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Generalized hyperinterpolation on the sphere and the Newman-shapiro operators. (English) Zbl 1002.41016

In this interesting paper, the author modifies Ian Sloan’s idea of hyperinterpolation on the sphere, to generalized hyperinterpolation. The advantage is that while hyperinterpolation diverges in the uniform norm, generalized hyperinterpolation converges in the uniform norm. Let \(r\geq 3\) and let \(S^{r-1}\) denote the unit sphere in \(\mathbb{R}^3\). Let \({\mathbf P}^r_\mu\) denote the polynomials of degree \(\leq\mu\) in \(r\) variables, restricted to the unit sphere. Let \(G_\mu\) denote the reproducing kernel for the homogeneous harmonic polynomials of degree \(\mu\) on \(S^{r-1}\). It admits a representation in terms of ultraspherical polynomials. Both hyperinterpolation and generalized hyperinterpolation are based on a sequence of quadrature rules on the sphere, \[ Q_\mu F=\sum^M_{j=1} A_jF(t_j), \] where \(F\) is continuous on \(S^{r-1}\), the nodes \(\{t_j\}\) belong to \(S^{r-1}\), and quite often the weights \(A_j\) are positive. If \[ Q_\mu F=\int_{S^{r-1}} Fd\omega\text{ for all }F\in {\mathbf P}^r_\mu, \] where \(\omega\) is a Lebesgue measure on \(S^{r-1}\), the rules are said to have exact degree \(\mu+1\). Generalized hyperinterpolation also involves an infinite matrix \(A=(a_{j,k})\) satisfying the following conditions: (i) \(a_{j,k} =0\) for \(k>j\); (ii) \(\lim_{j\to\infty} a_{j,k}=1\) for \(k\in\{0,1\}\); (iii) \(K_\mu= \sum^\mu_{k=0} a_{\mu,k} G_k\geq 0\) in \([-1,1]\). One may think of \(K_\mu\) as a smoothing of the reproducing kernel \(G_\mu\), for example a Cesàro mean. Now one can define the generalized hyperinterpolation operator \(L_\mu\) based on \(Q_\mu\) by \[ L_\mu F(t)= \sum^M_{j=0} A_jF(t_j) K_\mu(t_j\cdot t). \] The author proves that \(\{L_\mu\}\) are positive operators that converge in the uniform norm as \(\mu\to \infty\) for all \(F\) continuous on \(S^{r-1}\). As examples of the matrix \(A\), the author discusses Cesàro methods, and also the Newman-Shapiro summation method. This paper is of great interest to anyone working in approximation on the sphere.

MSC:

41A63 Multidimensional problems
41A10 Approximation by polynomials
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