On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications

  • T. Tuan Electric Power Univ., Hanoi, Vietnam

Abstract

UDC 517.5

We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$.
We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution.

We impose necessary and sufficient conditions on the kernel $h$ to ensure that the generalized convolution transform
$$
D_h\colon f\mapsto D_{h}[f] = \left(1-\dfrac{d^2}{dx^2}\right)(h\underset{F_s,K} \ast f)(x)
$$
is a unitary operator in $L_2(\Bbb R_+)$ (Watson-type theorem) and derive its inverse formula.
Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the $L_p$-norm.

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Published
15.02.2020
How to Cite
Tuan T. “On the Fourier Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Applications”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 2, Feb. 2020, pp. 267-79, http://umj.imath.kiev.ua/index.php/umj/article/view/2370.
Section
Research articles