On the Duality of Regular and Local Functions

Fischer, Jens

doi:10.20944/preprints201705.0175.v1

Cited by 4 publications

(26 citation statements)

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“…The theory that made this notion rigorous is the theory of distributions (generalized functions theory) summarized in Laurent Schwartz' encompassing two-volume work [7,8]. It has already become a standard setting in Fourier analysis [2,5,[9][10][11][12][13][14], wavelet theory [15] and beside mathematics [16][17][18][19][20][21][22][23][24][25][26][27][28][29], also in quantum physics [30][31][32][33] where its origin [34], the Dirac delta [35] can be found, and in electrical engineering [36,37] where the Dirac delta is used to formally describe sampling [38]. Laurent Schwartz' theory of distributions is also part of wider theories, such as those on pseudo-differential operators [39][40][41] or modulation spaces [42][43][44] including Feichtinger's algebra [45].…”

Section: The Fourier Transform and The Theory Of Infinitely Differentmentioning

confidence: 99%

“…The circumstance that convolutions and, correspondingly, multiplications among distributions cannot arbitrarily be applied, is sometimes considered a major disadvantage of Laurent Schwartz' distribution theory [46,47]. It is, however, not a disadvantage of the theory -it is rather owned to Heisenberg's uncertainty principle [29]. Intuitively it is clear that convolutions fail if they are not summable.…”

Section: The Fourier Transform and The Theory Of Infinitely Differentmentioning

confidence: 99%

“…Because of the fact that both, convolution and multiplication, may fail among arbitrary tempered distributions, we need to consider three important subspaces of the space S of tempered distributions. For a deeper understanding one may refer to [6,[18][19][20][21][22][23][24]28,29,[49][50][51][52][53][54]. We require the subspace O C of convolution operators in S , the subspace O M of multiplication operators in S and the Schwartz space S which consists of (ordinary) functions which are both, convolution and multiplication operators.…”

Section: Convolution Vs Multiplicationmentioning

confidence: 99%

“…In Table 1 we summarize -for the readers convenience -symbolic calculation rules including their validity scope. They are derived in [6,28,29]. The validities hold on tempered distributions and in particular on Lebesgue square-integrable functions under the same conditions.…”

Section: Validity Statementmentioning

confidence: 99%

“…For simplicity, we do not assume vector-valued functions in this study, i.e., we let n = 1 in t, σ ∈ R n or k, m ∈ Z n although the n-dimensional case looks very similar. For cases of n > 1, one may refer to our previous studies [6,28,29].…”

Section: Generalized Functionsmentioning

confidence: 99%

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Four Particular Cases of the Fourier Transform

Fischer¹

2018

Preprint

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In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

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Section: The Fourier Transform and The Theory Of Infinitely Differentmentioning

confidence: 99%

Section: The Fourier Transform and The Theory Of Infinitely Differentmentioning

confidence: 99%

Section: Convolution Vs Multiplicationmentioning

confidence: 99%

Section: Validity Statementmentioning

confidence: 99%

Section: Generalized Functionsmentioning

confidence: 99%

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Four Particular Cases of the Fourier Transform

Fischer¹

2018

Preprint

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Distribution Theory by Riemann Integrals

Feichtinger

Jakobsen²

2020

Industrial and Applied Mathematics

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It is the purpose of this article to outline a syllabus for a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably, these are needed for a deeper understanding of basic questions in signal analysis. Objects such as the Dirac delta and the Dirac comb should have a proper definition, and it should be possible to explain how one can reconstruct a band-limited function from its samples by means of simple series expansions. It should also be useful for graduate mathematics students who want to see how functional analysis can help to understand fairly practical problems, or teachers who want to offer a course related to the "Mathematical Foundations of Signal Processing" at their institutions. The course requires only an understanding of the basic terms from linear functional analysis, namely Banach spaces and their duals, bounded linear operators and a simple version of w *convergence. As a matter of fact we use a set of function spaces which is quite different from the collection of Lebesgue spaces L p (R d ), · p used normally. We thus avoid the use of Lebesgue integration theory. Furthermore we avoid topological vector spaces in the form of the Schwartz space.Although practically all the tools developed and presented can be realized in the context of LCA (locally compact Abelian) groups, i.e. in the most general setting where a (commutative) Fourier transform makes sense, we restrict our attention in the current presentation to the Euclidean setting, where we have (generalized) functions over R d . This allows us to make use of simple BUPUs (bounded, uniform partitions of unity), to apply dilation operators and occasionally to make use of concrete special functions such as the (Fourier invariant) standard Gaussian, given by g 0 (t) = exp(−π|t| 2 ).The problems of the overall current situation, with the separation of theoretical Fourier Analysis as carried out by (pure) mathematicians and Applied Fourier Analysis (as used in engineering applications) are getting bigger and bigger and therefore courses filling the gap are in strong need. This note provides an outline and may serve as a guideline. The first author has given similar courses over the last years at different schools (ETH Zürich, DTU Lyngby, TU Muenich, and currently Charlyes University Prague) and so one can claim that the outline is not just another theoretical contribution to the field.

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All-Optical Reinforcement Learning In Solitonic X-Junctions

Alonzo

Moscatelli

Bastiani

et al. 2018

Sci Rep

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Ethology has shown that animal groups or colonies can perform complex calculation distributing simple decision-making processes to the group members. For example ant colonies can optimize the trajectories towards the food by performing both a reinforcement (or a cancellation) of the pheromone traces and a switch from one path to another with stronger pheromone. Such ant’s processes can be implemented in a photonic hardware to reproduce stigmergic signal processing. We present innovative, completely integrated X-junctions realized using solitonic waveguides which can provide both ant’s decision-making processes. The proposed X-junctions can switch from symmetric (50/50) to asymmetric behaviors (80/20) using optical feedbacks, vanishing unused output channels or reinforcing the used ones.

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On the Duality of Regular and Local Functions

Abstract: Abstract:In this paper, we relate Poisson's summation formula to Heisenberg's uncertainty

Cited by 4 publications

References 3 publications

Four Particular Cases of the Fourier Transform

Four Particular Cases of the Fourier Transform

Distribution Theory by Riemann Integrals

All-Optical Reinforcement Learning In Solitonic X-Junctions

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