Approximation in Sobolev spaces by kernel expansions
The authors consider a complex-valued orthonormal basis \(\{ \varphi_j(x)\}_{j\in {\mathcal{J}}}\) in \(L_2(\Omega)\), where \({\mathcal{J}}\) is a countable index set, while \(\Omega\) is a bounded domain in \({\mathbb{R}}^n\) or a compact \(n\)-dimensional Riemannian manifold. Using the expansions of functions \(f\in L_2(\Omega)\) with respect to \(\{\varphi_j(x)\}_{j\in{\mathcal{J}}}\) \[ f = \sum_{j\in {\mathcal{J}}}\widehat{f}(j)\varphi_j, \quad \widehat f(j)=(f,\varphi_j)_2 \] for any sequence \(w=\{w_j\}_{j\in{\mathcal{J}}}\) of positive weights, they define Sobolev-type subspaces \({\mathcal{S}}_w \) of \(L_2(\Omega)\) in the following way: \[ {\mathcal{S}}_w=\big\{f= \sum_{j\in {\mathcal{J}}}\widehat f(j)\varphi_j , \;\|f\|_w^2=\sum_{j\in {\mathcal{J}}}\frac{|\widehat f(j)|^2}{w_j} <\infty\big\}. \] The Sobolev spaces are defined by using special weights depending on an order \(\sigma\). In the paper are studied approximation of functions \(f\in L_2(\Omega)\) by linear combinations of functions \(\Phi(.,y)\), where \(y \in \Omega\) and \(\Phi: \Omega\times\Omega\to{\mathbb{R}}\) is a symmetric positive definite kernel having an expansion \[ \Phi (x,y) = \sum_{j\in {\mathcal J}}\widehat \Phi(j)\varphi_j(x)\bar\varphi_j(y) \] with the coefficients \(\widehat\Phi(j)\) being strictly positive. The case of the \(n\)-sphere \({\mathbb{S}}^n\) and the \(n\)-torus \({\mathbb{T}}^n\) are considered in more detail.
scientific article; zbMATH DE number 1763608