On the optimal approximation of bounded linear functionals in Hilbert spaces with inner product invariant in rotation or translation

Abstract

Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions defined in a domain B ⊂ C or B ⊂ R m, invariant in rotation or translation (e.g. circle, circular annulus, ball, spherical shell, strip of the complex plane) and equipped with inner product invariant in rotation or translation are considered. The weights and error functional norms for optimal approximate rules based on nodes located angle-equidistant on concentric spheres or circles of B, for B invariant in rotation, and on nodes located equispaced on in B lying line, for B invariant in translation, are explicitly given in terms of the kernel function of the Hilbert space. A number of concrete Hilbert spaces satisfying the required conditions are listed.