Exponential-type Approximation in Multivariate Harmonic Hilbert Spaces

Abstract

Babuška introduced the concept of periodic Hilbert spaces for studying universally optimal quadrature formulas. Prager continued these investigations and discovered the relationship between optimal approximation of linear functionals on periodic Hilbert spaces and minimum norm interpolation ( optimal periodic interpolation ). In the case of a uniform mesh methods of periodic interpolation by translation are applicable and relations to periodic spline interpolation and approximation have been studied. It is the objective of this paper to introduce the concept of harmonic Hilbert space in a multivariate setting as an extension of periodic Hilbert space and to study approximation via Fourier partial integrals and exponential-type interpolation in these spaces.