Approximate identities and pointwise convergence

Abstract

We give two additional conditions on an approximate identity (or positive kernel) {K a } which insure that f*K a -+f a.e. if feL 1 on the line or circle. Where the convolution defines a function on the disc or a half-plane, as for the Poisson kernels or heat kernels, then the theorem gives automatically the paths toward a boundary point along which point wise convergence occurs.