2017
Том 69
№ 6

All Issues

On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms

Thao N. X., Virchenko N. A.

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Abstract

We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. These generalized convolutions satisfy the factorization equalities $$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0$$ We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 1, pp 89-101.

Citation Example: Thao N. X., Virchenko N. A. On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms // Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 81-91.

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