Harmonic blending approximation

Abstract

The concept of harmonic Hilbert space \(H_D({\mathbb R} ^n)\) was introduced in [2] as an extension of periodic Hilbert spaces [1], [2], [5], [6]. In [4] we introduced multivariate harmonic Hilbert spaces and studied approximation by exponential-type function in these spaces and derived error bounds in the uniform norm for special functions of exponential type which are defined by Fourier partial integrals \(S_b(f)\): \[S_b(f)(x)=\int _{ {\mathbb R} ^n } \chi _{[-b,b]}(t) F(t) \exp(i(t,x)) dt,\]\([-b,b]=[-b_1,b_1]\times ... \times [-b_n ,b_n], \quadb_1>0,...,b_n>0\), where\(F(t)\sim \left( {\textstyle\frac 1{2\pi}}\right) ^n\ \int_{{\mathbbR} ^n}f(x) \exp (-i(x,t))dx \ \in L_2({\mathbb R} ^n) \capL_1({\mathbb R} ^n)\)is the Fourier transform of \(f \in L_2({\mathbb R} ^n) \capC_0({\mathbb R} ^n)\). In this paper we will investigate more general approximation operators \(S_\psi \) in harmonic Hilbert spaces of tensor product type.