On the Duality of Regular and Local Functions

Abstract

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.

There Is Only One Fourier Transform

“…As a rule of thumb for their validity it is very useful to know that  functions being discretized should not grow faster than polynomials and  functions being periodized should decay to zero faster than polynomials, otherwise they will cease to converge. These requirements are furthermore dual to each other in the sense that if a function fulfills one requirement then its Fourier transform fulfills the other [3][4].…”

“…Note that the dual of discreteness (discrete functions) is not continuity (continuous functions) as usually taught but smoothness (smooth functions). Further details on these principles can be found in [4]. It remains to mention that both ⊥ ⊥ ⊥ and △△△ perform sampling, one in "time domain" and the other in "frequency domain" (Figure 4) which means that the classical sampling theorem must be respected twice (in time and frequency domain) otherwise aliasing effects occur; hence f(t) and (⊥⊥⊥△△△ N f)(t) would not represent the same function any more.…”

“…To keep things simple, we only treat the one-dimensional case (t ∈ R). For n-dimensional cases (t ∈ R These formulas are moreover rigorously based on generalized functions theory and part of a wider symbolic calculation scheme [3][4], mainly summarized in Table 1 and Table 2 and their relationship to four different Fourier transform variants, i.e., to F, F per, DTFT and DFT, is summarized in Table 3 and Table 4. As a rule of thumb for their validity it is very useful to know that  functions being discretized should not grow faster than polynomials and  functions being periodized should decay to zero faster than polynomials, otherwise they will cease to converge.…”

A Paradox of Unity

“…Discreteness arises as soon as we think of unity as if it where "taken out of its context" (embedded in zeros) and smoothness arises as soon as we think of unity as if it where "repeated in all dimensions" (padded by itself). Recall that we already encountered a discreteness-periodicity duality in [18] and a locality-globality duality in [19]. They are expressed in Poisson's summation formula and in Heisenberg's uncertainty principle, respectively.…”

“…discrete in both, time and frequency domain), i.e., to vectors, matricies or tensors (depending on whether they depend on one, two or more variables) by applying discretization and periodization, respectively [18]. Vice versa, fully discrete functions (vectors, matricies or tensors) can be converted back to fully smooth functions via inverse periodization (localization) and inverse discretization (regularization) [19]. It followed that the space of tempered distributions falls into four parts (or three if the two mixed spaces are combined to a space of "half-discrete" functions) [20]:…”