Quasi-interpolation on the 2-sphere using radial polynomials
Let \(\{\lambda_n\}_{k\in\mathbb{N}}\) be a certain multiplier sequence, and let \(K\) be an integrable radial (zonal) function on the 2-dimensional unit sphere in \(\mathbb{R}^3\) whose Fourier series in zonal polynomials is \[ K\sim\sum^\infty_{k=1} \lambda_k Z^{(k)}. \] Generalizing Sobolev's class \(W^\alpha_p\), the authors introduce for each \(p\in [1,\infty]\) a convolution class \(K* U_p\oplus \mathbb{R}\) and prove that \[ \sup\{\|f-T_{b-1}(f, \cdot) \|_\infty\mid f\in K* U_p\oplus \mathbb{R}\}\leq Cb^{1/2} \sum^\infty_{k= b}|\Delta^2 \lambda_k|k^{1+ 2/p} \] when \(b\to\infty\). In the above inequality \(T_{b-1}(f,\cdot)\) is a certain quasi-interpolation polynomial of degree \(\leq b-1\) assiged to equiangular grid points.
scientific article; zbMATH DE number 1417555