Approximate identities and pointwise convergence

Herbert Stanley Bear, Jr.

Vol. 81 (1979), No. 1, 17–27

DOI: 10.2140/pjm.1979.81.17

Abstract

We give two additional conditlons on an approximate identity (or positive kernel) {K_α} which insure that f^∗K_α → f a.e. if f ∈ L¹ 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 pointwise convergence occurs.

Mathematical Subject Classification 2000

Primary: 31A10

Milestones

Received: 8 May 1978

Published: 1 March 1979