Abstract

We prove the Plancherel theorem and establish related L 2-properties for the following singular integral transformation associated with the product of Euler’s Gamma-functions: $$[\mathcal{G}f](x) = P.V.\int\limits_0^\infty {\Gamma (i(x + \tau ))\Gamma (i(x - \tau ))f(\tau )d\tau ,} x \in \mathbb{R}.$$ The results are based on the corresponding properties for the Kontorovich-Lebedev and Mellin transformations. Necessary and sufficient conditions of the analytic continuability of the Gamma-product transform to the right half-plane are derived.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.