A theorem concerning approximation on the sphere
The authors prove the following interesting theorem concerning approximation on a unit sphere \(\sigma\) in \(R^ n\). Theorem: If \(f(\mu)\in H^ r_ p(\sigma)\), then \(\| f-T_{\nu}\|_{L_ p(\sigma)}\leq (C\| f\| /\nu^ r)H^ r_ p(\sigma)\) \((\nu =1,2,...)\) for a suitable sequence \(\{T_{\nu}(\mu)\}\) of spherical polynomials of order \(\nu\), where c is independent of f and \(\nu\). Conversely, if f(\(\mu)\), \(\mu\in \sigma\), is a function such that for some sequence \(\{T_{\nu}(\mu)\}\) of spherical polynomials of order \(\nu\), \(\| f-T_{\nu}\|_{L_ p(\sigma)}\leq K/\nu^ r\) \((\nu =1,2,...)\) then \(f\in H^ r_ p(\sigma)\) and \(\| f\|_{H^ r_ p(\sigma)}\leq C_ 1(\| f\|_{L_ p(\sigma)}+K),\) where K is a constant independent of \(\nu\) and \(C_ 1\) is independent of f and K.
approximation on a unit sphere \(\sigma \)
scientific article; zbMATH DE number 3901386