Tbilisi Mathematical Journal

Marvels of fractional calculus

P. K. Banerji and Deshna Loonker

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Abstract

This is an expository article that describes, in brief, one of the preeminent branch of applicable mathematics, roots of which lie in the nucleus of pure mathematics that ruled the research since past six decades. In writing this article though several important research papers were excised yet attempt is made to retain the beauty of fractional calculus. This article, accommodates Stieltjes transform and fractional integral operator on spaces of generalized functions, distributional Laplace-Hankel transform by fractional integral operators, and wavelet transform of fractional integrals for the integral Boehmians.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 1 (2017), 295-314.

Dates
Received: 2 July 2016
Accepted: 21 November 2016
First available in Project Euclid: 26 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1527300032

Digital Object Identifier
doi:10.1515/tmj-2017-0019

Mathematical Reviews number (MathSciNet)
MR3646818

Zentralblatt MATH identifier
1364.26008

Subjects
Primary: 26A33: Fractional derivatives and integrals
Secondary: 44A10: Laplace transform 44A15: Special transforms (Legendre, Hilbert, etc.) 46F10: Operations with distributions 46F99: None of the above, but in this section 42C40: Wavelets and other special systems 65T60: Wavelets

Keywords
Fractional calculus Laplace-Hankel transform Stieltjes transform Wavelet transform distribution spaces Boehmians

Citation

Banerji, P. K.; Loonker, Deshna. Marvels of fractional calculus. Tbilisi Math. J. 10 (2017), no. 1, 295--314. doi:10.1515/tmj-2017-0019. https://projecteuclid.org/euclid.tbilisi/1527300032


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References

  • Banerji, P. K. and Gehlot, Kuldeep Singh, Products of Stieltjes transform and fractional integrals on spaces of generalized functions, Bull Cal. Math. Soc. 88 (1996) 375 - 384.
  • Banerji, P. K., Loonker, Deshna and Debnath, Lokenath. Wavelet transform for integrable Boehmains, J. Math. Anal. Appl. 296 (2) (2004), 473-478.
  • Bochner, S. Vorlesungen über Fouriesche Integrale, Leipzig (1932).
  • Boehme, T. K. The support of Mikusiński Operators, Trans. Amer. Math. Soc. 176 (1973) 319-334.
  • Chui, C. K. and Wang, J. Z. On compactly supported spline wavelets and a duality principle, Proc. Amer. Math. Soc. 330 (1992), 903 - 915.
  • Chui, C. K. and Wang, J. Z. A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785 - 793.
  • Coifman, R. R., Meyer, Y., Wicherhauser, M. V. Wavelet analysis and signal processing in Wavelets and their applications (Ed. Ruskai et al.), Jones and Barttlett, Boston, (1992a), 153 - 178.
  • Coifman, R. R., Meyer, Y., Wicherhauser, M. V. Size properties of wavelet packets in Wavelets and their applications (Ed. Ruskai et al.), Jones and Barttlett, Boston (1992b), 453 -470.
  • Daubechies, I. Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math. 41 (1998), 909-996.
  • Daubechies, I., Grossman A. and Meyer, Y. Painless non - orthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.
  • Debnath, L. Wavelet Transform and their Applications, PINSA-A 64 (A) 6 (1998), 685-713.
  • Erdélyi, A. Mangnus, W., Oberhettinger, F. and Tricomi, F. G. Table of Integral Transforms, Vol. 2, Mc-Graw Hill Book Co., Inc., New York London (1984)
  • Gabor, D. Theory of communication, J. Inst. Electr. Engg. London, 93 (1946), 429 - 457.
  • Glaeske, H. J. and Saigo,M., Products of Laplace transform and fractional integrals on spaces of generalized functions, Math. Japon. 37 (2) (1992), 373 - 382
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Boston, London, (2006)
  • Lemarie, P. G. and Meyer, Y. Ondelettes á localisation exponentielle, Revista Mat. Iberoamer, 2 (1986), 1-18.
  • Loonker, Deshna and Banerji, P. K. Distributional Laplace - Hankel transform by Fractional integral operators, The Mathematics Student, 73 (1-4) (2004), 193 -197
  • Loonker, Deshna and Banerji, P. K. On the Laplace and the Fourier transformations of Fractional Integrals for Integrable Boehmians, Bull. Cal. Math. Soc., 99 (4) (2007), 345-354.
  • Loonker, Deshna, Banerji, P. K. and Kalla, S. L., Wavelet Transform of Fractional Integrals for Integrable Boehmians, Applications and Applied Mathematics 5 (1) (2010), 1-10.
  • Mallat, S. Multriesolution representation of Wavelets. Ph. D. Thesis, Univ. of Pennsylvania. (1988)
  • Mallat, S. Multiresolution approximations and wavelet orthonormal bases of, Trans. Amer. Math. Soc. 315 (1989), 69 -87.
  • Malvar, H.S. Lapped transforms for coefficients transforms/subband coding, IEEE Trans. Acoust. Speech Signal Processing 38 (1990a), 969 – 978.
  • Malvar, H. S. The design of two dimensional filters by transformations, Seventh Annual Princeton University Press, Princeton, New Jersey, 247 (1990b), 251 - 264.
  • McBride, A. C. Fractional Calculus and Integral Transforms of Generalized Functions, Research Notes in Mathematics, Vol. 31, Pitman Publishing Program, London, 1979.
  • McBride, A. C. and Roach, G. F. (Eds.) Fractional Calculus, Proceedings of the Second International Conference on Fractional Calculus, Research Notes in Mathematics, Vol. 138, Pitman Adavanced Publishing Program, London, 1985.
  • Meyer, Y. Wavelets and Operators, Cambridge University Press, Cambridge (1992)
  • Mikusiński, J. and Mikusinski, P. Quotients de suites et lecurs applications dans l' analyse fonctionnelle, C. R. Acad. Sc. Paris, 293, Series I, (1981), 463-464.
  • Mikusiński, P. Convergence of Boehmians, Japan. J. Math. 9 (1) (1983), 159-179.
  • Mikusiński, P. Boehmians and generalized functions, Acta Math. Hung., 51 (1988), 271-281.
  • Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Frational Differential Equations, John Wiley & Sons, Inc., New York- Toronto (1993).
  • Morlet, J. Arens, G., Fourgeau, E. and Giard, J. Wave propagation and sampling theory, Part I: Complex signal land scattering in multilayer media, J. Geophys. 47 (1982a), 203 -221.
  • Morlet, J. Arens, G., Fourgeau, E. and Giard, J. Wave propagation and sampling theory, Part II: Sampling theory and complex waves, J. Geophys. 47 (1982b), 222-236.
  • Nishimoto, K. (Ed.) Fractional Calculus and its Applications, Proceedings of the Third International Conference on Functional Calculus, Nihon Univ. Descartes Press, Koriyama, Japan, 1990.
  • Ross, B. (Ed.) Fractional Calculus and its Applications, Proceedings of the First International Conference on Fractional Calculus, Lecture Notes in Mathematics, Vol. 457, Springer Verlag, New York, 1975.
  • Saigo, M. A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. College General Ed., Kyushu Univ. 11 (1978), 135 - 143.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I. Fractional Integrals and Derivatives : Theory and Applications, Gordon & Breach Science Publishers, Australia, Canada (1993).
  • Schwartz, L. Théorie des Distributions, 2 Vols., Hermann, Paris (1950 - 1951).