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On the generalized convolution for F c , F s , and KL integral transforms

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Ukrainian Mathematical Journal Aims and scope

We study new generalized convolutions \( f\mathop{*}\limits^{\gamma } g \) with weight function γ(y) = y for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms in weighted function spaces with two parameters \( L\left( {{\mathbb{R}_{{ + }}},{x^{\alpha }}{e^{{ - \beta x}}}dx} \right) \). These generalized convolutions satisfy the factorization equalities

$$ {F_{{\left\{ {\begin{array}{*{20}{c}} s \\ c \\ \end{array} } \right\}}}}{\left( {f\mathop{*}\limits^{\gamma } g} \right)_{{\left\{ {\begin{array}{*{20}{c}} 1 \\ 2 \\ \end{array} } \right\}}}}(y) = y\left( {{F_{{\left\{ {\begin{array}{*{20}{c}} c \\ s \\ \end{array} } \right\}}}}f} \right)(y)\left( {{K_{{iy}}}g} \right)\;\;\;\forall y > 0. $$

We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels.

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References

  1. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  2. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York (1953).

    Google Scholar 

  3. H. Bateman and A. Erdélyi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, New York (1954).

    Google Scholar 

  4. J. J. Betancor and B. J. Gonzalez, “Spaces of L p -type and the Hankel convolution,” Proc. Amer. Math. Soc., 129, No. 1, 219–228 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. I. M. Ryzhik and I. S. Gradstein, Table of Integrals, Sums, Series and Products [in Russian], Moscow (1951).

  6. R. J. Marks II, I. A. Gravague, and J. M. Davis, “A generalized Fourier transform and convolution on time scales,” J. Math. Anal. Appl., 340, 901–919 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. J. Miana, “Convolutions, Fourier trigonometric transforms and applications,” Int. Transforms Special Funct., 16, No. 7, 583–585 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Garcia-Vicente, J. M. Delgado, and C. Peraza, “Experimental determination of the convolution kernel for the study of spatial response of a detector,” Med. Phys., 25, 202–207 (1998).

    Article  Google Scholar 

  9. F. Garcia-Vicente, J. M. Delgado, and C. Rodriguez, “Exact analytical solution of the convolution integral equation for a general profile fitting function and Gaussian detector kernel,” Phys. Med. Rick., (2000).

  10. H. H. Kagiwada and R. Kalaba, “Integral equations via imbedding methods,” Appl. Math. Comput., No. 6, 111–120 (1974).

  11. V. A. Kakichev, “On convolution for integral transforms,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 53–62 (1967).

  12. V. A. Kakichev and Nguyen Xuan Thao, “On the design method for the generalized integral convolution,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 1, 31–40 (1998).

  13. Nguyen Xuan Thao, V. A. Kakichev, and Vu Kim Tuan, “On the generalized convolution for Fourier cosine and sine transforms,” East-West J. Math., 1, 85–90 (1998).

    MathSciNet  MATH  Google Scholar 

  14. Nguyen Xuan Thao, Vu Kim Tuan, and Nguyen Minh Khoa, “On the generalized convolution with a weight function for the Fourier cosine and sine transforms,” Frac. Cal. Appl. Anal., 7, No. 3, 323 –337 (2004).

  15. M. G. Krein, “On a new method for solving linear integral equations of the first and second kinds,” Dokl. Akad. Nauk SSSR, 100, 413–416 (1955).

    MathSciNet  Google Scholar 

  16. I. N. Sneddon, Fourier Transform, McGraw-Hill, New York (1951).

    Google Scholar 

  17. I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York (1972).

    MATH  Google Scholar 

  18. Vu Kim Tuan, “Integral transforms of Fourier cosine convolution type,” J. Math. Anal. Appl., 229, No. 2, 519–529 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  19. J. N. Tsitsiklis and B. C. Levy, Integral Equations and Resolvents of Toeplitz Plus Hankel Kernels, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Ser./Rep. No. LIDS-P 1170 (1981).

  20. S. B. Yakubovich and L. E. Britvina, “Convolution related to the Fourier and Kontorovich–Lebedev transforms revisited,” Int. Transforms Special Funct., 21, No. 4, 259–276 (2010).

    Article  MathSciNet  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 1, pp. 81–91, January, 2012.

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Thao, N.X., Virchenko, N.O. On the generalized convolution for F c , F s , and KL integral transforms. Ukr Math J 64, 89–101 (2012). https://doi.org/10.1007/s11253-012-0631-3

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  • DOI: https://doi.org/10.1007/s11253-012-0631-3

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