On the Convergence of Fourier Representations and Schwartz Distributions

Abstract

<p>Fourier theory is one of the most important tools used ubiquitously for understanding the spectral content of a signal, extracting and interpreting information from signals, and transmitting, processing, and analyzing the signals and systems. Undergraduate engineering students are exposed to these concepts, usually in their second year, to build their foundation in the areas of signal processing and communication engineering. However, the popular signal processing literature \cite{book1,book2,book3,book4} does not offer a clear explanation regarding the convergence or the existence of Fourier representations for certain well-known signals. Because of this subtle gap, it becomes hard for young students to assimilate the Fourier theory with clarity, and they are forced to be familiar with some of these concepts without understanding them. To bring clarity to the existence and the convergence of Fourier representation, including Fourier series and transform, lecture notes were published recently in IEEE Signal Processing Magazine's September 2022 issue \cite{PSDCs}. This lecture note is in the continuation with technical details added from yet another mathematical topic of distribution theory that connects delta Dirac functions with the Fourier theory. </p> <p>The distribution theory by Schwartz in 1945 is one of the great revolutions in mathematical function analysis. It is considered as a completion of differential calculus, similar to how the revolutionary measure theory or Lebesgue integration theory proposed in 1903, is considered as a completion of integral calculus. Both these theories unlocked new paradigms of mathematical development. </p> <p>Although distribution theory is a powerful tool for understanding Fourier theory, it is ignored in engineering textbooks. In this lecture note, we utilize the concepts of this theory to show how some signals that fail to exhibit FT in the conventional sense can have FT in the distributional sense.</p> <p><br></p>