Sampling multipliers and the Poisson Summation Formula

Benedetto, John J.; Zimmermann, Georg

doi:10.1007/bf02648881

Cited by 68 publications

(56 citation statements)

References 11 publications

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“…(2) and d is a positive real number. This is a well-known result and holds, for instance, if f (x) is an integrable function of bounded variation (Benedetto and Zimmermann, 1997). The following two relations are alternative versions of the summation formula:…”

Section: Appendix a The Poisson Summation Formulamentioning

confidence: 91%

Limiting similarity, species packing, and the shape of competition kernels

Leimar

Sasaki

Doebeli

et al. 2013

Journal of Theoretical Biology

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A traditional question in community ecology is whether species' traits are distributed as more-or-less regularly spaced clusters. Interspecific competition has been suggested to play a role in such structuring of communities. The seminal theoretical work on limiting similarity and species packing, presented four decades ago by Robert MacArthur, Richard Levins and Robert May, has recently been extended. There is now a deeper understanding of how competitive interactions influence community structure, for instance, how the shape of competition kernels can determine the clustering of species' traits. Competition is typically weaker for greater phenotypic difference, and the shape of the dependence defines a competition kernel. The clustering tendencies of kernels interact with other effects, such as variation in resource availability along a niche axis, but the kernel shape can have a decisive influence on community structure. Here we review and further extend the recent developments and evaluate their importance.

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Section: Appendix a The Poisson Summation Formulamentioning

confidence: 91%

Limiting similarity, species packing, and the shape of competition kernels

Leimar

Sasaki

Doebeli

et al. 2013

Journal of Theoretical Biology

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“…Gröchenig obtained in [11] a similar result in connection with the uncertainty principle. In the 1-D case, when d = 1, it is also known [4,36] that PSF holds pointwise when f,f ∈ L 1 (R) ∩ C(R) andf has bounded total variation. Many other forms of the PSF on L 1 (R) can also be found in the extensive work of Benedetto and Zimmermann [4].…”

Section: Introductionmentioning

confidence: 99%

“…In the 1-D case, when d = 1, it is also known [4,36] that PSF holds pointwise when f,f ∈ L 1 (R) ∩ C(R) andf has bounded total variation. Many other forms of the PSF on L 1 (R) can also be found in the extensive work of Benedetto and Zimmermann [4]. The main limitation of all the above versions of the PSF is that they do not apply to functions that might be growing, such as realizations of a non-stationary stochastic process, the typical example being Brownian motion [31].…”

Section: Introductionmentioning

confidence: 99%

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

Nguyen

Unser

Ward

2016

J Fourier Anal Appl

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The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of L 1 signals. This L 1 assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that d/2 + ε derivatives of the signal are bounded by some polynomial in the L 2 sense.Communicated by Eric Todd Quinto.

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“…5 For a given threshold ε (trunc) (for example, around floating-point rounding error), determine a corresponding integer B such that…”

Section: Computing Segment Of Interest -Practicementioning

confidence: 99%

A Framework for Low-Communication 1-D FFT

Tang

Park

Kim

et al. 2013

Scientific Programming

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Abstract. In high-performance computing on distributed-memory systems, communication often represents a significant part of the overall execution time. The relative cost of communication will certainly continue to rise as compute-density growth follows the current technology and industry trends. Design of lower-communication alternatives to fundamental computational algorithms has become an important field of research. For distributed 1-D FFT, communication cost has hitherto remained high as all industry-standard implementations perform three all-to-all internode data exchanges (also called global transposes). These communication steps indeed dominate execution time. In this paper, we present a mathematical framework from which many single-all-to-all and easy-to-implement 1-D FFT algorithms can be derived. For large-scale problems, our implementation can be twice as fast as leading FFT libraries on state-of-the-art computer clusters. Moreover, our framework allows tradeoff between accuracy and performance, further boosting performance if reduced accuracy is acceptable.

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Sampling multipliers and the Poisson Summation Formula

Cited by 68 publications

References 11 publications

Limiting similarity, species packing, and the shape of competition kernels

Limiting similarity, species packing, and the shape of competition kernels

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

A Framework for Low-Communication 1-D FFT

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