Details

Autor(en) / Beteiligte

• GRIEBEL, MICHAEL
• RIEGER, CHRISTIAN
• ZWICKNAGL, BARBARA

Titel

MULTISCALE APPROXIMATION AND REPRODUCING KERNEL HILBERT SPACE METHODS

Ist Teil von

• SIAM journal on numerical analysis, 2015-01, Vol.53 (2), p.852-873

Ort / Verlag

Society for Industrial and Applied Mathematics

Erscheinungsjahr

2015

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We consider reproducing kernels K : Ω × Ω → R in multiscale series expansion form, i.e., kernels of the form $\mathrm{K}(\mathrm{x},\mathrm{y})={\mathrm{\Sigma }}_{\mathrm{\ell}\in \mathrm{\mathbb{N}}}{\mathrm{\lambda }}_{\mathrm{\ell}}{\mathrm{\Sigma }}_{\mathrm{j}\in {\mathrm{I}}_{\mathrm{\ell}}}{\mathrm{\phi }}_{\mathrm{\ell},\mathrm{j}}\left(\mathrm{x}\right){\mathrm{\phi }}_{\mathrm{\ell},\mathrm{j}}\left(\mathrm{y}\right)$ with weights λl and structurally simple basis functions {φl,i}. Here, we deal with basis functions such as polynomials or frame systems, where, for l ∈ N, the index set Il is finite or countable. We derive relations between approximation properties of spaces based on basis functions {φl,j : 1 ≤ l ≤ L, j ∈ Il} and spaces spanned by translates of the kernel span {K(x1,·),..., K(xN,·)} with XN := {x1,..., xN} ⊂ Ω if the truncation index L is appropriately coupled to the discrete set XN. An analysis of a numerically feasible approximation from trial spaces span {KL(x1,·),..., KL(xN,·)} based on finitely truncated series kernels of the form ${\mathrm{K}}^{\mathrm{L}}(\mathrm{x},\mathrm{y}):={\mathrm{\Sigma }}_{\mathrm{l}=1}^{\mathrm{L}}{\mathrm{\lambda }}_{\mathrm{\ell}}{\mathrm{\Sigma }}_{\mathrm{j}\in {\mathrm{I}}_{\mathrm{\ell}}}{\mathrm{\phi }}_{\mathrm{\ell},\mathrm{j}}\left(\mathrm{x}\right){\mathrm{\phi }}_{\mathrm{\ell},\mathrm{j}}\left(\mathrm{y}\right)$ is provided, where the truncation index L is chosen sufficiently large depending on the point set XN. Furthermore, Bernstein-type inverse estimates and derivative-free sampling inequalities for kernel-based spaces are obtained from estimates for spaces based on the basis functions {φl,j : 1 ≤ l ≤ L, j ∈ Il}.