In:
Georgian Mathematical Journal, Walter de Gruyter GmbH, Vol. 0, No. 0 ( 2023-04-06)
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 ) ⟩ f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangle 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 \mathbb{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 ) L^{p}(\mathbb{R}^{2}) , where 1 〈 p ≤ 2 1 〈 p\leq 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 ) \mathrm{KP}(\mathbb{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. Exchange 45 (2020), 2, 283–326]. A question arises about the dual space of the new space KP ( R 2 ) \mathrm{KP}(\mathbb{R}^{2}) .
Type of Medium:
Online Resource
ISSN:
1072-947X
,
1572-9176
DOI:
10.1515/gmj-2023-2008
Language:
English
Publisher:
Walter de Gruyter GmbH
Publication Date:
2023
detail.hit.zdb_id:
2043991-X
SSG:
17,1
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