Logarithmic uncertainty principle, convolution theorem related to continuous fractional wavelet transform and its properties on a generalized Sobolev space

2017 ◽  
Vol 15 (05) ◽  
pp. 1750050 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Sobolev Space ◽  
Inversion Formula ◽  
Uncertainty Relation ◽  
Fractional Fourier Transform ◽  
Convolution Theorem ◽  
Inner Product ◽  
Fractional Wavelet Transform ◽  
Product Relation

The continuous fractional wavelet transform (CFrWT) is a nontrivial generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FrFT) domain. Firstly, the Riemann–Lebesgue lemma for the FrFT is derived, and secondly, the CFrWT in terms of the FrFT is introduced. Based on the CFrWT, a different proof of the inner product relation and the inversion formula of the CFrWT are provided. Thereafter, a logarithmic uncertainty relation for the CFrWT is investigated and the convolution theorem related to the CFrWT is established using the convolution of the FrFT. The CFrWT on a generalized Sobolev space is introduced and its important properties are presented.

scholarly journals Fractional Spectral Graph Wavelets and Their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Jiasong Wu ◽  
Fuzhi Wu ◽  
Qihan Yang ◽  
Yan Zhang ◽  
Xilin Liu ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Fractional Fourier Transform ◽  
Weighted Graphs ◽  
Extended Version ◽  
Spectral Graph ◽  
Potential Applications ◽  
Fractional Wavelet Transform ◽  
Fourier Series Approximation ◽  
Spectral Graph Wavelets

One of the key challenges in the area of signal processing on graphs is to design transforms and dictionary methods to identify and exploit structure in signals on weighted graphs. In this paper, we first generalize graph Fourier transform (GFT) to spectral graph fractional Fourier transform (SGFRFT), which is then used to define a novel transform named spectral graph fractional wavelet transform (SGFRWT), which is a generalized and extended version of spectral graph wavelet transform (SGWT). A fast algorithm for SGFRWT is also derived and implemented based on Fourier series approximation. Some potential applications of SGFRWT are also presented.

scholarly journals On the Multilinear Fractional Transforms

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 740
Author(s):  
Öznur Kulak
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Fractional Fourier Transform ◽  
Type Inequality ◽  
Young Inequality ◽  
Lebesgue Spaces ◽  
Schwartz Functions ◽  
Fractional Wavelet Transform ◽  
Fractional Transforms ◽  
Complex Valued

In this paper we first introduce multilinear fractional wavelet transform on Rn×R+n using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces.

The Continuous Fractional Wavelet Transform on W-Type Spaces

2018 ◽  
Vol 85 (3-4) ◽  
pp. 377
Author(s):  
Anuj Kumar ◽  
S. K. Upadhyay
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Fractional Fourier Transform ◽  
Euclidean Spaces ◽  
Continuous Maps ◽  
Type Spaces ◽  
Fractional Wavelet Transform

An n-dimensional continuous fractional wavelet transform involving <em>n</em>-dimensional fractional Fourier transform is studied and its properties are obtained on Gel'fand and Shilov spaces of type <em>W<sub>M</sub></em>(R<sup>n</sup>), <em>W</em><sup>Ω</sup> (C<sup>n</sup>) and W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup>). It is shown that continuous fractional wavelet transform, W<sup>α</sup><sub>ψ</sub>Φ : W<sub>M</sub>(R<sup>n</sup>) → W<sub>M</sub>(R<sup>n</sup> × R<sub>+</sub>), W<sup>α</sup><sub>ψ</sub>Φ : W<sup>Ω</sup> (C<sup>n</sup>) → W<sup>Ω</sup> (C<sup>n</sup> × R<sub>+</sub>) and W<sup>α</sup><sub>ψ</sub>Φ : W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup>) → W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup> × R<sub>+</sub>) are linear and continuous maps, where R<sup>n</sup> and C<sup>n</sup> are the usual Euclidean spaces.

A note on continuous fractional wavelet transform in ℝn

2021 ◽  
Author(s):  
Amit K. Verma ◽  
Bivek Gupta
Keyword(s):  
Wavelet Transform ◽  
Reproducing Kernel ◽  
Morrey Space ◽  
Inner Product ◽  
Dimensional Euclidean Space ◽  
Reconstruction Formula ◽  
Local Uncertainty ◽  
Fractional Wavelet Transform ◽  
Uncertainty Inequality ◽  
Product Relation

In this paper, we study the continuous fractional wavelet transform (CFrWT) in [Formula: see text]-dimensional Euclidean space [Formula: see text] with scaling parameter [Formula: see text] such that [Formula: see text]. We obtain inner product relation and reconstruction formula for the CFrWT depending on two wavelets along with the reproducing kernel function, involving two wavelets, for the image space of CFrWT. We obtain Heisenberg’s uncertainty inequality and Local uncertainty inequality for the CFrWT. Finally, we prove the boundedness of CFrWT on the Morrey space [Formula: see text] and estimate [Formula: see text]-distance of the CFrWT of two argument functions with respect to different wavelets.

Image encryption based on Fractional wavelet transform, Arnold transform with the double random phases in HSV color domain

2020 ◽  
Vol 13 ◽  
Author(s):  
Aarushi Shrivastava ◽  
Janki Ballabh Sharma ◽  
Sunil Dutt Purohit
Keyword(s):  
Wavelet Transform ◽  
Image Encryption ◽  
Fractional Fourier Transform ◽  
Color Image ◽  
Random Phase ◽  
Arnold Transform ◽  
Time Frequency ◽  
Key Space ◽  
Secret Keys ◽  
Fractional Wavelet Transform

Objective: In the recent multimedia technology images play an integral role in communication. Here in this paper, we propose a new color image encryption method using FWT (Fractional Wavelet transform), double random phases and Arnold transform in HSV color domain. Methods: Firstly the image is changed into the HSV domain and the encoding is done using the FWT which is the combination of the fractional Fourier transform with wavelet transform and the two random phase masks are used in the double random phase encoding. In this one inverse DWT is taken at the end in order to obtain the encrypted image. To scramble the matrices the Arnold transform is used with different iterative values. The fractional order of FRFT, the wavelet family and the iterative numbers of Arnold transform are used as various secret keys in order to enhance the level of security of the proposed method. Results: The performance of the scheme is analyzed through its PSNR and SSIM values, key space, entropy, statistical analysis which demonstrates its effectiveness and feasibility of the proposed technique. Stimulation result verifies its robustness in comparison to nearby schemes. Conclusion: This method develops the better security, enlarged and sensitive key space with improved PSNR and SSIM. FWT reflecting time frequency information adds on to its flexibility with additional variables and making it more suitable for secure transmission.

scholarly journals Generalized convolution theorem associated with fractional Fourier transform

2012 ◽  
Vol 14 (13) ◽  
pp. 1340-1351 ◽  
Author(s):  
Jun Shi ◽  
Xuejun Sha ◽  
Xiaocheng Song ◽  
Naitong Zhang
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Convolution Theorem ◽  
Generalized Convolution

Product of continuous fractional wave packet transforms

2019 ◽  
Vol 17 (04) ◽  
pp. 1950021
Author(s):  
Akhilesh Prasad ◽  
Praveen Kumar ◽  
Tanuj Kumar
Keyword(s):  
Fourier Transform ◽  
Wave Packet ◽  
Inversion Formula ◽  
Fractional Fourier Transform

In this paper, we investigated the fractional Fourier transform (FrFT) of the continuous fractional wave packet transform and studied some properties of continuous fractional wave packet transform. The product of continuous fractional wave packet transforms (CFrWPTs) is defined. Parseval’s relation and an inversion formula for product of CFrWPT are obtained. An example is also given.

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