Approximation on the Sphere by Weighted Fourier Expansions

A standard procedure to approximate a function f in an inner product space is to consider the Fourier series of the function with respect to an orthogonal system. The basic general results on this topic can be found in many references in the literature, for example, [2, Chapter VIII]. It is well known that even in the case in which K is a closed interval there always exists a function f in C(K) for which the corresponding Fourier series does not converge to f with respect to the uniform norm. Thus, in this and other cases, the common solution is to consider weighted expansions and to study convergence based on the choice of the weights. Here is a list of problems that emerges: how to choose the weights in order to guarantee convergence for every function in the space, to study orders of convergence, how to choose the weights so that the operators given by the truncated Fourier series inherit properties of other known operators, and so forth. In this paper, we consider some of the problems above in the case when K = Sm, the unit sphere in Rm+1. The focus is on convergence but we intend to study the analysis of convergence orders in a forthcoming paper. For functions defined on Sm, orthogonality depends upon dσm, the usual surface measure on Sm. The surface area of Sm will be written as σm. The uniform norm is then given by