Extremal functions in de Branges and Euclidean spaces
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In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on RNden">de">RN. These extremal functions minimize the L1(RN,|x|2谓+2−Ndx)den">de">L1(RN,|x|2+2Ndx)-distance to the original function, where 谓>−1den">de">>1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function F(x)=e−位|x|den">de">F(x)=e|x|, where 位>0den">de">>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 位>0den">de">>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−1den">de">SN1 with an axis of symmetry.

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