Abstract
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.
Highlights
The Lebesgue integral has strong implications in Fourier Analysis
The implications from these results are that the classical Fourier transform F p ( f )(s) for f in a dense subspace of L p (R) is represented by a Lebesgue integral, is a continuous function, except at s = 0, and it vanishes at infinity as o (|s|−1 )
An integral representation of the Fourier transform is obtained on the subspace L p (R) ∩ BV0 (R) ∩
Summary
The Lebesgue integral has strong implications in Fourier Analysis. Integration theory had an important development in the last half-century. In [2] it was proved that, for subsets of p-integrable functions, 1 < p ≤ 2, the classical Fourier transform, F p ( f )(s), is equal a.e. to a Henstock–Kurzweil integral, which has a pointwise expression for any s 6= 0, see Remark 1 and Theorem 2 This representation allows for analyzing more properties related to the Fourier transform, as continuity or asymptotic behavior. This opens up the possibility to consider a strictly larger class of functions where some calculus remains valid This happens for extensions of the Fourier transform operator. To a strictly larger class of functions This is facilitated through the use of the Henstock–Kurzweil integration theory in the framework of Fourier Analysis.
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