A review on Harmonic Wavelets and their fractional extension <s> Some properties of the Harmonic wavelets in Fourier domain <s> A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform. <s> BIB001 </s> A review on Harmonic Wavelets and their fractional extension <s> Some properties of the Harmonic wavelets in Fourier domain <s> Abstract A non-differentiable model of the LC-electric circuit described by a local fractional differential equation of fractal dimensional order is addressed in this article. From the fractal electrodynamics point of view, the relaxation oscillator, defined on Cantor sets in LC-electric circuit, and its exact solution using the local fractional Laplace transform are obtained. Comparative results among local fractional derivative, Riemann–Liouville fractional derivative and conventional derivative are discussed. Local fractional calculus is proposed as a new tool suitable for the study of a large class of electric circuits. <s> BIB002
It is clear from  that the reconstruction of a function f (x) it is impossible when its Fourier transform f (ω) is not dened. Moreover, the function (to be reconstructed) must be concentrated around the origin (like a pulse) and should rapidly decay to zero. The reconstruction can be done also for periodic functions, or x 0 = 0, by using the so-called periodized harmonic wavelets  BIB001 8] ). Among all functions f (x) some of them are constant under harmonic wavelet map (28) . In fact, we have that, Theorem 5. For a non trivial function f (x) = 0 the corresponding wavelet coecients  , in general, vanish when either In particular, it can be seen that the wavelet coecients  trivially vanish when  Proof: For instance from (26) 1 , for cos (2kπx) it is from where by the change of variable 2πx = ξ and taking into account  there follows According to   and, because of (11) χ(2π + 2πh) = 1, 0 < h < 1 so that ϕ 0 k (2πh) = 0, ∀h = 0. There follows that α h = 0, as well as the remaining wavelet coecients of cos(2kπx) (with k ∈ Z and k = 0) are trivially vanishing. Analogously, it can be shown that all wavelet coecients of cos(2kπx) (∀k ∈ Z) are zero. As a consequence, a given function f (x), for which the coecients (26) are dened, admits the same wavelet coecients of BIB002 or (by a simple tranformation) in terms of complex exponentials, so that the wavelet coecients of f (x) are dened unless an additional trigonometric series (the coecients A h , B h , C h being constant) as in (38).