Extremal functions in de Branges and Euclidean spaces II

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.