<p>This paper explores Parseval-Goldstein and Abelian theorems for the Mehler-Fock-Clifford transform acting on functions defined over the interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left( \frac{1}{4}, \infty \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mi>∞</mi> </mfenced> </math></EquationSource> </InlineEquation>. We establish integral identities within the framework of weighted Lebesgue spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{1}\left( \left( \frac{1}{4}, \infty \right) ; P_{-\frac{1}{2}}(2\sqrt{x})m(x)\,dx\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mfenced close=")" open="("> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mi>∞</mi> </mfenced> <mo>;</mo> <msub> <mi>P</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, as well as in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^{\infty }\left( \left( \frac{1}{4}, \infty \right) \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mfenced close=")" open="("> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mi>∞</mi> </mfenced> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{1}\left( \left( \frac{1}{4}, \infty \right) \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mfenced close=")" open="("> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mi>∞</mi> </mfenced> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we derive Abelian-type results that describe the asymptotic behaviour of distributions and generalized functions under the Mehler-Fock-Clifford transform.</p>

The Mehler–Fock–Clifford transform framework for Parseval–Goldstein and abelian theorems in weighted lebesgue spaces

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2026年2月23日

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Jeetendrasingh Maan,
Akhilesh Prasad

摘要

This paper explores Parseval-Goldstein and Abelian theorems for the Mehler-Fock-Clifford transform acting on functions defined over the interval $\left( \frac{1}{4}, \infty \right) $ $(\frac{1}{4}, \infty)$ . We establish integral identities within the framework of weighted Lebesgue spaces $L^{1}\left( \left( \frac{1}{4}, \infty \right) ; P_{-\frac{1}{2}}(2\sqrt{x})m(x)\,dx\right) $ $L^{1} ((\frac{1}{4}, \infty); P_{- \frac{1}{2}} (2 \sqrt{x}) m (x) d x)$ , as well as in $L^{\infty }\left( \left( \frac{1}{4}, \infty \right) \right) $ $L^{\infty} ((\frac{1}{4}, \infty))$ and $L^{1}\left( \left( \frac{1}{4}, \infty \right) \right) $ $L^{1} ((\frac{1}{4}, \infty))$ . Furthermore, we derive Abelian-type results that describe the asymptotic behaviour of distributions and generalized functions under the Mehler-Fock-Clifford transform.