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Gasquet, Claude; Witomski, Patrick

doi:10.1007/978-1-4612-1598-1_2

Cited by 2 publications

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“…A classical theorem in Lebesgue's theory is about differentiability under the integral sign [12,29]. The following result is a generalization of this, first we introduce the concept of a generalized absolutely continuous function in the restricted sense.…”

Section: Differentiability Of the Fourier Transformmentioning

confidence: 99%

Fourier Analysis with Generalized Integration

2020

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We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

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Section: Differentiability Of the Fourier Transformmentioning

confidence: 99%