1 Introduction
For
$a\in \mathbb R,$
consider the translation operator
$T_a$
acting on
$ L^2(\mathbb R) $
by
$(T_af)(x)= f(x-a).$
Recall that a generalized shift-invariant system (GSI system for short) is a system of functions of the form
$\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
, where J is a countable index set,
$\{\phi _j\}_{j\in J} \subset L^2(\mathbb R)$
, and
$\{c_j\}_{j\in J}\subset \mathbb R_+$
. More information can be found in the papers [Reference Hernandez, Labate and Weiss5, Reference Ron and Shen7].
In applications of GSI systems, we typically need that the functions
$\phi _j$
have compact support, either in the time domain or in the frequency domain. The purpose of this paper is to derive conditions under which a given GSI system
$\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
can be approximated by a GSI system
$\{T_{c_j k} \widetilde {\phi _j}\}_{j\in J, k\in \mathbb Z}$
with generators
$\widetilde {\phi _j}$
having compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (lower frame bound) for the system
$\{T_{c_j k} (\phi _j- \widetilde {\phi _j})\}_{j\in J, k\in \mathbb Z}.$
Recall that
$B>0$
is a Bessel bound for
$\{T_{c_j k} (\phi _j- \widetilde {\phi _j})\}_{j\in J, k\in \mathbb Z}$
if
$$ \begin{align} \sum_{j\in J} \sum_{k\in \mathbb Z} | \langle f, T_{c_j k} (\phi_j- \widetilde{\phi_j})\rangle|^2 \le B\, || f||^2, \forall f\in L^2(\mathbb R). \end{align} $$
The rationale behind this condition is that it provides stability. For example, if the system
$\{T_{c_j k} \phi \}_{j\in J, k\in \mathbb Z}$
is a frame, small values of the Bessel bound imply that
$\{T_{c_j k} \widetilde {\phi _j}\}_{j\in J, k\in \mathbb Z}$
is also a frame, and that the synthesis operators and frame operators for the two systems are close (see [Reference Christensen and Hasannasab3] for a more detailed discussion).
We will show that under suitable conditions, this Bessel bound can be controlled in terms of parameters associated with the given GSI system, and also be made arbitrarily small by appropriate choices of the approximating functions
$\widetilde {\phi _j}$
. This places our paper in the context of classical approximation theory for frames; indeed, assuming that the given GSI system is a frame, the results provide us with conditions ensuring that also the approximating GSI system will be a frame.
The paper is organized as follows. In the rest of the introduction, we set the stage by collecting results we need from the literature and providing further motivation. The new results appear in Section 2. First, in Section 2.1, we consider the case of a system
$\{T_{c k} \phi \}_{ k\in \mathbb Z}$
generated by a single function
$\phi .$
The results in Section 2.1 are based on a decay condition on the Fourier transform
$\widehat {\phi }$
of the function
$\phi ,$
and no relation between the parameter c and the function
$\phi $
is required. In Section 2.2, we obtain general results for GSI systems
$\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
by applying Section 2.1 to the functions
$\phi _j$
“one by one.” The advantage of the obtained results is that they are very general: they apply to any GSI system where the functions
$\phi _j$
satisfy individual decay conditions, and still no relationship between
$\phi _j$
and the parameters
$c_j$
is required. In Section 2.3, a different type of decay condition, involving as well the functions
$\phi _j$
as the parameters
$c_j,$
is introduced. For GSI systems
$\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
satisfying the introduced condition, we show how to obtain an alternative way of approximation, which simultaneously takes care of all the generators
$\{\phi _j\}_{j\in J}$
in the GSI system. In particular, we prove that the results apply to wavelet-type systems. Finally, in Section 2.4, we analyze GSI systems of the same type, assuming now that the set of parameters
$\{c_j\}_{j\in J}$
is relatively separated.
We define the Fourier transform of
$f\in L^1(\mathbb R)$
by
$$ \begin{align*} \widehat{f}(\gamma)= \int_{-\infty}^\infty f(x)\, e^{-2 \pi i x \gamma}dx, \end{align*} $$
with the usual extension to
$ L^2(\mathbb R) .$
Our approach is based on the following result; it originally appeared in [Reference Christensen and Rahimi4] as a generalization of a result in [Reference Labate, Weiss and Wilson6].
Lemma 1.1 Consider a GSI system
$\{T_{c_{j}k}\phi _{j}\}_{j\in J, k\in \mathbb Z},$
and assume that
$$ \begin{align} B:=\sup_{\gamma\in \mathbb{R}}\sum_{j\in J}\sum_{k\in \mathbb Z} \frac1{c_j} | \widehat{\phi_j}(\gamma)\widehat{\phi_j}(\gamma- k/{c_j})|<\infty. \end{align} $$
Then
$\{T_{c_{j}k}\phi _{j}\}_{j\in J, k\in \mathbb Z}$
is a Bessel sequence with bound B.
Given a GSI system
$\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z},$
the approximating GSI system will be defined by considering functions
$\widetilde {\phi _j}$
that are defined via suitable truncations of the functions
$\phi _j;$
that is, the Fourier transform of
$\widetilde {\phi _j}$
will have the form
$\widehat {\phi _j} \chi _{K_j}$
for some compact set
$K_j\subset \mathbb R.$
We will estimate the Bessel bound of the GSI system
$\{T_{c_{j}k} (\phi _j-\widetilde {\phi _j}) \}_{j\in J, k\in \mathbb Z}$
. Usually, the exact truncation will depend on the index
$j,$
as illustrated by the following example.
Example 1.2 Consider the GSI system
$\{ T_{c_jk} \phi \}_{j\in \mathbb N, k\in \mathbb Z},$
where
$\phi (x):= e^{-x^2}$
and
$c_j=1$
for all
$j\in \mathbb N;$
that is, the system is an infinite repetition of the shift-invariant system
$\{T_k \phi \}_{k\in \mathbb Z}.$
Note that
$\widehat {\phi }(\gamma )=\sqrt {\pi }e^{-\pi ^2 \gamma ^2}$
. Applying (1.1), in order to approximate the GSI system
$\{ T_{c_jk} \phi \}_{j\in \mathbb N, k\in \mathbb Z}$
with a truncated GSI system
$\{ T_{c_jk} \widetilde {\phi _j}\}_{j\in \mathbb N, k\in \mathbb Z},$
within a given Bessel bound
$\epsilon ,$
we need to find functions
$\widetilde {\phi _j}$
such that
$$ \begin{align} \sup_{||f||=1} \sum_{j \in \mathbb N} \sum_{k\in \mathbb Z} | \langle f, T_k(\phi-\widetilde{\phi_j}) \rangle|^2 \le \epsilon. \end{align} $$
The natural way to obtain (1.3) would be to approximate each system
$\{T_k \phi \}_{k\in \mathbb Z}$
individually. In other words, for each
$j\in \mathbb N,$
we will consider a function
$\widetilde {\phi _j}$
defined via
$\widehat {\widetilde \phi _j} := \widehat {\phi _j} \chi _{[-K_j,K_j]},$
and show that
$K_j>0$
can be chosen such that
$$ \begin{align} \sup_{||f||=1} \sum_{k\in \mathbb Z} | \langle f, T_k(\phi-\widetilde{\phi_j}) \rangle|^2 \le \epsilon/2^j. \end{align} $$
Note that by Lemma 1.1,
$$ \begin{align*} \sup_{||f||=1} \sum_{k\in \mathbb Z} | \langle f, T_k(\phi-\widetilde{\phi_j}) \rangle|^2 &\leq \sup_{\gamma \in \mathbb R} \sum_{k\in \mathbb Z} \bigg|(\widehat{\phi}-\widehat{\widetilde{\phi}_j})(\gamma) (\widehat{\phi}-\widehat{\widetilde{\phi}_j})(\gamma -k)\bigg|\\ &\leq \sup_{\gamma\in\mathbb R} \bigg|(\widehat{\phi}-\widehat{\widetilde{\phi}_j})(\gamma)\bigg| \left(\sup_{\gamma\in\mathbb R}\sum_{k\in \mathbb Z} \bigg|(\widehat{\phi}-\widehat{\widetilde{\phi}_j})(\gamma -k)\bigg|\right) \nonumber\\ & \leq \sqrt{\pi} e^{-\pi^2 K_j^2}\left(\sum_{k=0}^\infty \bigg| \widehat{\phi}(K_j+k)\bigg| +\sum_{k=0}^\infty \bigg| \widehat{\phi}(-K_j-k)\bigg| \right) \nonumber\\ & \leq 2\pi e^{-\pi^2K_j^2} \sum_{k=0}^\infty e^{-\pi^2(K_j+k)^2} \\ &\leq 2\pi e^{-\pi^2 K_j^2} \sum_{k=0}^\infty e^{-\pi^2(K_j^2+2k K_j)} \leq \frac{2\pi e^{-2\pi^2 K_j^2}}{1-e^{-2\pi^2 K_j}}. \end{align*} $$
Now, fix any
$K\geq \max \{\ln (32/\epsilon ),\ln 32\}$
and take
$K_j:=\sqrt {Kj/(2\pi ^2)}.$
A direct calculation shows that then (1.4) holds. Hence, we have obtained the desired estimate in (1.3).
Note that for this construction to work, it is essential that the support size of
$\phi _j$
in the Fourier domain is allowed to depend on
$j\in \mathbb N;$
in fact, it is necessary that
$K_j\to \infty $
as
$j\to \infty .$
Let us finally remind the reader about a classical perturbation result (see, e.g., Corollary 22.1.5 in [Reference Christensen2]), which shows the relevance of the results in the current paper in the frame context: it shows that a perturbation of a frame with a sufficiently small Bessel bound again yields a frame. For notational convenience, we formulate the result in the setting of a frame in a general Hilbert space.
Lemma 1.3 Assume that
$\{f_k\}_{k\in I}$
is a frame in the Hilbert space
${\cal H},$
with frame bounds
$A,B.$
Consider any sequence
$\{g_k\}_{k\in I}$
in
${\cal H},$
and assume that
$ \{f_k-g_k\}_{k\in I}$
is a Bessel sequence with Bessel bound
$R<A.$
Then
$ \{g_k\}_{k\in I}$
is a frame with frame bounds
$A(1- \sqrt {R/A})^2, B(1+ \sqrt {R/B})^2.$
It is known that perturbations in the sense of Lemma 1.3 preserve other key features, e.g., the excess of the frame.
2 The results
Motivated by the considerations in Example 1.2, we will apply the following convention in the entire paper.
Standing convention: Given
$\phi _j\in L^2(\mathbb R) $
and any
$K_j>0$
, define the function
$\widetilde \phi _j \in L^2(\mathbb R) $
by
$$ \begin{align} \widehat{\widetilde\phi_j} := \widehat{\phi_j} \chi_{[-K_j,K_j]}. \end{align} $$
In the entire paper, we will use the short notation
$\Phi := \{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
and
$\widetilde {\Phi }:= \{T_{c_j k} \widetilde {\phi _j}\}_{j\in J, k\in \mathbb Z}.$
Furthermore, we will denote the (optimal) Bessel bound for
$\{T_{c_j k} (\phi _j- \widetilde {\phi _j})\}_{j\in J, k\in \mathbb Z}$
by
$B(\Phi , \widetilde {\Phi }).$
With this notation, our goal is to estimate
$B(\Phi , \widetilde {\Phi })$
in terms of parameters related to the given GSI system
$\Phi .$
In particular, for the considered GSI systems, we will provide explicit values for the cutoff points
$K_j$
ensuring that
$B(\Phi , \widetilde {\Phi })$
stays below a desired threshold.
2.1 The case
$\{T_{c k} \phi \}_{ k\in \mathbb Z}$
We will first consider the case of a (shift-invariant) system generated by a single function
$\phi ,$
i.e., a system of the form
$\Phi =\{T_{c k} \phi \}_{ k\in \mathbb Z}$
for some
$\phi \in L^2(\mathbb R) $
and some
$c>0.$
Theorem 2.1 For a shift-invariant system
$\Phi = \{T_{c k} \phi \}_{ k\in \mathbb Z}$
, assume that there exist
$E>0$
and
$\sigma>0$
such that
$$ \begin{align} \left|\widehat{\phi} (\gamma)\right| \leq \dfrac{E}{1+|\gamma|^{1+\sigma}},\, \gamma\in\mathbb R. \end{align} $$
Let
$K\geq \sigma /c$
and consider a function
$\widetilde \phi $
defined via
$\widehat {\widetilde \phi }= \widehat {\phi } \chi _{[-K,K]}$
. Then, with
$\widetilde {\Phi }= \{T_{c k} \widetilde {\phi }\}_{ k\in \mathbb Z}$
,
$$ \begin{align} B(\Phi, \widetilde{\Phi}) \le \frac{4E^2}{\sigma K^{1+2\sigma}}. \end{align} $$
Proof By Lemma 1.1,
$$ \begin{align*} B(\Phi, \widetilde{\Phi}) &\leq \sup_{\gamma \in \mathbb R} \sum_{k\in \mathbb Z} \bigg|c^{-1}(\widehat{\phi}-\widehat{\widetilde{\phi}})(\gamma) (\widehat{\phi}-\widehat{\widetilde{\phi}})(\gamma -k/c)\bigg| \\ &\le \left(\sup_{\gamma \in \mathbb R} \bigg|c^{-1}(\widehat{\phi}-\widehat{\widetilde{\phi}})(\gamma)\bigg| \right) \left(\sup_{\gamma \in \mathbb R} \sum_{k\in \mathbb Z} \bigg|(\widehat{\phi}-\widehat{\widetilde{\phi}})(\gamma -k/c)\bigg|\right). \end{align*} $$
Let
$u(\gamma ):=E/(1+|\gamma |^{1+\sigma })$
. Then
$$ \begin{align*} B(\Phi, \widetilde{\Phi}) &\le \frac{E}{c K^{1+\sigma}}\left( \sum_{k=0}^\infty u(K+k/c)+\sum_{k=0}^\infty u(-K-k/c) \right) \nonumber\\&\le \frac{E}{c K^{1+\sigma}} \left(\sum_{k=0}^\infty \frac{2E}{(K+k/c)^{1+\sigma}} \right) \nonumber\\&\le \frac{2E^2}{c K^{1+\sigma}} \left(\frac{1}{K^{1+\sigma}}+c\int_K^\infty \frac{1}{x^{1+\sigma}}dx \right) \\&\le \frac{2E^2}{c K^{1+\sigma}} \left(\frac{1}{K^{1+\sigma}}+\frac{c}{\sigma K^\sigma} \right)\\&= \frac{2E^2}{\sigma K^{1+2\sigma}} \left(\frac{\sigma}{cK}+1 \right). \\&\le \frac{4E^2}{\sigma K^{1+2\sigma}}, \end{align*} $$
where we used
$K\geq \sigma /c$
in the last inequality.
2.2 The general case
$\{T_{c_j k} \phi _j\}_{j\in \mathbb N, k\in \mathbb Z}$
The approach in Section 2.1 gives a natural way of approximating a general GSI system
$\{T_{c_j k} \phi _j\}_{j\in \mathbb N, k\in \mathbb Z},$
subject to appropriate decay conditions on each of the generators
$\phi _j.$
Indeed, applying Theorem 2.1 “one by one” to the functions
$\phi _j$
, we obtain the following theorem.
Theorem 2.2 For a GSI system
$\Phi =\{T_{c_j k} \phi _j\}_{j\in \mathbb N, k\in \mathbb Z}$
, assume that for
$j\in \mathbb N$
, there exist
$E_j>0$
and
$\sigma _j>0$
such that
$$ \begin{align*} \left|\widehat{\phi_j} (\gamma)\right| \leq \dfrac{E_j}{1+|\gamma|^{1+\sigma_j}},\, \gamma\in\mathbb R. \end{align*} $$
Then
$$ \begin{align} B(\Phi, \widetilde{\Phi}) \le \sum_{j=1}^\infty \frac{4E_j^2}{\sigma_j K_j^{1+2\sigma_j}}. \end{align} $$
In particular, if we for any given
$\epsilon>0$
take
$$ \begin{align} K_j\ge \max\left\{\frac{\sigma_j}{c_j}, \left( \frac{4E_j^2 2^j}{\epsilon \sigma_j} \right)^{\frac{1}{1+2\sigma_j}} \right\}, \end{align} $$
then
$$ \begin{align*} B(\Phi, \widetilde{\Phi})\le \epsilon. \end{align*} $$
Proof For
$j\in \mathbb N,$
let
$\Phi _j:= \{T_{c_j k} \phi _j\}_{k\in \mathbb Z}$
and
$\widetilde {\Phi _j}:= \{T_{c_j k} \widetilde {\phi _j}\}_{k\in \mathbb Z}.$
Then, by (2.5) and Theorem 2.1,
$$ \begin{align} B(\Phi_j, \widetilde\Phi_j)\leq \frac{4E_j^2}{\sigma_j K_j^{1+2\sigma_j}}. \end{align} $$
This immediately leads to (2.4). Finally, due to (2.5), we have that
$B(\Phi _j, \widetilde \Phi _j)\leq \frac {\epsilon }{ 2^j}$
and hence
$B(\Phi , \widetilde {\Phi })\le \epsilon ,$
as desired.
We will now provide an application of Theorem 2.2 to a Gabor-type system. For this purpose, we need the following elementary result.
Lemma 2.3 Let
$k\in \mathbb R$
. Then
$$ \begin{align*} \frac{1}{1+|\gamma-k|^2}\leq \frac{2(1+k^2)}{1+|\gamma|^2},\ \forall \gamma\in\mathbb R. \end{align*} $$
Proof Let
$f(\gamma ):=2(1+k^2)(1+(\gamma -k)^2)-(1+\gamma ^2)$
. It is enough to show that
$f(\gamma )\ge 0$
for
$\gamma \in \mathbb R$
; this follows from
$$ \begin{align*} f(\gamma)&=2+2k^2+2(\gamma-k)^2+2k^2(\gamma-k)^2-1-\gamma^2\\ &\ge 1+2k^2+2(\gamma-k)^2-\gamma^2\\ &= (\gamma-2k)^2+1>0.\\[-34pt] \end{align*} $$
Example 2.4 Let
$a,b>0$
and assume that
$g\in L^2(\mathbb R)$
satisfies
$$ \begin{align*} \left|\widehat{g} (\gamma)\right| \leq \dfrac{E}{1+|\gamma|^{2}},\, \gamma\in\mathbb R, \end{align*} $$
for some
$E>0$
. Consider a GSI system
$\Phi =\{T_{c_j k}\phi _j\}_{j \in \mathbb N, k\in \mathbb Z}$
, where
$c_j=a$
for all
$j\in \mathbb N$
and
$$ \begin{align*}\phi_j=\left\{ \begin{array}{ll} E_{\frac{(1-j)b}{2}} g, & j\in 2\mathbb N-1,\\ E_{\frac{jb}{2}} g, & j\in 2\mathbb N. \end{array} \right. \end{align*} $$
Then we see
$\{T_{c_j k}\phi _j\}_{j \in \mathbb N, k\in \mathbb Z}=\{T_{na}E_{mb}{g}\}_{m,n \in \mathbb Z}$
. By Lemma 2.3, if
$j\in 2\mathbb N-1$
, then
$$ \begin{align*} \left|\widehat{\phi_j}(\gamma)\right| =\left|T_{\frac{(1-j)b}{2}} \widehat{g}(\gamma)\right| \le \dfrac{E}{1+|\gamma-\frac{(1-j)b}{2}|^{2}} \le \dfrac{2E(1+(\frac{(1-j)b}{2})^2)}{1+|\gamma|^2}; \end{align*} $$
if
$j\in 2\mathbb N$
, then
$$ \begin{align*} \left|\widehat{\phi_j}(\gamma)\right| =\left|T_{\frac{jb}{2}} \widehat{g}(\gamma)\right| \le \dfrac{E}{1+|\gamma-\frac{jb}{2}|^{2}} \le \dfrac{2E(1+(\frac{jb}{2})^2)}{1+|\gamma|^2}. \end{align*} $$
A direct calculation shows that for
$j\in \mathbb N$
,
$$ \begin{align*} \left|\widehat{\phi_j}(\gamma)\right| \le \dfrac{2E(1+(\frac{(1-j)b}{2})^2+(\frac{jb}{2})^2)}{1+|\gamma|^2} \le \dfrac{E(2+(jb)^2)}{1+|\gamma|^2}. \end{align*} $$
Choose
$E_j:=E(2+(jb)^2)$
. Let
$\epsilon>0$
and for
$j\in \mathbb N$
, take
$K_j$
as in (2.5) with
$\sigma _j=1$
. By Theorem 2.2,
$B(\Phi , \widetilde {\Phi })\le \epsilon $
, as desired.
At first glance, it seems cumbersome that the result in Theorem 2.2 is obtained by considering each system
$\Phi _j:= \{T_{c_j k} \phi _j\}_{k\in \mathbb Z}$
individually and requiring “better and better approximations for increasing
$j.$
” However, to the best understanding of the authors, this is the only way a general result just based on decay conditions on
$\phi _j$
can be obtained. Indeed, the general setting of a GSI system is very broad and as illustrated already by Example 1.2 the functions
$\phi _j$
and the parameters
$c_j$
are in general unrelated, forcing us to make individual choices for the parameters
$K_j.$
In the next section, we will show that more universal ways of choosing the parameters
$K_j$
can be obtained by imposing decay conditions on the functions
$\phi _j$
that also involve the parameters
$c_j.$
2.3 A condition relating
$\{c_j\}_{j\in J}$
and
$\{\phi _j\}_{j\in J}$
In this section, we continue the analysis of a GSI system
$\Phi = \{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z},$
but we apply a different decay condition which involves as well the generators
$\phi _j$
as the parameters
$c_j.$
We begin with a general result, which does not yet specify how to choose the “cutoff” points
$K_j$
in order to obtain a desired approximation. In the subsequent examples, we will see how to choose these parameters in a number of concrete cases, e.g., for wavelet-type examples.
Theorem 2.5 For a GSI system
$\Phi =\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
, assume that there exist
$E>0$
and
$\sigma>0$
such that for
$j\in J$
,
$$ \begin{align} \bigg| c_j^{-1/2}\widehat{\phi_j} (\gamma/{c_j}) \bigg| \leq \dfrac{E}{1+|\gamma|^{1+\sigma}},\, \gamma\in\mathbb R. \end{align} $$
Consider now a truncated GSI system
$\widetilde {\Phi }=\{T_{c_j k} \widetilde {\phi _j}\}_{j\in J, k\in \mathbb Z}$
as in the general setup, and let
$J_{i}=\{j\in J|\, K_j\leq K_{i}\}$
for
$i\in J$
. Then
$$ \begin{align} B(\Phi, \widetilde{\Phi}) \le 2E^2 \left(\sup_{j\in J}\frac{1}{\sigma (c_jK_j)^{\sigma}}\left( \frac{\sigma}{c_jK_j}+1\right)\right) \left(\sup_{i\in J} \frac{1}{(K_{i})^{1+\sigma}}\sum_{k\in J_{i}} \frac{1}{(c_k)^{1+\sigma}} \right). \end{align} $$
Proof For notational convenience, let
$p_j(x) := \phi _j(x)- \widetilde \phi _j(x)$
for
$j\in J$
and
$u(\gamma ):= E/(1+|\gamma |^{1+\sigma })$
. We first note that
$\widehat {p_j}(\gamma ) = \widehat {\phi _j} (\gamma ) \chi _{[-K_j,K_j]^c}(\gamma ),$
and
$$ \begin{align} B(\Phi, \widetilde{\Phi}) &\le \sup_{\gamma \in \mathbb R} \sum_{j\in J}\sum_{k\in \mathbb Z} \left|c_j^{-1} \widehat{p_j} (\gamma) \widehat{p_j}(\gamma -k/c_j)\right| \nonumber \\ &\le \left(\sup_{\gamma \in \mathbb R, j\in J} \sum_{k \in \mathbb Z} \bigg|c_j^{-1/2} \widehat{p_j}(\gamma-k/c_j)\bigg|\right) \left(\sup_{\gamma\in \mathbb R} \sum_{j\in J} \bigg| c_j^{-1/2} \widehat{p_j}(\gamma)\bigg|\right). \end{align} $$
We now proceed with a number of estimates that finally yields the result.
Estimate of
$ \sum _{k \in \mathbb Z} \bigg | c_j^{-1/2} \widehat {p_j}(\gamma -k/c_j)\bigg |$
: Fix
$\gamma \in \mathbb R$
and
$j \in J$
. Then the contribution from
$\gamma -k/c_j$
hitting the interval
$]-\infty ,-K_j]\cup [K_j,\infty [$
is at most
$$ \begin{align} \sum_{k \in \mathbb Z} \bigg| c_j^{-1/2} \widehat{p_j}(\gamma-k/c_j)\bigg| &\leq \sum_{k=0}^\infty u(c_j K_j+k) +\sum_{k=0}^\infty u(-c_j K_j-k) \nonumber \\ &\leq 2\sum_{k=0}^{\infty } \frac{E}{\left(c_jK_j+k\right)^{1+\sigma}} \nonumber \\ &\leq \frac{2E}{(c_jK_j)^{1+\sigma}}+ \int_{c_jK_j}^\infty \frac{2E}{t^{1+\sigma}}dt \nonumber \\ &= 2E \left(\frac{1}{(c_jK_j)^{1+\sigma}}+\frac{1}{\sigma (c_jK_j)^{\sigma}} \right) \nonumber \\ &= \frac{2E}{\sigma (c_jK_j)^{\sigma}}\left( \frac{\sigma}{c_jK_j}+1\right). \end{align} $$
This leads to
$$ \begin{align} \sup_{\gamma \in \mathbb R, j\in J} \sum_{k \in \mathbb Z} \bigg|c_j^{-1/2} \widehat{p_j}(\gamma-k/c_j) \bigg| \leq \sup_{j\in J}\frac{2E}{\sigma (c_jK_j)^{\sigma}}\left( \frac{\sigma}{c_jK_j}+1\right). \end{align} $$
Estimate of
$\sum _{j\in J} \bigg | c_j^{-1/2} \widehat {p_j}(\gamma )\bigg |$
: Fix
$\gamma \in \mathbb R\setminus \{0\}$
. Note that
$\widehat {p_j}(\gamma ) = \widehat {\phi _j} (\gamma ) \chi _{[-K_j,K_j]^c}(\gamma )$
. If
$|\gamma |\leq K_j$
for all
$j\in J$
, then
$\widehat {p_j}(\gamma )=0$
. Now, assume that there exists
$j\in J$
such that
$K_j < |\gamma |$
. Choose
$j_0\in J$
such that
$K_{j_0}=\max _{j\in J}\{K_j | \, K_j < |\gamma |\}.$
Note that
$J_{j_0}=\{j\in J|\, K_j\leq K_{j_0}\}$
. Then we have
$$ \begin{align*} \sum_{j\in J} \bigg| c_j^{-1/2} \widehat{p_j}(\gamma)\bigg| &\le \sum_{j\in J_{j_0}} u(c_j \gamma)\leq \sum_{j\in J_{j_0}} u(c_j K_{j_0}) \\ &\leq \sum_{j\in J_{j_0}}^{\infty } \frac{E}{\left(c_j K_{j_0}\right)^{1+\sigma}} =\frac{E}{(K_{j_0})^{1+\sigma}}\sum_{j\in J_{j_0}} \frac{1}{(c_j)^{1+\sigma}}. \end{align*} $$
This leads to
$$ \begin{align} \sup_{\gamma\in \mathbb R}\sum_{j\in J} \bigg| c_j^{-1/2} \widehat{p_j}(\gamma)\bigg| \leq \sup_{i\in J} \frac{E}{(K_{i})^{1+\sigma}}\sum_{j\in J_{i}} \frac{1}{(c_j)^{1+\sigma}}. \end{align} $$
The result now follows from (2.9) together with (2.11) and (2.12).
We will now consider a number of concrete manifestations of Theorem 2.5. We first prove that for the case
$c_j=j,$
the decay condition (2.7) implies that we can choose a cutoff point K that is independent of
$j\in J.$
Example 2.6 Consider a GSI system
$\Phi =\{T_{c_j k}\phi _j\}_{j \in \mathbb N, k\in \mathbb Z}$
, where
$\{c_j\}_{j\in \mathbb N}=\{j\}_{j\in \mathbb N}$
. Assume that
$\phi _j$
,
$j\in \mathbb N$
, satisfies (2.7) with
$E>0$
and
$\sigma =1$
. Let
$K\geq 1$
and take
$\widetilde \phi _j \in L^2(\mathbb R) $
as in (2.1) with
$K_j=K$
for
$j\in \mathbb N$
. Then
$$ \begin{align*}J_{i}=\{j\in \mathbb N|\, K_j\leq K_{i}\}=\mathbb N,\, i\in \mathbb N.\end{align*} $$
By Theorem 2.5, we have
$$ \begin{align*} B(\Phi, \widetilde{\Phi}) &\le 2E^2 \left(\sup_{j\in \mathbb N}\frac{1}{ jK}\left( \frac{1}{jK}+1\right)\right) \left( \sup_{i\in \mathbb N} \frac{1}{K^{2}}\sum_{k\in \mathbb N} \frac{1}{k^{2}}\right)\\ &= 2E^2 \left(\frac{1}{K}\left( \frac{1}{K}+1\right)\right) \left(\frac{1}{K^{2}}\sum_{k\in \mathbb N} \frac{1}{k^{2}}\right)\leq \frac{2\pi^2E^2}{3K^3}, \end{align*} $$
where we used
$K\geq 1$
and
$\sum _{k\in \mathbb N} 1/k^2 = \pi ^2/6$
in the last inequality.
In the next example, we consider the case where
$c_j=a^j$
for some
$a\neq 1.$
The result will be formulated for
$a>1,$
but obviously we obtain a corresponding result for
$a<1$
by “running through the system in the opposite order.” The example covers classical wavelet-type examples. Indeed, defining the scaling operator
$D_a: L^2(\mathbb R) \to L^2(\mathbb R) $
by
$(D_af)(x)= a^{1/2}f(ax),$
the wavelet system
$\{D_{a^j}T_{k}\phi \}_{j,k\in \mathbb Z}$
equals the GSI system
$\{T_{c_j k}\phi _j\}_{j \in J, k\in \mathbb Z}$
, where
$c_j=a^{-j}$
and
$\phi _j= D_{a^j}\phi .$
In this particular case, the decay conditions stated in condition (2.7) reduce to the single condition
$$ \begin{align} \left|\widehat{\phi} (\gamma)\right| \leq \dfrac{E}{1+|\gamma|^{1+\sigma}},\, \gamma\in\mathbb R. \end{align} $$
Note that this condition appeared already in (2.2).
Example 2.7 Consider a GSI system
$\Phi =\{T_{c_j k}\phi _j\}_{j \in \mathbb Z, k\in \mathbb Z}$
, where
$\{c_j\}_{j\in \mathbb Z}=\{a^{-j}\}_{j\in \mathbb Z}$
for some
$a> 1.$
Assume that
$\phi _j$
,
$j\in \mathbb Z$
, satisfies (2.7) with
$E>0$
and
$\sigma =1$
. Let
$K>0$
and take
$\widetilde \phi _j \in L^2(\mathbb R) $
as in (2.1) with
$K_j=a^j K$
for
$j\in \mathbb Z$
. Then
$$ \begin{align*}J_{i}=\{j\in \mathbb Z|\, K_j\leq K_{i}\}=\{j\in \mathbb Z|\, j\leq i\}, \, i\in \mathbb Z. \end{align*} $$
By Theorem 2.5, we have
$$ \begin{align*} B(\Phi, \widetilde{\Phi})&\le 2E^2 \left(\sup_{j\in \mathbb Z}\frac{1}{K}\left( \frac{1}{K}+1\right)\right) \left(\sup_{i\in \mathbb Z} \frac{1}{(a^i K)^2}\sum_{k=-\infty}^i \frac{1}{(a^{-k})^2}\right)\\ &\le \frac{4E^2}{K}\left(\frac{1}{K^{2}}\sum_{k=0}^\infty \frac{1}{(a^{2})^k}\right) =\frac{4E^2a^2}{K^3(a^2-1)}. \end{align*} $$
This example shows that Theorem 2.5 generalizes previous results from the literature (see [Reference Benavente, Christensen, Hasannasab, Kim, Kim and Kovac1]).
2.4 Relatively separated sets
$\{c_j\}_{j\in J}$
In this section, we will continue the analysis of GSI systems
$\Phi =\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z}$
satisfying the decay condition (2.7); however, we will apply different assumptions on the set
$\{c_j\}_{j\in J}.$
Recall that a set
$\{c_j\}_{j\in J} \subset \mathbb R_+$
is separated if
$\inf _{i,j\in J, i \neq j} |c_i-c_j|>0.$
The set
$\{c_j\}_{j\in J}$
is relatively separated if it is a finite union of separated sets.
Let us now consider a relatively separated set
$\{c_j\}_{j\in J} \subset \mathbb R_+$
and a corresponding GSI system
$\Phi =\{T_{c_j k} \phi _j\}_{j\in J, k\in \mathbb Z};$
that is, for some
$N\in \mathbb N$
, we can write
$$ \begin{align} \{T_{c_jk}\phi_j\}_{j\in J, k\in \mathbb Z} =\bigcup_{\ell=1}^N \{T_{c^\ell_jk}\phi^{(\ell)}_j\}_{j\in J^{(\ell)},k\in \mathbb Z}, \end{align} $$
where
$\{ c^{(\ell )}_j \}_{j\in J^{(\ell )}}$
is separated for each
$\ell \in \{1,2,\ldots ,N\}$
. Let
$\delta $
be a joint separation constant, meaning that for each
$\ell \in \{1, 2, \dots , N\},$
we have
$ |c_i^{(\ell )} - c_j^{(\ell )} | \ge \delta $
for
$i\neq j.$
This setup implies that for each
$\ell \in \{1, 2, \dots , N\},$
any interval of the form
$[n \delta , (n+1)\delta [, n=0,1,2,\ldots ,$
contains at most one parameter
$c_i^{(\ell )}.$
For technical reasons, we will now consider an enlarged GSI system where each such interval contains precisely one parameter
$c_i^{(\ell )}.$
Definition 2.8 Consider a GSI system
$\Phi $
of the form (2.14), with separation constant
$\delta>0.$
The corresponding enlarged GSI system
$ \{T_{c_jk}\phi _j\}_{j\in \widetilde {J}, k\in \mathbb Z}$
is defined by adding parameters
$c_j, j\in \widetilde {J} \mathop{\setminus} J$
and corresponding functions
$\phi _j$
as follows:
-
(i) If for some
$\ell \in \{1, 2, \dots , N\}$
an interval of the form
$[(n-1) \delta , n\delta [, n\in \mathbb N,$
does not contain any parameter
$c_i^\ell ,$
define a parameter
$c_j, j\in \widetilde {J} \mathop{\setminus} J,$
by adding the point
$n\delta - \delta /2.$
-
(ii) For
$j\in \widetilde {J} \mathop{\setminus} J,$
let
$\phi _j:=0.$
Formulated in words, the enlarged GSI system contains all the elements of the given GSI system, plus a number of added zero function. We immediately obtain the following consequence of this observation.
Lemma 2.9 Consider a GSI system
$\Phi $
of the form (2.14), with separation constant
$\delta>0.$
Then the corresponding enlarged set of parameters
$\{c_j\}_{j\in \widetilde {J}}$
is relatively separated with separation constant
$\delta /2.$
The GSI systems
$\{T_{c_j k}(\phi _j- \widetilde {\phi _j})\}_{j\in J, k\in \mathbb Z}$
and
$\{T_{c_j k} (\phi _j- \widetilde {\phi _j})\}_{j\in \widetilde {J}, k\in \mathbb Z}$
have identical Bessel bounds.
We will now prove that for a GSI system of the form (2.14), with separation constant
$\delta>0,$
our decay condition implies that a uniform cutoff point can be chosen for the functions
$\phi _j, \, j\in J.$
Theorem 2.10 Consider a GSI system of the form (2.14), with separation constant
$\delta>0.$
Let
$\delta _0= \min _{j\in J}\{\delta /2,c_j\}.$
Assume that
$\phi _j$
,
$j\in J$
, satisfies (2.7) with
$E>0$
and
$\sigma>0,$
that is,
$$ \begin{align*} \bigg|{c_j}^{-1/2}\widehat{\phi_j} (\gamma/{c_j})\bigg| \leq \dfrac{E}{1+|\gamma|^{1+\sigma}},\, \gamma\in\mathbb R. \end{align*} $$
Let
$K\geq \sigma /\delta _0$
and for each
$j\in \mathbb N,$
consider a function
$\widetilde {\phi _j}$
defined via
$\widehat {\widetilde \phi _j} := \widehat {\phi _j} \chi _{[-K,K]}$
. Then
$$ \begin{align} B(\Phi, \widetilde{\Phi}) \le \frac{4NE^2}{\sigma (\delta_0 K)^{1+2\sigma}} \left(2+\frac{1}{2\delta_0} \right). \end{align} $$
Proof Consider the corresponding enlarged GSI system
$\bigcup _{\ell =1}^N \{T_{c^{(\ell )}_jk}\phi ^{(\ell )}_j\}_{j\in \mathbb N,k\in \mathbb Z}$
with separation constant
$\delta /2$
. For the first interval, instead of
$[0,\delta /2[$
, if
$\delta _0<\delta /2$
, then we take
$[\delta _0, \delta /2[$
; if
$\delta _0=\delta /2$
, then we take
$\{\delta /2\}.$
We then note that
$c^{(\ell )}_1\in [\delta _0, \delta /2]$
and
$c^{(\ell )}_k\in [(k-1)\delta /2,k\delta /2[$
for
$k=2,3,\ldots .$
Fix
$\ell \in \{1,2,\ldots ,N\}$
and let
$\Phi ^{(\ell )}:=\{T_{c^{(\ell )}_j k} {\phi ^{(\ell )}_j}\}_{j\in \mathbb N, k\in \mathbb Z}$
and
$\widetilde {\Phi }^{(\ell )}:= \{T_{c^{(\ell )}_j k} \widetilde {\phi _j}\}_{j\in \mathbb N, k\in \mathbb Z}$
, By Theorem 2.5, we have
$$ \begin{align*} B(\Phi^{(\ell)}, \widetilde{\Phi}^{(\ell)})&\leq 2E^2 \left(\sup_{j\in \mathbb N}\frac{1}{\sigma (c_j^{(\ell)} K)^{\sigma}} \left( \frac{\sigma}{c^{(\ell)}_jK}+1\right)\right) \left(\sup_{i\in \mathbb N} \frac{1}{K^{1+\sigma}}\sum_{k\in \mathbb N} \left( \frac{1}{c^{(\ell)}_k} \right)^{1+\sigma}\right)\\ &\leq 2E^2 \left(\frac{1}{\sigma (\delta_0 K)^{\sigma}} \left( \frac{\sigma}{\delta_0 K}+1\right)\right) \left( \frac{1}{K^{1+\sigma}}\left(\frac{1}{\delta^{1+\sigma}_0}+ \sum_{k=2}^\infty \left( \frac{2}{(k-1)\delta} \right)^{1+\sigma}\right)\right)\\ &\le \frac{4E^2}{\sigma (\delta_0 K)^{\sigma}} \left( \frac{1}{K^{1+\sigma}}\left(\frac{1}{\delta^{1+\sigma}_0}+ \left(\frac{2}{\delta}\right)^{1+\sigma}\sum_{k=1}^\infty \frac{1}{k^{1+\sigma}}\right)\right)\\ &\le \frac{4E^2}{\sigma (\delta_0 K)^{\sigma}} \left( \frac{1}{K^{1+\sigma}}\left(\frac{1}{\delta^{1+\sigma}_0}+ \left(\frac{2}{\delta}\right)^{1+\sigma} \left(1+\frac{1}{\delta} \right)\right)\right)\\ &\le \frac{4E^2}{\sigma (\delta_0 K)^{\sigma}} \left( \frac{1}{K^{1+\sigma}}\left(\frac{1}{\delta^{1+\sigma}_0}+ \frac{1}{\delta_0^{1+\sigma}}\left(1+\frac{1}{2\delta_0} \right)\right)\right)\\ &= \frac{4E^2}{\sigma (\delta_0 K)^{1+2\sigma}} \left(2+\frac{1}{2\delta_0} \right). \end{align*} $$
Since
$B(\Phi , \widetilde {\Phi })\leq \sum _{\ell =1}^N B(\Phi ^{(\ell )}, \widetilde {\Phi }^{(\ell )})$
, (2.15) holds.
Example 2.11 Let
$N\in \mathbb N$
. Consider a GSI system
$$ \begin{align*}\{T_{c_jk}\phi_j\}_{j\in J,k\in \mathbb Z}=\bigcup_{\ell=1}^N \{T_{c^{(\ell)}_jk}\phi^{(\ell)}_j\}_{j\in \mathbb N,k\in \mathbb Z},\end{align*} $$
where
$c^{(\ell )}_j\in [j,j+1[$
for
$j\in \mathbb N$
and
$\ell \in \{1,2,\ldots ,N\}$
. Assume that
$\phi ^{(\ell )}_j$
,
$j\in \mathbb N$
and
$\ell \in \{1,2,\ldots ,N\}$
, satisfies (2.7) with
$E>0$
and
$\sigma =1$
. That is,
$$ \begin{align*} \bigg|\left(c^{(\ell)}_j\right)^{-1/2}\widehat{\phi^{(\ell)}_j} \left(\gamma/c^{(\ell)}_j \right)\bigg| \leq \dfrac{E}{1+|\gamma|^{1+\sigma}},\, \gamma\in\mathbb R. \end{align*} $$
Let
$K\geq 1$
and take
$\widetilde \phi _j \in L^2(\mathbb R) $
as in (2.1) with
$K_j=K$
for
$j\in \mathbb N$
. Then
$J_{i}=\mathbb N,\, i\in \mathbb N.$
Fix
$\ell \in \{1,2,\ldots , N\}$
. By Theorem 2.5, we have
$$ \begin{align*} B(\Phi^{(\ell)}, \widetilde{\Phi}^{(\ell)}) &\le 2E^2 \left(\sup_{j\in \mathbb N}\frac{1}{c^{(\ell)}_jK}\left( \frac{1}{c^{(\ell)}_jK}+1\right)\right) \left( \sup_{i\in \mathbb N} \frac{1}{K^{2}}\sum_{k\in \mathbb N} \frac{1}{(c^{(\ell)}_k)^{2}}\right)\\ &\le 2E^2 \left(\sup_{j\in \mathbb N}\frac{1}{ jK}\left( \frac{1}{jK}+1\right)\right) \left( \sup_{i\in \mathbb N} \frac{1}{K^{2}}\sum_{k\in \mathbb N} \frac{1}{k^{2}}\right)\\ &\le\frac{2\pi^2E^2}{3K^3}. \end{align*} $$
Hence,
$$ \begin{align*} B(\Phi, \widetilde{\Phi}) \le \sum_{\ell=1}^N B(\Phi^{(\ell)}, \widetilde{\Phi}^{(\ell)}) \leq \frac{2\pi^2NE^2}{3K^3}. \end{align*} $$
