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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 $\Cal K\in C(S^d\times S^d),$ where $S^d$ is the unit sphere in $\Bbb R^{d+1}$. If the kernel $\Cal K$ is rotational invariant on $S^d,$ then there is a function $\phi :[-1,1]\to \Bbb R$ such that $\Cal K(x,y) =\phi (xy),$ where $xy =\langle x,y\rangle$ denotes the usual inner product in $\Bbb R^{d+1}.$ The function $\phi$ is called a zonal function and $\Cal K$ 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 $\Cal H_k^{(0)}$ the linear space of all homogeneous harmonic polynomials of degree $k$ and let $d_k = \dim \Cal H_k^{(0)}.$ If $\{Y_{k,\mu} : \mu =1,...,d_k\},$ is an orthonormal basis of $\Cal H_k^{(0)},$ then a zonal function $\phi$ admits the expansion $\phi(xy) = \sum_{k=0}^\infty a_k\sum_{\mu =1}^{d_k}Y_{k,\mu}(x)Y_{k,\mu}(y). $ Denote by $PH_\phi$ the linear space of all finite linear combinations of zonal shifts of $\phi$ equipped with the inner product $\langle f,g\rangle = \sum_{\xi\in \Xi}\sum_{\zeta\in \Theta} c_\xi d_\zeta \phi(\xi \zeta ),$ for $f= \sum_{\xi\in \Xi} c_\xi \phi(\xi \cdot)$ and $g =\sum_{\zeta\in \Theta} d_\zeta\phi (\zeta\cdot),$ where $\Xi,\Theta$ are finite subsets of $S^d.$ The completion of $PH_\phi$ is called the native space $\Cal N_\phi$ and it is a reproducing kernel Hilbert space with kernel $\phi.$ Considering two zonal functions, $\phi$ as above, and $\psi $ with the expansion coefficients $(b_k)_{k=0}^\infty,$ then $\Cal N_\phi\subset \Cal N_\psi,$ provided $0\leq a_k\leq b_k,\,k=0,1,...,$ but this embedding is not isometric in general. The authors define a multiplier operator $T:\Cal N_\phi \to \Cal N_\psi$ and prove that, for a finite subset $\Xi$ of $S^d,$ every $f\in \Cal N_\psi$ has a unique best approximation element $s_\phi[f]= \sum_{\xi\in \Xi}c_\xi \phi(\cdot \xi) $ in $\phi_\Xi =\text{span}\{\phi(\cdot \xi) : \xi \in \Xi\},$ and that the coefficients $c_\xi,\, \xi\in \Xi,$ are determined by the interpolation condition $T(s_\phi[f]) =T(f)\vert _{\,\Xi}.$ They give also an estimate of the error, namely $$ \Vert \psi_x-s_\phi[\psi_x]\Vert _{\Cal N_\psi} \leq C\left(\sum_{k>L}^\infty b_k/d_k\right)^{1/2}, $$ where $\psi_x =\psi(x\cdot), \, L$ satisfies $h(\Xi) \leq 1/(2L),\, h(\Xi)$ being the mesh of the finite set $\Xi\subset S^d, $ and the constant $C$ is independent of $x$.

MSC:
41A30Approximation by other special function classes
41A65Abstract approximation theory
41A05Interpolation (approximations and expansions)
41A58Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
42C15General harmonic expansions, frames
33C55Spherical harmonics
46E22Hilbert spaces with reproducing kernels
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