The Journal of the Indian Mathematical Society

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The Continuous Fractional Wavelet Transform on W-Type Spaces


Affiliations
1 Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi - 221005, India
2 DST-CIMS, Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi - 221005, India
     

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An n-dimensional continuous fractional wavelet transform involving n-dimensional fractional Fourier transform is studied and its properties are obtained on Gel'fand and Shilov spaces of type WM(Rn), WΩ (Cn) and WΩM (Cn). It is shown that continuous fractional wavelet transform, WαψΦ : WM(Rn) → WM(Rn × R+), WαψΦ : WΩ (Cn) → WΩ (Cn × R+) and WαψΦ : WΩM (Cn) → WΩM (Cn × R+) are linear and continuous maps, where Rn and Cn are the usual Euclidean spaces.

Keywords

Fractional Fourier Transform, Fractional Wavelet Transform, Convex Functions, Gel'fand and Shilov Spaces.
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  • A. Friedman, Generalized Functions and Partial Differential Equations Prentice Hall, Englewood Cliffs, N.J., (1963).

  • B.L. Gurevich, New Types of Test Function spaces and spaces of Generalized functions and the Cauchy problem for Operator Equations (in Russian), Dissertation, Kharkov, (1956).

  • I.M. Gel'fand, G.E. Shilov, Generalized functions Academic Press, New York, Vol.3, (1967).

  • A. Prasad, S. Manna, A Mahato, V.K. Singh, The generalized continuous wavelet transform associated with fractional Fourier transform , Journal of Computational and Applied Mathematics, 259 (2014), 660-671 .

  • A. Prasad, A. Mahato, The fractional wavelet transform on spaces of type W, Integral Transforms and Special Functions, 24 (2013), 239-250.

  • R.S. Pathak, The wavelet transform of distributions, Tohoku Math, 56 (2004), 411-421.

  • R.S. Pathak, G. Pandey, Wavelet transform on spaces of type W, Rocky Mountain Journal of Mathematics, 39 (2009), 619-631.

  • J. Shi , N. Zhang, X. Liu, A novel fractional wavelet transform and its applications, Sci. China inf. Sci. 55 (2012), 1270-1279.

  • S.K. Upadhyay, R.N. Yadav, L. Debnath, The n-dimentional continuous wavelet transformation on Gel'fand and Shilov type spaces, Surveys in mathematics and its applications, 4 (2009), 239-252.


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  • The Continuous Fractional Wavelet Transform on W-Type Spaces

Abstract Views: 218  |  PDF Views: 2

Authors

Anuj Kumar
Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi - 221005, India
S. K. Upadhyay
DST-CIMS, Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi - 221005, India

Abstract


An n-dimensional continuous fractional wavelet transform involving n-dimensional fractional Fourier transform is studied and its properties are obtained on Gel'fand and Shilov spaces of type WM(Rn), WΩ (Cn) and WΩM (Cn). It is shown that continuous fractional wavelet transform, WαψΦ : WM(Rn) → WM(Rn × R+), WαψΦ : WΩ (Cn) → WΩ (Cn × R+) and WαψΦ : WΩM (Cn) → WΩM (Cn × R+) are linear and continuous maps, where Rn and Cn are the usual Euclidean spaces.

Keywords


Fractional Fourier Transform, Fractional Wavelet Transform, Convex Functions, Gel'fand and Shilov Spaces.

References