The Fourier transform of thick distributions

Estrada, Ricardo; Vindas, Jasson; Yang, Yunyun

doi:10.1142/s0219530520500074

Cited by 2 publications

(1 citation statement)

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“…We can hence cite in [19,20] the definition of the FT as the limit of a sequence of functions integrated on a finite domain, or [60] for a two-sided Laplace transform defined on a space larger than that of tempered distributions, and similarly in [3] for the directional short-time Fourier transform of exponentialtype distributions. In the same direction we can inscribe the works [2,7,13,32,46,54,51,16,17] on ultradistributions, hyperfunctions and thick distributions.…”

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confidence: 95%

A Fourier transform for all generalized functions

Akbarali¹,

Tiwari²,

Giordano³

2021

Preprint

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Using the existence of infinite numbers k in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to [−k, k] n instead of (−∞, ∞) n . In order to realize this idea, the space of generalized functions we consider is that of generalized smooth functions (GSF), an extension of classical distribution theory sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition, a good integration theory, and several classical theorems of calculus. Even if the final transform depends on k, we obtain a new notion that applies to all GSF, in particular to all Schwartz's distributions and to all Colombeau generalized functions, without growth restrictions. We prove that this FT generalizes several classical properties of the ordinary FT, and in this way we also overcome the difficulties of FT in Colombeau's settings. Differences in some formulas, such as in the transform of derivatives, reveal to be meaningful since allow to obtain also non-tempered global unique solutions of differential equations.

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