On the Integral Transformation Associated with the Product of Gamma-Functions

Semyon B. Yakubovich

Abstract: We introduce the following integral transformation $$ \Phi(z)=2^{z-2}\int_{\R_+}f(\tau)\Gamma\biggl({z+i\tau\over2}\biggr) \Gamma\biggl({z-i\tau\over 2}\biggr)\,d\tau, $$ where $z=x+iy$, $x>0$, $y\in\R$, $\Gamma(z)$ is Euler's Gamma-function. 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 half-plane 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 Kontorovich--Lebedev transforms.

Keywords: Gamma-function; Hardy spaces; Plancherel theorem; Mellin transform; Hilbert transform; Kontorovich--Lebedev transform; Fourier transform; Parseval equality; singular integral.

Classification (MSC2000): 44A20, 44A15, 42B20, 33B15.

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