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 $S_b(f)$:

$$S_b(f)(x)={\int }_{{\mathbb{R}}^n}{\chi }_{[-b,b]}(t)F(t)\exp (i(t,x))dt,$$
$[-b,b]=[-b_1,b_1]\times ...\times [-b_n,b_n],b_1>0,...,b_n>0$, where
$F(t)\sim {\left(\frac{1}{2\pi }\right)}^n{\int }_{{\mathbb{R}}^n}f(x)\exp (-i(x,t))dx\in L_2({\mathbb{R}}^n)\cap L_1({\mathbb{R}}^n)$
is the Fourier transform of $f\in L_2({\mathbb{R}}^n)\cap C_0({\mathbb{R}}^n)$. In this paper we will investigate more general approximation operators $S_{\psi }$ in harmonic Hilbert spaces of tensor product type.