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

T.  Tuan

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.

References

Adams, Robert A.; Fournier, John J. F. Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. xiv+305 pp. ISBN: 0-12-044143-8 MR2424078

Abramowitz, Milton; Stegun, Irene A. Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964 {rm xiv}+1046 pp. MR0167642

Al-Musallam, F.; Vu, Kim Tuan. Integral transforms related to a generalized convolution. Results Math. 38 (2000), no. 3-4, 197--208. doi: 10.1007/BF03322007

Grigoriev, Y. N.; Ibragimov, N. H.; Kovalev, V. F.; Meleshko, S. V. Symmetries of integro-differential equations. With applications in mechanics and plasma physics. Lecture Notes in Physics, 806. Springer, Dordrecht, 2010. xiv+305 pp. ISBN: 978-90-481-3796-1 doi: 10.1007/978-90-481-3797-8

Sneddon, Ian N. Fourier Transforms. McGraw-Hill Book Co., Inc., New York, Toronto, London, 1951. {rm xii}+542 pp. MR0041963

Britvina, L. E. A class of integral transforms related to the Fourier cosine convolution. Integral Transforms Spec. Funct. 16 (2005), no. 5-6, 379--389. doi: 10.1080/10652460412331320395

Glaeske, H.-J.; Prudnikov, A. P.; Skyrnik, K. A. Operational calculus and related topics. Analytical Methods and Special Functions, 10. Chapman & Hall/CRC, Boca Raton, FL, 2006. xvi+403 pp. ISBN: 978-1-58488-649-5; 1-58488-649-8 doi: 10.1201/9781420011494

Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Интегралы и ряды. (Russian) [[Integrals and series]] Дополнительные главы. [Supplementary chapters] ``Nauka'', Moscow, 1986. 800 pp. MR0888165

Nguyen Xuan Thao. On the polyconvolution with the weight function for the Fourier cosine, Fourier sine, and the Kontorovich-Lebedev integral transforms. Math. Probl. Eng. 2010, Art. ID 709607, 16 pp. MR2660082

Titchmarsh, E. C. Introduction to the theory of Fourier integrals. Third edition. Chelsea Publishing Co., New York, 1986. {rm x}+394 pp. ISBN: 0-8284-0324-4 MR0942661

Trinh Tuan; Nguyen Xuan Thao; Nguyen Van Mau. On the generalized convolution for the Fourier sine and the Kontorovich-Lebedev transforms. Acta Math. Vietnam. 35 (2010), no. 2, 303--317. MR2731331

Tuan, Vu Kim. Integral transforms of Fourier cosine convolution type. J. Math. Anal. Appl. 229 (1999), no. 2, 519--529. doi: 10.1006/jmaa.1998.6177

Yakubovich, S. B. Index transforms. With a foreword by H. M. Srivastava. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. {rm xiv}+248 pp. ISBN: 981-02-2216-5 doi: 10.1142/9789812831064

Yakubovich, Semyon B. Integral transforms of the Kontorovich-Lebedev convolution type. Collect. Math. 54 (2003), no. 2, 99--110. MR1995135

Yakubovich, S. B.; Britvina, L. E. Convolutions related to the Fourier and Kontorovich-Lebedev transforms revisited. Integral Transforms Spec. Funct. 21 (2010), no. 3-4, 259--276. doi: 10.1080/10652460903101919

Wimp, Jet. A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 33--40. doi: 10.1017/S0013091500011202