There Is Only One Fourier Transform

Jv, Fischer

doi:10.20944/preprints201712.0173.v1

Cited by 2 publications

(11 citation statements)

References 14 publications

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“…Section 4 provides an introduction to the notations used and previous results. Section 5 presents a justification for Section 6 where regularization and localization coincide in the distributional sense [20][21][22] or deriving a digital radar image from the continuous case (the landscape being imaged) via operations of localization (windowing), discretization (sampling), convolution (defocusing) and deconvolution (focusing) [23].…”

Section: Introductionmentioning

confidence: 99%

On the Duality of Regular and Local Functions

Fischer

2017

Mathematics

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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.

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Section: Introductionmentioning

confidence: 99%

On the Duality of Regular and Local Functions

Fischer

2017

Mathematics

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“…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]:…”

Section: Preliminariesmentioning

confidence: 99%

“…However, there is no possibility whatsoever to absolutly determine which of the two spaces should be called positive or negative dimensional. Their orientability rather behaves like a Möbius band [20]. A minus sign therefore simply expresses that it is the Fourier transform of another space or, equivalently, that "time" is running in an opposite direction.…”

Section: Negative Dimensional Spacesmentioning

confidence: 99%

“…Such identifications will at least be done when they are Fourier transformed (using the DFT). They represent discrete periodic functions [18][19][20] and therewith build a bridge from distribution theory to linear algebra. Zero-padding is moreover depicted in Figure 5.…”

Section: Zero-paddingmentioning

confidence: 99%

“…Square brackets indicate the fact that a periodic function Fourier transformed yields a discrete function see (17), (18) or [20] for details.…”

Section: Corollary 3 Let δ Be Dirac's Delta Thenmentioning

confidence: 99%

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A Paradox of Unity

Fischer¹

2018

Preprint

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Abstract:In previous studies we found that generalized functions can be smooth, discrete, periodic or discrete periodic and they can either be local or global and they are regular or generalized functions. We also saw that these properties were related to Poisson's summation formula on one hand and to Heisenberg's uncertainty principle on the other. In this paper, we interlink these studies and show that scalars (real or complex numbers) considered as trivial functions are discrete and periodic, local and global as well as regular and generalized, simultaneously. However, this is also a paradox because it means that Dirac's δ and 1 (its Fourier transform) coincide. They both are unity. We show that δ and 1 coincide in the sense of scalars (real or complex numbers) but they differ in the sense of (generalized) functions. This result can moreover be related to Max Born's principle of reciprocity. It also answers an open question in present-day quantum mechanics because it means that the Dirac delta squared is simply delta.

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There Is Only One Fourier Transform

Cited by 2 publications

References 14 publications

On the Duality of Regular and Local Functions

On the Duality of Regular and Local Functions

A Paradox of Unity

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