Extremal functions in de Branges and Euclidean spaces

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.

References in corpus (1)

• Bounding $\zeta (s)$ in the critical strip

Cited by in corpus (1)

• One-sided Band-limited Approximations of Some Radial Functions