Positive Sampling in Wavelet Subspaces

This paper is devoted to the discussion of a “hybrid” sampling series, a series of translates of a nonnegative summability function used in place of an orthogonal scaling function. The coefficients in the series are taken to be sampled values of the function to be approximated. This enables one to avoid the integration which arises in the other series. The approximations based on this hybrid series have certain desirable convergence properties: they are locally uniformly convergent for locally continuous functions, they have quadratic uniform convergence rate for functions in certain Sobolev spaces, they are locally bounded when the function is locally bounded and therefore, in particular, Gibbs' phenomenon is avoided. Numerical experiments are given to illustrate the theoretical results and to compare these approximations with the scaling function approximations.