Fourier Analysis with Generalized Integration

Juan H. Arredondo, Manuel Bernal and María Guadalupe Morales
Additional contact information
Juan H. Arredondo: Departamento de Matemáticas, Universidad Autónoma Metropolitana—Iztapalapa, Av. San Rafael Atlixco 186, México City 09340, Mexico
Manuel Bernal: Departamento de Matemáticas, Universidad Autónoma Metropolitana—Iztapalapa, Av. San Rafael Atlixco 186, México City 09340, Mexico
María Guadalupe Morales: Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Mathematics, 2020, vol. 8, issue 7, 1-16

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.

Keywords: fourier transform; Henstock-Kurzweil integral; bounded variation function; ? p spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/7/1199/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/7/1199/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:7:p:1199-:d:387741

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().