Abstract
In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.
Highlights
Regularization is a popular trick in applied mathematics; see [1] for example
For example, to integrate functions which could not be integrated otherwise, if we think of “locally integrable” functions or if we think of the Short-Time Fourier Transform (STFT), capable of analyzing infinitely extended signals
The theorem below (Figure 3) appears in a wider sense. It even holds within the space of tempered distributions and is directly related to Heisenberg’s uncertainty principle
Summary
Regularization is a popular trick in applied mathematics; see [1] for example. It is the technique “to approximate functions by more differentiable ones” [2]. Regularization (Figure 1) and localization (Figure 2) appear to be quite different, a connection between these operations is no surprise. It is rather ubiquitous in the literature. The theorem below (Figure 3) appears in a wider sense It even holds within the space of tempered distributions and is directly related to Heisenberg’s uncertainty principle. While discretization and periodization map functions from smoothness towards discreteness in both time and frequency domain, the operations investigated in this study, regularization and localization, map functions from discreteness towards smoothness in both time and frequency domain (Figure 4 and Figure 5).
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