Abstract and Applied Analysis

Abstract and Applied Analysis / 2013 / Article
Special Issue

Scaling, Self-Similarity, and Systems of Fractional Order

View this Special Issue

Research Article | Open Access

Volume 2013 |Article ID 725416 | https://doi.org/10.1155/2013/725416

Xiao-Jun Yang, Dumitru Baleanu, H. M. Srivastava, J. A. Tenreiro Machado, "On Local Fractional Continuous Wavelet Transform", Abstract and Applied Analysis, vol. 2013, Article ID 725416, 5 pages, 2013. https://doi.org/10.1155/2013/725416

On Local Fractional Continuous Wavelet Transform

Academic Editor: Carlo Cattani
Received25 Oct 2013
Accepted10 Nov 2013
Published25 Nov 2013

Abstract

We introduce a new wavelet transform within the framework of the local fractional calculus. An illustrative example of local fractional wavelet transform is also presented.

1. Introduction

Wavelet transforms have been applied successfully in the areas of signals analysis, data compression, and sound processing (see, for details, [16] and the references cited therein). Although there is scaled and shifted versions of a mother wavelet, the daughter wavelets are structured as follows (see [35]): where is the dyadic dilation, is the dyadic position, and is the normalization factor. The expression of a one-dimensional wavelet transform for a given continuous signal is given by and the reconstruction formula becomes where

Recently, fractional wavelet transform, as a generalization of the classical wavelet transform, was proposed in [7]. The one-dimensional fractal wavelet transform of a continuous signal has the following form: where denotes a bulk optics kernel.

The reconstructing formula of the input is defined as given by the following expression:

We notice that the fractional wavelet transforms was applied to image encryption [8], to the simultaneous spectral analysis in [9], and to the composite signals in [10, 11]. For other definition of fractional wavelet transform, see [12] and the references cited therein.

Keeping in mind the study of the fractal signals (local fractional continuous signals), a new local fractional wavelet transform was developed in [13] based upon the local fractional Fourier transform [14] via local fractional calculus [1518]. In this paper, we investigate the local fractional Fourier transform to deal with the local fractional wavelet transforms by implementing the local fractional calculus.

The organization of the paper is as follows. Section 2 presents the concept of local fractional Fourier transform and wavelet. Section 3 discusses the derivation of the local fractional continuous wavelet transform. Section 4 studies the wave space and Section 5 present an illustrative example. Finally, Section 6 outlines the main conclusions of our present investigation.

2. Local Fractional Fourier Transform and Wavelet

Let be local fractional continuous function, which is denoted as follows (see [18]): The space of local fractional continuous functions , under , is given by (see [13]) where the operator is local fractional operator.

The space norm on is defined by for . This is infinite for and .

The local fractional Fourier transforms in fractal space is defined as follows (see [13, 14]): Its inverse is formulated as follows (see [13, 14]): Let and let When the function is called a local fractional wavelet [13].

Let . Then, we have so that where and .

3. Local Fractional Continuous Wavelet Transform

Let . Then, we arrive at the following relation: where , , and .

Similarly, we get

Taking in place of , we obtain In the special case when , we have the following relation: such that Hence, there exists the following relation: In general, we also deduce the following identities: Now, we establish the following relations: Hence, the local fractional continuous wavelet transform takes the following form (see [13]): And the inversion formula of local fractional continuous wavelet transform is derived as follows (see [14]): where

4. The Wavelet Space

In order to differ the classical wavelets from fractional wavelets, here we formulate a wavelet space as follows. In fact, a wavelet space is defined by When the fractal dimension is equal to 1, from (27), we deduce (see [35]) where is continuous and .

Taking the fractal dimension , we derive a formula given by with , where is a local fractional continuous function.

5. An Illustrative Example

In order to construct the local fractional continuous wavelet, we suppose that is times the local fractional differentiable function.

We define the local fractional wavelet by means of the following expression: where the differential operator is the local fractional operator proposed by Yang [18] (for other definition, see [19] and the references cited therein).

Then, we get Let us consider the following nondifferentiable signal, namely, For , we obtain For , we obtain In view of (33)-(34), we get a local fractional wavelet given by Following (35), we obtain In view of (15), taking and , we have for integers .

Hence, we get the following equation: We thus conclude that

6. Concluding Remarks and Observations

A novel local fractional wavelet transformation was investigated by using Fourier transform based upon local fractional calculus. This transform has been found to be advantageous in dealing with the functions in fractal space. The wave space is considered and an illustrative example is shown.

Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

References

  1. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Transactions on Information Theory, vol. 36, no. 5, pp. 961–1005, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. R. K. Martinet, J. Morlet, and A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Journal of Pattern Recognition and Artificial Intelligence, vol. 1, no. 2, pp. 273–302, 1987. View at: Publisher Site | Google Scholar
  3. C. K. Chui, An Introduction to Wavelets, Academic Press, San Diego, Calif, USA, 1992. View at: MathSciNet
  4. L. Debnath, Wavelet Transforms and Their Application, Birkhäuser, Boston, Mass, USA, 2002. View at: Publisher Site | MathSciNet
  5. C. Cattani, “Harmonic wavelet solution of Poisson's problem,” Balkan Journal of Geometry and its Applications, vol. 13, no. 1, pp. 27–37, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, World Scientific, 2007. View at: Publisher Site | MathSciNet
  7. D. Mendlovic, Z. Zalevsky, D. Mas, J. García, and C. Ferreira, “Fractional wavelet transform,” Applied Optics, vol. 36, no. 20, pp. 4801–4806, 1997. View at: Publisher Site | Google Scholar
  8. L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Optics Communications, vol. 254, no. 4–6, pp. 361–367, 2005. View at: Publisher Site | Google Scholar
  9. E. Dinç, F. Demirkaya, D. Baleanu, Y. Kadioǧlu, and E. Kadioǧlu, “New approach for simultaneous spectral analysis of a complex mixture using the fractional wavelet transform,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 812–818, 2010. View at: Publisher Site | Google Scholar
  10. E. Dinç and D. Baleanu, “Fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two-component mixture,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1701–1708, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. E. Dinç, D. Baleanu, and K. Taş, “Fractional wavelet analysis of the composite signals of two-component mixture by multivariate spectral calibration,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1283–1290, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. J. Shi, N. Zhang, and X. Liu, “A novel fractional wavelet transform and its applications,” Science China Information Sciences, vol. 55, no. 6, pp. 1270–1279, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher, Hong Kong, China, 2011.
  14. X. J. Yang, D. Baleanu, and J. T. A. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, vol. 2013, no. 1, article 131, 2013. View at: Google Scholar
  15. X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013. View at: Publisher Site | Google Scholar
  16. A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2013, Article ID 351057, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  17. W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,” Fixed Point Theory and Applications, vol. 2013, article 89, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  18. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  19. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006. View at: MathSciNet

Copyright © 2013 Xiao-Jun Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

905 Views | 675 Downloads | 9 Citations
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.