Extremal functions in de Branges and Euclidean spaces II

TL;DR

This paper develops a Gaussian subordination framework to construct optimal one-sided approximations of multidimensional functions, extending classical extremal problems with applications in analysis and number theory using de Branges spaces and Laplace transform techniques.

Contribution

It introduces a novel Gaussian subordination method for extremal approximation in multiple dimensions, leveraging de Branges spaces and new interpolation tools.

Findings

Optimized Gaussian majorants and minorants with Fourier transforms supported on Euclidean balls.

Unified approach for a broad class of radial functions via parameter integration.

Applications to inequalities and periodic function analogues.

Abstract

This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of Beurling and Selberg and provide a variety of applications in analysis and analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the Gaussian ${\bf x} \mapsto e^{-\pi \lambda |{\bf x}|^2}$, where $\lambda >0$ is a free parameter, by functions with distributional Fourier transforms supported on Euclidean balls, optimizing weighted $L^1$-errors. By integrating the parameter $\lambda$ against suitable measures, we solve the analogous problem for a wide class of radial functions. Applications to inequalities and periodic analogues are discussed. The constructions presented here rely on the theory of de Branges spaces of entire functions and on new interpolations tools derived from the theory of Laplace transforms of Laguerre-P\'{o}lya functions.