On the Duality of Discrete and Periodic Functions

Fischer, Jens V. (2015): On the Duality of Discrete and Periodic Functions. In: Mathematics, Vol. 3, No. 2: pp. 299-318

Abstract

Although versions of Poisson's Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 2227-7390
• Language: English