The Titchmarsh Theorem

Abstract

About 50 years ago, E. C. Titchmarsh discovered, by the occasion of investigating zeros of some analytical functions, an interesting theorem on convolution. That theorem plays an important role in modern Analysis and is actually called the Titchmarsh Convolution Theorem. The original proof of Titchmarsh is difficult and involves deep theorems on analytical functions. Simpler proofs consisting in examinating the rate of growth of analytical or harmonic functions were given by M. Crum [6] in 1941 and J. Dufresnoy [8] in 1947. At present there exist a large number of proofs, based on various methods.KeywordsInitial PointBanach AlgebraInteresting TheoremDeep TheoremBochner Integrable FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.