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A VARIATION ON SELBERG’S APPROXIMATION PROBLEM

Part of: Linear function spaces and their duals Special classes of linear operators Harmonic analysis in one variable Entire and meromorphic functions, and related topics Spaces and algebras of analytic functions

Published online by Cambridge University Press:  14 August 2014

Michael Kelly
Michael Kelly*
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A. email mkelly@math.utexas.edu

Abstract

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Let ${\it\alpha}\in \mathbb{C}$ in the upper half-plane and let $I$ be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of $I$ that vanishes at the point ${\it\alpha}$. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishing is easily adapted to provide analogous “majorants with vanishing” for any Beurling–Selberg majorant.

MSC classification

Primary: 42A05: Trigonometric polynomials, inequalities, extremal problems 42A85: Convolution, factorization 30D15: Special classes of entire functions and growth estimates 30H99: None of the above, but in this section 46E22: Hilbert spaces with reproducing kernels (=
Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)

Information

Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 213 - 235
Copyright
Copyright © University College London 2014 

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