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Ukrainian Mathematical Journal

, Volume 64, Issue 1, pp 89–101 | Cite as

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

  • N. X. Thao
  • N. O. Virchenko
Article
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.

Keywords

Integral Equation Weight Function Factorization Identity Convolution Operator Factorization Equality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
  2. 2.
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York (1953).Google Scholar
  3. 3.
    H. Bateman and A. Erdélyi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, New York (1954).Google Scholar
  4. 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).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    I. M. Ryzhik and I. S. Gradstein, Table of Integrals, Sums, Series and Products [in Russian], Moscow (1951).Google Scholar
  6. 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).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    P. J. Miana, “Convolutions, Fourier trigonometric transforms and applications,” Int. Transforms Special Funct., 16, No. 7, 583–585 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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).CrossRefGoogle Scholar
  9. 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).Google Scholar
  10. 10.
    H. H. Kagiwada and R. Kalaba, “Integral equations via imbedding methods,” Appl. Math. Comput., No. 6, 111–120 (1974).Google Scholar
  11. 11.
    V. A. Kakichev, “On convolution for integral transforms,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 53–62 (1967).Google Scholar
  12. 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).Google Scholar
  13. 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).MathSciNetzbMATHGoogle Scholar
  14. 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).Google Scholar
  15. 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).MathSciNetGoogle Scholar
  16. 16.
    I. N. Sneddon, Fourier Transform, McGraw-Hill, New York (1951).Google Scholar
  17. 17.
    I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York (1972).zbMATHGoogle Scholar
  18. 18.
    Vu Kim Tuan, “Integral transforms of Fourier cosine convolution type,” J. Math. Anal. Appl., 229, No. 2, 519–529 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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).Google Scholar
  20. 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).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • N. X. Thao
    • 1
  • N. O. Virchenko
    • 2
  1. 1.HanoiVietnam
  2. 2.KievUkraine

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