Integral transforms of the Kontorovich-Lebedev convolution type. (English) Zbl 1067.44004

Yakubovich, Semyon B.

Collect. Math. 54, No. 2, 99-110 (2003).

Summary: We deal with a class of integral transformations of the form \[ f(x)\to{1\over 2x} \prod^\infty_{n=1} \Biggl(1+ {x(x-{d\over dx}- x{d^2\over dx^2})\over (2n- 1)^2}\Biggr) \int_{\mathbb{R}^2_+} e^{-{1\over 2}(x{u^2+ y^2\over uy}+{yu\over x})}f(u) h(y)\,du\,dy,\;x\in\mathbb{R}_+ \] in \(L_2(\mathbb{R}_+; x\,dx)\), which is associated with the Kontorovich-Lebedev operator \[ K_{i\tau}[f]= \int^\infty_0 K_{i\tau}(x) f(x)\,dx,\quad \tau\in\mathbb{R}_+. \] Necessary and sufficient conditions on \(h\) to establish that the transformation is unitary in \(L_2(\mathbb{R}_+; x\,dx)\) are obtained. A reciprocal inversion formula and an example of the unitary convolution transformation are given.

MSC:

44A15	Special integral transforms (Legendre, Hilbert, etc.)
44A35	Convolution as an integral transform

Keywords:

Kontorovich-Lebedev transform; convolution; Plancherel theory; Watson theorem; Parseval equality; reciprocal inversion formula