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On a Class of Integral Equations Involving Kernels of Cosine and Sine Type

  • L. P. CastroEmail author
  • R. C. Guerra
  • N. M. Tuan
Chapter

Abstract

We consider a class of integral equations characterized by kernels of cosine and sine type and study their solvability. Moreover, we analyse the integral operator T, which is generating those equations, by identifying its characteristic polynomial, characterizing its invertibility, spectrum, Parseval-type identity and involution properties. Additionally, a new convolution is here introduced, associated with T, for which we deduce a corresponding factorization property.

Notes

Acknowledgements

This work was supported in part by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. The second named author was also supported by FCT through the Ph.D. scholarship PD/BD/114187/2016. The third named author was supported partially by the Viet Nam National Foundation for Science and Technology Developments (NAFOSTED).

References

  1. [AnEtAl16]
    Anh, P.K., Castro, L.P., Thao, P.T., Tuan, N.M.: Two new convolutions for the fractional Fourier transform. Wirel. Pers. Commun. 92, 623–637 (2017)CrossRefGoogle Scholar
  2. [Br86]
    Bracewell, R.N.: The Hartley Transform. Oxford University Press, Oxford (1986)zbMATHGoogle Scholar
  3. [CaSp00]
    Castro, L.P., Speck, F.-O.: Relations between convolution type operators on intervals and on the half-line. Integr. Equ. Oper. Theory 37, 169–207 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [CaZh05]
    Castro, L.P., Zhang, B.: Invertibility of convolution operators arising in problems of wave diffraction by a strip with reactance and Dirichlet conditions. Z. Anal. Anwend. 24, 545–560 (2005)CrossRefzbMATHGoogle Scholar
  5. [Ga90]
    Gakhov, F.D.: Boundary Value Problems. Dover, New York (1990)zbMATHGoogle Scholar
  6. [GiEtAl09]
    Giang, B.T., Mau, N.V., Tuan, N.M.: Operational properties of two integral transforms of Fourier type and their convolutions. Integr. Equ. Oper. Theory 65, 363–386 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Li00]
    Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  8. [OzEtAl01]
    Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)Google Scholar
  9. [PR88]
    Przeworska-Rolewicz, D.: Algebraic Analysis. PWN-Polish Scientific Publishers, Warsawa (1988)CrossRefzbMATHGoogle Scholar
  10. [PR01]
    Przeworska-Rolewicz, D.: Some open questions in algebraic analysis. In: Abe, J.M., Tanaka, S. (eds.) Unsolved Problems on Mathematics for the 21st Century, pp. 109–126, 1st edn. IOS Press, Amsterdam (2001)Google Scholar
  11. [Th98]
    Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  12. [Ti86]
    Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Chelsea, New York (1986)zbMATHGoogle Scholar
  13. [TuHu12a]
    Tuan, N.M., Huyen, N.T.T.: The Hermite functions related to infinite series of generalized convolutions and applications. Complex Anal. Oper. Theory 6, 219–236 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [TuHu12b]
    Tuan, N.M., Huyen, N.T.T.: Applications of generalized convolutions associated with the Fourier and Hartley transforms. J. Integr. Equ. Appl. 24, 111–130 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CIDMAUniversity of AveiroAveiroPortugal
  2. 2.Vietnam National UniversityHanoiVietnam

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