We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation.
- Authors
Mendoza, Francisco J.; Arredondo, Juan H.; Sánchez-Perales, Salvador; Flores-Medina, Oswaldo; Torres-Teutle, Edgar
- Abstract
In the classical Fourier analysis, the representation of the double Fourier transform as the integral of f (x , y) exp (− i ⟨ (x , y) , (s 1 , s 2) ⟩ is usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions on R 2 . We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined on L p (R 2) , where 1 < p ≤ 2 , yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functions KP (R 2) which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane, Real Anal. Exchange45 (2020), 2, 283–326]. A question arises about the dual space of the new space KP (R 2) .
- Subjects
INTEGRABLE functions; FUNCTIONS of bounded variation; LEBESGUE integral; INTEGRAL transforms; FOURIER integrals; FOURIER analysis; FOURIER transforms
- Publication
Georgian Mathematical Journal, 2023, Vol 30, Issue 3, p403
- ISSN
1072-947X
- Publication type
Article
- DOI
10.1515/gmj-2023-2008