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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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References:

[1] Bochner, S., Hilbert distances and positive definite functions, Ann. of Math., 42, 647-656 (1941)
[2] Chen, D.; Menegatto, V. A.; Sun, X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 131, 2733-2740 (2003) · Zbl 1125.43300
[3] Ditzian, Z., Fractional derivatives and best approximation, Acta. Math. Hungar., 81, 4, 323-348 (1998) · Zbl 0962.41017
[4] Dyn, N.; Narcowich, F.; Ward, J., Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold, Constr. Approx., 15, 175-208 (1999) · Zbl 0948.41015
[5] von Golitschek, M.; Light, W., Interpolation by polynomials and radial basis functions on spheres, Constr. Approx., 17, 1-18 (2001) · Zbl 0983.41002
[6] Jetter, K.; Stöckler, J.; Ward, J. D., Error estimates for scattered data interpolations on spheres, Math. Comp., 68, 733-747 (1999) · Zbl 1042.41003
[7] J. Levesley, W. A. Light, D. Rogozin, X. Sun, A simple approach to variational theory for interpolation on spheres, International Theory of Numerical Analysis, vol. 132, Birkhauser, Verlag Base/Switerzland, 1999.; J. Levesley, W. A. Light, D. Rogozin, X. Sun, A simple approach to variational theory for interpolation on spheres, International Theory of Numerical Analysis, vol. 132, Birkhauser, Verlag Base/Switerzland, 1999.
[8] Menegatto, V. A., Strictly positive definite kernels on the Hilbert spheres, Appl. Anal., 55, 114-119 (1994)
[9] Menegatto, V. A., Interpolation on the complex Hilbert sphere, Approx. Theory Appl., 11, 1-9 (1995)
[10] Narcowich, F. J., Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Appl., 190, 165-193 (1995) · Zbl 0859.58032
[11] Narcowich, F. J.; Schaback, R.; Ward, J. D., Approximation in Sobolev spaces by kernel expansions, J. Approx. Theory, 114, 70-83 (2002) · Zbl 1022.46023
[12] F.J. Narcowich, J.D. Ward, Scattered data interpolation on sphere: error estimates and locally supported basis functions, SIAM J. Math. Anal. 36 (2004) 284-300.; F.J. Narcowich, J.D. Ward, Scattered data interpolation on sphere: error estimates and locally supported basis functions, SIAM J. Math. Anal. 36 (2004) 284-300. · Zbl 1081.41014
[13] Ragozin, D.; Levesley, J., The fundamentality of translates of spherical functions on compact homogeneous spaces, J. Approx. Theory, 103, 252-268 (2000) · Zbl 0947.41017
[14] Ron, A.; Sun, X., Strictly positive definite functions on spheres, Math. Comp., 65, 1513-1530 (1996) · Zbl 0853.42018
[15] Schoenberg, I. J., Positive definite functions on spheres, Duke Math. J., 9, 96-108 (1942) · Zbl 0063.06808
[16] Schaback, R., Improved error bounds for scattered interpolation by radial basis functions, Math. Comp., 68, 201-216 (1999) · Zbl 0917.41011
[17] X. Sun, Strictly positive definite functions on the unit circle, Math. Comp. 74 (2005) 709-721.; X. Sun, Strictly positive definite functions on the unit circle, Math. Comp. 74 (2005) 709-721. · Zbl 1068.42008
[18] Sun, X.; Cheney, E. W., Fundamental sets of continuous functions on spheres, Constr. Approx., 13, 245-250 (1997) · Zbl 0886.41016
[19] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0232.42007
[20] G. Szegő, Orthogonal Polynomials, fourth ed., American Mathematical Society Colloquium Publication, vol. 23, American Mathematical Society, Providence, RI, 1959.; G. Szegő, Orthogonal Polynomials, fourth ed., American Mathematical Society Colloquium Publication, vol. 23, American Mathematical Society, Providence, RI, 1959. · Zbl 0089.27501
[21] Morton, Tanya M.; Neamtu, M., Error bounds for solving pseudo-differential equations on spheres by collocation with zonal kernels, J. Approx. Theory, 114, 242-268 (2002) · Zbl 1004.65108
[22] Xu, Y.; Cheney, E. W., Strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 116, 977-981 (1992) · Zbl 0787.43005
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