]{x1D706}[\unicode[STIX]{x1D712}_{r},{\mathcal{D}}_{N}]\ & =\ & \displaystyle \unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}A_{l}(\unicode[STIX]{x1D712}_{r})\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}],\\ \unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]\ & =\ & \displaystyle \unicode[STIX]{x1D705}(d,d_{0})\mathop{\sum }_{l\geqslant 1}l^{-2}M_{l}A_{l}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D711}_{l}[{\mathcal{D}}_{N}],\end{array}\right\}\end{eqnarray}$$

where ${\mathcal{D}}_{N}\subset Q(d,d_{0})$ is an arbitrary $N$ -point subset. The series (4.69) converge and all their terms are non-negative.

Now we are in position to prove Theorems 2.2 and 2.3.

Proof of Theorem 2.3.

First of all, we notice that

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{Q}(\#\{B(r,y)\cap {\mathcal{D}}_{N}\})^{2}\,d\unicode[STIX]{x1D707}(y)=\int _{Q}\bigg(\mathop{\sum }_{y_{1}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D712}(B(r,y),y_{1})\bigg)^{2}\,d\unicode[STIX]{x1D707}(y)\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{Q}\bigg(\mathop{\sum }_{y_{1}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D712}(B(r,y_{1}),y)\bigg)^{2}\,d\unicode[STIX]{x1D707}(y)=\mathop{\sum }_{y_{1},y_{2}\in {\mathcal{D}}_{N}}\unicode[STIX]{x1D707}(B(r,y_{1})\cap B(r,y_{2})),\nonumber\end{eqnarray}$$

where formula (1.26) has been used. Comparing these equalities with (1.29), we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{r}[{\mathcal{D}}_{N}]<\int _{Q}(\#\{B(r,y)\cap {\mathcal{D}}_{N}\})^{2}\,d\unicode[STIX]{x1D707}(y)=\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r].\end{eqnarray}$$

Therefore,

(4.72) $$\begin{eqnarray}\unicode[STIX]{x1D706}[\unicode[STIX]{x1D712}_{r},{\mathcal{D}}_{N}]=\int _{0}^{r}\unicode[STIX]{x1D706}_{u}[{\mathcal{D}}_{N}]\,du<r\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r],\end{eqnarray}$$

since $\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r]$ is a non-decreasing function of  $r$ . Substituting (4.72) into (4.71), we obtain

(4.73) $$\begin{eqnarray}\unicode[STIX]{x1D706}[\unicode[STIX]{x1D709},{\mathcal{D}}_{N}]<Cr^{-d+1}\unicode[STIX]{x1D708}_{1}[{\mathcal{D}}_{N},r].\end{eqnarray}$$

If $r=Lt^{-1}$ , then the bound (4.73) coincides with the bound (2.30).◻