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

J. Approximation Theory 133, No. 2, 269-283 (2005); corrigendum and open questions ibid. 138, No. 1, 124-127 (2006).
A kernel is a function $$\mathcal K\in C(S^d\times S^d),$$ where $$S^d$$ is the unit sphere in $$\mathbb R^{d+1}$$. If the kernel $$\mathcal K$$ is rotational invariant on $$S^d,$$ then there is a function $$\phi :[-1,1]\to \mathbb R$$ such that $$\mathcal K(x,y) =\phi (xy),$$ where $$xy =\langle x,y\rangle$$ denotes the usual inner product in $$\mathbb R^{d+1}.$$ The function $$\phi$$ is called a zonal function and $$\mathcal K$$ 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 $$\mathcal H_k^{(0)}$$ the linear space of all homogeneous harmonic polynomials of degree $$k$$ and let $$d_k = \dim \mathcal H_k^{(0)}.$$ If $$\{Y_{k,\mu} : \mu =1,...,d_k\},$$ is an orthonormal basis of $$\mathcal H_k^{(0)},$$ 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 $$\mathcal N_\phi$$ 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 $$\mathcal N_\phi\subset \mathcal N_\psi,$$ 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 $$T:\mathcal N_\phi \to \mathcal N_\psi$$ and prove that, for a finite subset $$\Xi$$ of $$S^d,$$ every $$f\in \mathcal N_\psi$$ 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 $$T(s_\phi[f]) =T(f)| _{\,\Xi}.$$ They give also an estimate of the error, namely $\| \psi_x-s_\phi[\psi_x]\| _{\mathcal N_\psi} \leq C\left(\sum_{k>L}^\infty b_k/d_k\right)^{1/2},$ 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:

 41A30 Approximation by other special function classes 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A05 Interpolation in approximation theory 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 (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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