Article
MSC:
33C10,
44A35,
45E10,
45J05,
47A30,
47B15 | MR 3083524 | Zbl 06221241 | DOI: 10.1007/s10492-013-0023-5
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Keywords:
convolution; Hölder inequality; Young's theorem; Watson's theorem; unitary; Fourier cosine; Kontorovich-Lebedev; transform; integro-differential equation
convolution; Hölder inequality; Young's theorem; Watson's theorem; unitary; Fourier cosine; Kontorovich-Lebedev; transform; integro-differential equation
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.
References:
[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
[2] Adams, R. A., Fournier, J. J. F.: Sobolev Spaces, 2nd ed. Pure and Applied Mathematics 140. Academic Press New York (2003). MR 2424078
[3] Al-Musallam, F., Tuan, V. K.: Integral transforms related to a generalized convolution. Result. Math. 38 (2000), 197-208. DOI 10.1007/BF03322007 | MR 1797712 | Zbl 0970.44004
[4] Britvina, L. E.: A class of integral transforms related to the Fourier cosine convolution. Integral Transforms Spec. Funct. 16 (2005), 379-389. DOI 10.1080/10652460412331320395 | MR 2138055 | Zbl 1085.42003
[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
[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
[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
[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
[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
[10] Sneddon, I. N.: Fourier Transforms. McGray-Hill Book Company New York (1950). MR 0041963 | Zbl 0038.26801
[11] Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Third edition. Chelsea Publishing Co. New York (1986). MR 0942661
[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. MR 2391937 | Zbl 1142.44007
[13] Tuan, V. K.: Integral transforms of Fourier cosine convolution type. J. Math. Anal. Appl. 229 (1999), 519-529. DOI 10.1006/jmaa.1998.6177 | MR 1666432 | Zbl 0920.46035
[14] Wimp, J.: A class of integral transforms. Proc. Edinb. Math. Soc., II. Ser. 14 (1964), 33-40. DOI 10.1017/S0013091500011202 | MR 0164204 | Zbl 0127.05701
[15] Yakubovich, S. B.: Integral transforms of the Kontorovich-Lebedev convolution type. Collect. Math. 54 (2003), 99-110. MR 1995135 | Zbl 1067.44004
[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. DOI 10.1080/10652460903101919 | MR 2604157 | Zbl 1191.44002
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