On the factorization theorem for the tensor product of integrable distributions
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The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
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}) .
The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
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}) .
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