We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms
in a two-parameter function space \( {L}_p^{\alpha, \beta}\left({\mathrm{\mathbb{R}}}_{+}\right) \). We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution. We also impose necessary and sufficient conditions on the kernel h to ensure that the generalized convolution transform
is a unitary operator in L2(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 Lp-norm.
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References
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edn., Academic Press, New York; Elsevier Sciences, Amsterdam (2003).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, 55, Washington, D.C. (1964).
F. Al-Musallam and V. K. Tuan, “Integral transforms related to a generalized convolution,” Results Math., 38, 197–208 (2000).
Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, and S. V. Meleshko, “Symmetries of integrodifferential equations with applications in mechanics and plasma physics,” Lect. Notes Phys., 806 (2010).
I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York (1951).
H. Bateman and A. Erdelyi, Table of Integral Transforms, Vol. 1, McGraw-Hill , New York, etc. (1954).
L. E. Britvina, “A class of integral transforms related to the Fourier cosine convolution,” Integral Transforms Spec. Funct., 16, No. 5-6, 379–389 (2005).
H. J. Glaeske, A. P. Prudnikov, and K. A. Skornik, Operational Calculus and Related Topics, Chapman & Hall/CRC, Boca Raton, etc. (2006).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions, Gordon & Breach, New York–London (1986).
N. X. Thao and N. T. Hong, “Fourier sine-cosine convolution inequalities and applications,” in: Proc. of the XIII Internat. Scientific Kravchuk Conference, Ukraine (2010), pp. 28–29.
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3rd edn., Chelsea Publ., New York (1986).
T. Tuan, N. X. Thao, and N. V. Mau, “On the generalized convolution for the Fourier sine and the Kontorovich–Lebedev transforms,” Acta Math. Vietnam., 35, No. 2, 303–317 (2010).
V. K. Tuan, “Integral transforms of Fourier cosine convolution type,” J. Math. Anal. Appl., 229, 519–529 (1999).
S. B. Yakubovich, Index Transforms, World Scientific, Singapore, etc. (1996).
S. B. Yakubovich, “Integral transforms of the Kontorovich–Lebedev convolution type,” Collect. Math., 54, No. 2, 99–110 (2003).
S. B. Yakubovich and L. E. Britvina, “Convolution related to the Fourier and Kontorovich–Lebedev transforms revisited,” Integral Transforms Spec. Funct., 21, No. 4, 259–276 (2010).
J. Wimp, “A class of integral transforms,” Proc. Edinburgh Math. Soc. (2), 14, No. 1, 33–40 (1964).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 267–279, February, 2020.
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Tuan, T. On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications. Ukr Math J 72, 302–316 (2020). https://doi.org/10.1007/s11253-020-01782-1
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DOI: https://doi.org/10.1007/s11253-020-01782-1