# Article

 Title: On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms (English) Author: Hong, Nguyen Thanh Author: Tuan, Trinh Author: Thao, Nguyen Xuan Language: English Journal: Applications of Mathematics ISSN: 0862-7940 Volume: 58 Issue: 4 Year: 2013 Pages: 473-486 Summary lang: English . Category: math . Summary: We deal with several classes of integral transformations of the form $$\label {generalformula} f(x)\rightarrow D\int _{\mathbb R_+^2} \frac 1u ({\rm e}^{-u\cosh (x+v)}+{\rm e}^{-u\cosh (x-v)}) h(u)f(v) {\rm d}u {\rm d} v,$$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p(\mathbb R_+)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2(\mathbb R_+)$ and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type. (English) Keyword: convolution Keyword: Hölder inequality Keyword: Young's theorem Keyword: Watson's theorem Keyword: unitary Keyword: Fourier cosine Keyword: Kontorovich-Lebedev Keyword: transform Keyword: integro-differential equation MSC: 33C10 MSC: 44A35 MSC: 45E10 MSC: 45J05 MSC: 47A30 MSC: 47B15 idZBL: Zbl 06221241 idMR: MR3083524 DOI: 10.1007/s10492-013-0023-5 . Date available: 2013-07-18T15:20:53Z Last updated: 2015-09-07 Stable URL: http://hdl.handle.net/10338.dmlcz/143341 . Reference: [1] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables.U.S. Department of Commerce Washington (1964). Zbl 0171.38503 Reference: [2] Adams, R. A., Fournier, J. J. F.: Sobolev Spaces, 2nd ed. Pure and Applied Mathematics 140.Academic Press New York (2003). MR 2424078 Reference: [3] Al-Musallam, F., Tuan, V. K.: Integral transforms related to a generalized convolution.Result. Math. 38 (2000), 197-208. Zbl 0970.44004, MR 1797712, 10.1007/BF03322007 Reference: [4] Britvina, L. E.: A class of integral transforms related to the Fourier cosine convolution.Integral Transforms Spec. Funct. 16 (2005), 379-389. Zbl 1085.42003, MR 2138055, 10.1080/10652460412331320395 Reference: [5] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Tables of Integral Transforms, Vol. I. Bateman Manuscript Project. California Institute of Technology.McGraw-Hill Book Co. New York (1954). MR 0061695 Reference: [6] Glaeske, H.-J., Prudnikov, A. P., Skórnik, K. A.: Operational Calculus and Related Topics. Analytical Methods and Special Functions 10.Chapman & Hall/CRC Boca Raton (2006). MR 2254107 Reference: [7] 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). MR 2662653 Reference: [8] Najmark, M. A.: Normed Algebras. Translated from the Second Russian Edition by Leo F. Boron. 3rd Completely Revised American Ed. Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics.Wolters-Noordhoff Publishing Groningen (1972). MR 0438123 Reference: [9] Prudnikov, A. P., Brychkov, Y. A., Marichev, O. I.: Integrals and Series Vol. 2: Special Functions. Transl. from the Russian by N. M. Queen.Gordon & Breach Science Publishers New York (1986). MR 0874987 Reference: [10] Sneddon, I. N.: Fourier Transforms.McGray-Hill Book Company New York (1950). Zbl 0038.26801, MR 0041963 Reference: [11] Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Third edition.Chelsea Publishing Co. New York (1986). MR 0942661 Reference: [12] Tuan, T.: On the generalized convolution with a weight function for the Fourier cosine and the inverse Kontorovich-Lebedev integral transformations.Nonlinear Funct. Anal. Appl. 12 (2007), 325-341. Zbl 1142.44007, MR 2391937 Reference: [13] Tuan, V. K.: Integral transforms of Fourier cosine convolution type.J. Math. Anal. Appl. 229 (1999), 519-529. Zbl 0920.46035, MR 1666432, 10.1006/jmaa.1998.6177 Reference: [14] Wimp, J.: A class of integral transforms.Proc. Edinb. Math. Soc., II. Ser. 14 (1964), 33-40. Zbl 0127.05701, MR 0164204, 10.1017/S0013091500011202 Reference: [15] Yakubovich, S. B.: Integral transforms of the Kontorovich-Lebedev convolution type.Collect. Math. 54 (2003), 99-110. Zbl 1067.44004, MR 1995135 Reference: [16] Yakubovich, S. B., Britvina, L. E.: Convolutions related to the Fourier and Kontorovich-Lebedev transforms revisited.Integral Transforms Spec. Funct. 21 (2010), 259-276. Zbl 1191.44002, MR 2604157, 10.1080/10652460903101919 .

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