PORTUGALIAE MATHEMATICA Vol. 60, No. 3, pp. 337351 (2003) 

On the Integral Transformation Associated with the Product of GammaFunctionsSemyon B. YakubovichDepartment of Pure Mathematics, Faculty of Sciences, University of Porto,Campo Alegre st., 687, 4169007 Porto  PORTUGAL Email: syakubov@fc.up.pt Abstract: We introduce the following integral transformation $$ \Phi(z)=2^{z2}\int_{\R_+}f(\tau)\Gamma\biggl({z+i\tau\over2}\biggr) \Gamma\biggl({zi\tau\over 2}\biggr)\,d\tau, $$ where $z=x+iy$, $x>0$, $y\in\R$, $\Gamma(z)$ is Euler's Gammafunction. Boundedness and analytic properties are investigated. The Bochner representation theorem is proved for functions $f\in L^*(\R_+)$, whose Fourier cosine transforms lie in $L_1(\R_+)$. It is shown, that this transform is an analytic function in the right halfplane and belongs to the Hardy space $\H_2$. When $x\to 0$ it has boundary values from $L_2(\R)$. Plancherel type theorem is established by using its relationships with the Mellin and KontorovichLebedev transforms. Keywords: Gammafunction; Hardy spaces; Plancherel theorem; Mellin transform; Hilbert transform; KontorovichLebedev transform; Fourier transform; Parseval equality; singular integral. Classification (MSC2000): 44A20, 44A15, 42B20, 33B15. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
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