A family of pseudo-differential operators on the Schwartz space associated with the fractional Fourier transform

In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically.

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“…We now recall the following facts from previous studies (see also other studies), which are useful in our present investigation.…”

Section: Introductionmentioning

confidence: 76%

“…We apply the theory of fractional Fourier transform of order α for

0 < α ≦ 1

, and α is taken such that

\frac{1}{α}

is always an integer. This theory of the fractional wavelet transform and its properties were studied recently in the study of Upadhyay and Khatterwani (see also other studies).…”

Section: Fractional Wavelet Transformmentioning

confidence: 91%

A certain family of fractional wavelet transformations

Srivástava

Khatterwani

Upadhyay

2019

Math Methods in App Sciences

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“…In order to justify the inversion of the order of integration with respect to a and t, we first perform the integration in the region [(a, t) : |a| > , a, t ∈ R n ], invert the order of integration and then let → 0. This existence of the triple integral in terms of b, a and t in Equation (14) is proved by using the Plancherel theorem with respect to the variable b. Thus, by using…”

Section: Lemmamentioning

confidence: 92%

Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

et al. 2019

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In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

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“…Luchko et al ( [15]) gave a novel definition of FrFT and the corresponding fractional wavelet transform have been studied in [25], [27]. For more information on the fractional Fourier transform introduced in [15], we refer the reader to [13], [26].…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Fractional Wavelet Transforms and Uncertainty Principles

Kaur¹,

Gupta²,

Verma³

2022

Preprint

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In this paper, we have given a new definition of continuous fractional wavelet transform in R N, namely the multidimensional fractional wavelet transform (MFrWT) and studied some of the basic properties along with the inner product relation and the reconstruction formula. We have also shown that the range of the proposed transform is a reproducing kernel Hilbert space and obtain the associated kernel. We have obtained the uncertainty principle like Heisenberg's uncertainty principle, logarithmic uncertainty principle and local uncertainty principle of the multidimensional fractional Fourier transform (MFrFT). Based on these uncertainty principles of the MFrFT we have obtained the corresponding uncertainty principles i.e., Heisenberg's, logarithmic and local uncertainty principles for the proposed MFrWT.

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A family of pseudo-differential operators on the Schwartz space associated with the fractional Fourier transform

Cited by 17 publications

References 8 publications

A certain family of fractional wavelet transformations

A certain family of fractional wavelet transformations

Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

Multidimensional Fractional Wavelet Transforms and Uncertainty Principles

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