An introduction to distribution theory for signals analysis. Part III. Periodic tempered distributions

This article continues the exposition in Parts I [1] and II [2] of certain concepts in the theory of tempered distributions which are useful for a thorough understanding of signal theory. In this series of articles, we strive to present a mathematically rigorous exposition of these ideas to supplement the imprecise treatment of distribution theory encountered in the signal processing literature. Part I contains an overview of the subject and discusses some well-known applications of tempered distributions in signals analysis. Subsequent parts delve more deeply into specialized topics. Thus, Part II focuses on the convolution of tempered distributions, and here in Part III we consider periodic tempered distributions and generalized Fourier series. Part III upholds the style of Parts I and II by maintaining a high level of mathematical rigor, while resisting the temptation to become unnecessarily abstract. Thus, for example, it is not necessary to invoke the notion of topological vector spaces from general distribution theory when the simpler notion of normed linear space will suffice. We also provide very detailed proofs which will hopefully make the material more accessible to physicists, engineers, computer scientists and other non-mathematicians with an interest in this subject.