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This paper presents the Gaussian subordination framework to generate optimalone-sided approximations to multidimensional real-valued functions by functionsof prescribed exponential type. Such extremal problems date back to the worksof Beurling and Selberg and provide a variety of applications in analysis andanalytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) theGaussian ${\bf x} \mapsto e^{-\pi \lambda |{\bf x}|^2}$, where $\lambda >0$ isa free parameter, by functions with distributional Fourier transforms supportedon Euclidean balls, optimizing weighted $L^1$-errors. By integrating theparameter $\lambda$ against suitable measures, we solve the analogous problemfor a wide class of radial functions. Applications to inequalities and periodicanalogues are discussed. The constructions presented here rely on the theory ofde Branges spaces of entire functions and on new interpolations tools derivedfrom the theory of Laplace transforms of Laguerre-P\'{o}lya functions.

Extremal functions in de Branges and Euclidean spaces II