Definability and Strength of Fourier Series Transform Obtained by Employing Parametric Integration

Abstract

Regardless of the term coinage, this article is a story of adding variables that act in a similar way to the “s” in the Laplace transform. This idea is similar to the avoiding of points in improper integration, but instead of taking limits, the endpoints of each maximum integrable interval are made to be variables, but they still preserve the orders. The result is here called a parametric integral. It is applied here to Fourier series transform as an example. Therefore, some functions that formerly had no Fourier series expansion may no longer exist if one employs parametric integration. The resulting transform has advantages in terms of definability and strength. In definability because the transform exists for more functions than even for the well-known Fourier transform. It gives hope for solving more problems that may be of certain practical interests, especially when endpoint variables in the solution are replaced by respective limiting processes. The possibility that a solution still contains non-eliminable variables may lead to interesting non-standard analysis with the parameters as new “numbers”, some of which are given here. The parameters may also serve in determining the partial inverses of the transform. Keywords: Fourier series expansion, Fourier series transform, half-range expansion, Laplace transform, parametric integration. https://doi.org/10.55463/issn.1674-2974.50.8.13