Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

Nguyen, Ha Q.; Unser, Michaël; Ward, John Paul

doi:10.1007/s00041-016-9475-9

Cited by 10 publications

(10 citation statements)

References 34 publications

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“…It is clear that otherwise the corresponding generalized function already belongs to one of these two spaces. This validity statement is furthermore obtained in good agreement with many other publications devoted to this topic, including [4,[55][56][57][58][59], except for the fact that, in this as well as in our previous studies, we generally do not move away from the overall principle that all functions must be infinitely differentiable. It is the default, tacit assumption in distribution theory.…”

Section: Validity Statementsupporting

confidence: 92%

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: Validity Statementsupporting

confidence: 92%

Four Particular Cases of the Fourier Transform

Fischer¹

2018

Preprint

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“…It is a given fact for all functions in generalized function spaces. Also recall that O M is the space of multiplication operators in S and O C is the space of convolution operators in S according to Laurent Schwartz' theory of distributions [14,16,[24][25][26][27][29][30][31].…”

Section: Preliminariesmentioning

confidence: 99%

“…Regularizations are treated in many mathematical textbooks [2,11,[24][25][26]31] and scientific papers 189 [1,[34][35][36][37][38][39][40][41][42]. They are also known in terms of "smooth cutoff functions" [2,8], "regularizers" [1,34,35,37],…”

mentioning

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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“…where the operator T y,τ is given in (27). Repeating the manipulations in the proof of Proposition 2, we obtain the counterpart of (34):…”

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confidence: 88%

Approximation of Non-decaying Signals from Shift-Invariant Subspaces

Nguyen¹,

Unser

2018

J Fourier Anal Appl

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In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted-L p spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang-Fix theory to show that, for ddimensional signals whose derivatives up to order L are all in some weighted-L p space, the weighted norm of the approximation error can be made to go down as O(h L ) when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang-Fix conditions of order L. We show that the O(h L ) behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of d/p + ε, for arbitrary ε > 0. This extra amount of derivatives is used to make sure that the direct sampling is stable.

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Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

Cited by 10 publications

References 34 publications

Four Particular Cases of the Fourier Transform

Four Particular Cases of the Fourier Transform

On the Duality of Regular and Local Functions

Approximation of Non-decaying Signals from Shift-Invariant Subspaces

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