Fourier Analysis with Generalized Integration
| Authors | |
|---|---|
| Year of publication | 2020 |
| Type | Article in Periodical |
| Magazine / Source | Mathematics |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.3390/math8071199 |
| Doi | http://dx.doi.org/10.3390/math8071199 |
| Keywords | fourier transform; Henstock-Kurzweil integral; bounded variation function; L-p spaces |
| Description | We generalize the classic Fourier transform operator F-p by using the Henstock-Kurzweil integral theory. It is shown that the operator equals the HK-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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