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Approximation in rough native spaces by shifts of smooth kernels on spheres. (English) Zbl 1082.41018

A kernel is a function $𝒦\in C\left({S}^{d}×{S}^{d}\right),$ where ${S}^{d}$ is the unit sphere in ${ℝ}^{d+1}$. If the kernel $𝒦$ is rotational invariant on ${S}^{d},$ then there is a function $\phi :\left[-1,1\right]\to ℝ$ such that $𝒦\left(x,y\right)=\phi \left(xy\right),$ where $xy=〈x,y〉$ denotes the usual inner product in ${ℝ}^{d+1}·$ The function $\phi$ is called a zonal function and $𝒦$ a zonal kernel. It is positively definite if and only if $\phi$ is positively definite. In the following $\phi$ is supposed strictly positively definite. Denote by ${ℋ}_{k}^{\left(0\right)}$ the linear space of all homogeneous harmonic polynomials of degree $k$ and let ${d}_{k}=dim{ℋ}_{k}^{\left(0\right)}·$ If $\left\{{Y}_{k,\mu }:\mu =1,···,{d}_{k}\right\},$ is an orthonormal basis of ${ℋ}_{k}^{\left(0\right)},$ then a zonal function $\phi$ admits the expansion $\phi \left(xy\right)={\sum }_{k=0}^{\infty }{a}_{k}{\sum }_{\mu =1}^{{d}_{k}}{Y}_{k,\mu }\left(x\right){Y}_{k,\mu }\left(y\right)·$

Denote by $P{H}_{\phi }$ the linear space of all finite linear combinations of zonal shifts of $\phi$ equipped with the inner product $〈f,g〉={\sum }_{\xi \in {\Xi }}{\sum }_{\zeta \in {\Theta }}{c}_{\xi }{d}_{\zeta }\phi \left(\xi \zeta \right),$ for $f={\sum }_{\xi \in {\Xi }}{c}_{\xi }\phi \left(\xi ·\right)$ and $g={\sum }_{\zeta \in {\Theta }}{d}_{\zeta }\phi \left(\zeta ·\right),$ where ${\Xi },{\Theta }$ are finite subsets of ${S}^{d}·$ The completion of $P{H}_{\phi }$ is called the native space ${𝒩}_{\phi }$ and it is a reproducing kernel Hilbert space with kernel $\phi ·$ Considering two zonal functions, $\phi$ as above, and $\psi$ with the expansion coefficients ${\left({b}_{k}\right)}_{k=0}^{\infty },$ then ${𝒩}_{\phi }\subset {𝒩}_{\psi },$ provided $0\le {a}_{k}\le {b}_{k},\phantom{\rule{0.166667em}{0ex}}k=0,1,···,$ but this embedding is not isometric in general. The authors define a multiplier operator $T:{𝒩}_{\phi }\to {𝒩}_{\psi }$ and prove that, for a finite subset ${\Xi }$ of ${S}^{d},$ every $f\in {𝒩}_{\psi }$ has a unique best approximation element ${s}_{\phi }\left[f\right]={\sum }_{\xi \in {\Xi }}{c}_{\xi }\phi \left(·\xi \right)$ in ${\phi }_{{\Xi }}=\text{span}\left\{\phi \left(·\xi \right):\xi \in {\Xi }\right\},$ and that the coefficients ${c}_{\xi },\phantom{\rule{0.166667em}{0ex}}\xi \in {\Xi },$ are determined by the interpolation condition $T\left({s}_{\phi }\left[f\right]\right){=T\left(f\right)|}_{\phantom{\rule{0.166667em}{0ex}}{\Xi }}·$ They give also an estimate of the error, namely

$\parallel {\psi }_{x}-{s}_{\phi }\left[{\psi }_{x}\right]{\parallel }_{{𝒩}_{\psi }}\le C{\left(\sum _{k>L}^{\infty }{b}_{k}/{d}_{k}\right)}^{1/2},$

where ${\psi }_{x}=\psi \left(x·\right),\phantom{\rule{0.166667em}{0ex}}L$ satisfies $h\left({\Xi }\right)\le 1/\left(2L\right),\phantom{\rule{0.166667em}{0ex}}h\left({\Xi }\right)$ being the mesh of the finite set ${\Xi }\subset {S}^{d},$ and the constant $C$ is independent of $x$.

##### MSC:
 41A30 Approximation by other special function classes 41A65 Abstract approximation theory 41A05 Interpolation (approximations and expansions) 41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 42C15 General harmonic expansions, frames 33C55 Spherical harmonics 46E22 Hilbert spaces with reproducing kernels