Extremal functions in de Branges and Euclidean spaces

TL;DR

This paper develops optimal extremal functions of exponential type for radial functions in multiple dimensions, using advanced interpolation in de Branges spaces, with applications to inequalities and polynomial approximations.

Contribution

It introduces new interpolation methods in de Branges spaces to construct extremal functions for multidimensional radial functions, extending classical one-dimensional results.

Findings

Constructed extremal majorants and minorants minimizing weighted L1 distance.

Extended results to a broad class of radial functions via parameter integration.

Applied findings to multidimensional inequalities and polynomial approximation problems.

Abstract

In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on $\mathbb{R}^N$. These extremal functions minimize the $L^1(\mathbb{R}^N, |x|^{2\nu + 2 - N}dx)$-distance to the original function, where $\nu >-1$ is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function $\mathcal{F}_{\lambda}(x) = e^{-\lambda|x|}$, where $\lambda >0$, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter $\lambda >0$ against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere $\mathbb{S}^{N-1}$ with an axis of symmetry.