EXTREMAL FUNCTIONS IN DE BRANGES AND EUCLIDEAN SPACES, II

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 Betiding and Selberg and provide a variety of applicalions in analysis and analytic number theory. Here we majorize and minorize (on R-N) the Gaussian x -> e(-pi lambda vertical bar x vertical bar 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 A 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 interpolation tools derived from the theory of Laplace transforms of Laguerre-Polya functions.