Convolutions related to the Fourier and Kontorovich-Lebedev transforms revisited. (English) Zbl 1191.44002

Yakubovich, S. B.; Britvina, L. E.

Integral Transforms Spec. Funct. 21, No. 3-4, 259-276 (2010).

The paper is devoted to commutative convolutions of the form
\[ (f\ast g)(x)=\frac{1}{2\pi x}\int_{\mathbb R^2_+} f(\tau)g(\theta)(e^{-x\cosh(\tau-\theta)}\pm e^{-x\cosh(\tau+\theta)})\,d\tau \,d\theta,\quad x>0, \]
and to the noncommutative convolutions
\[ (f\ast g)(x)= \frac{1}{2}\int_{\mathbb R^2_+} f(\tau)g(\theta)(e^{-\theta\cosh(\tau-x)}\pm e^{-\theta\cosh(\tau+x)})\,d\tau \,d\theta,\quad x>0. \]
For these convolutions, mapping properties in the weighted \(L_p\)-spaces are obtained and factorization equalities are established. Applications to a solvability of the corresponding class of convolution integral equations are demonstrated.

Reviewer: Vitaly Vladimirovich Volchkov (Donetsk)

MSC:

44A35	Convolution as an integral transform
44A15	Special integral transforms (Legendre, Hilbert, etc.)
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Keywords:

Fourier transform; Kontorovich-Lebedev transform; modified Bessel function; convolution integral equations; Plancherel theorem; commutative convolutions; noncommutative convolutions

References:

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