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