2014/05/06 by Emanuel Carneiro, Friedrich Littmann
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Mathematical functions and polynomials #math.CA #msc:41A05 #msc:41A30 #msc:41A63 #msc:46E22
paper · pdf · doi:10.1016/j.aim.2014.04.007
published as Adv. Math. 260 (2014), 281-349
openalex publication_date 2014/05/06 · arxiv created 2014/06/20 · arxiv updated 2014/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on ℝN. These extremal functions minimize the L1(ℝN, |x|2ν+ 2 - Ndx)-distance to the original function, where ν>-1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function Fλ(x) = e-λ|x|, where λ>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 λ>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 \mathbbSN-1 with an axis of symmetry.