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 RN. These extremal functions minimize the L1(RN,|x|2ν+2−Ndx)-distance to the original function, where ν>−1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function Fλ(x)=e−λ|x|, where λ>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 λ>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 SN−1 with an axis of symmetry.