Mother wavelet functions generalized through q-exponentials

Abstract

We generalize some widely used mother wavelets by means of the q-exponential function $e_q^x\equiv [1+(1-q)x{]}^{1/(1-q)}$ ($q\in \mathbb{R}$, $e_1^x=e^x$) that emerges from nonextensive statistical mechanics. Particularly, we define extended versions of the mexican hat and the Morlet wavelets. We also introduce new wavelets that are $q$-generalizations of the trigonometric functions. All cases reduce to the usual ones as $q\to 1$. Within nonextensive statistical mechanics, departures from unity of the entropic index q are expected in the presence of long-range interactions, long-term memory, multi-fractal structures, among others. Consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q. We exemplify with the WTMM Method for mono- and multi-fractal self-affine signals.

LaTeX, 23 pages, 7 figures (13 eps files), new version to be published in J. Phys. A: Math. Gen

References in corpus (7)

• Interpretation of the nonextensitivity parameter $q$ in some applications of Tsallis statistics and Lévy distributions
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Thermodynamic Description of the Relaxation of Two-Dimensional Euler Turbulence Using Tsallis Statistics
• Fluxes of cosmic rays: A delicately balanced stationary state
• Statistical-thermodynamical foundations of anomalous diffusion
• Nonextensive statistical mechanics - Applications to nuclear and high energy physics
• q-Stirling's formula in Tsallis statistics

Cited by in corpus (5)

• Quasi-additivity of Tsallis entropies and correlated subsystems
• Probability distribution of (Schwämmle and Tsallis) two-parameter entropies and the Lambert W-function
• Fourier Analysis and q-Gaussian Functions: Analytical and Numerical Results
• Extensivity and nonextensivity of two-parameter entropies
• Generalized Exponential Function and some of its Applications to Complex Systems