A review on Harmonic Wavelets and their fractional extension <s> Scaling function 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> Scaling function in Fourier domain <s> In this paper, harmonic wavelets, which are analytically defined and band limited, are studied, together with their differentiable properties. Harmonic wavelets were recently applied to the solution of evolution problems and, more generally, to describe evolution operators. In order to consider the evolution of a solitary profile (and to focus on the localization property of wavelets), it seems to be more expedient to make use of functions with limited compact support (either in space or in frequency). The connection coefficients of harmonic wavelets are explicitly computed (in the following) at any order, and characterized by some recursive formulas. In particular, they are functionally and finitely defined by a simple formula for any order of the basis derivatives. <s> BIB002 </s> A review on Harmonic Wavelets and their fractional extension <s> Scaling function in Fourier domain <s> A new scheme for solving the Vlasov equation using a compactly supported wavelets basis is proposed. We use a numerical method which minimizes the numerical diffusion and conserves a reasonable time computing cost. So we introduce a representation in a compactly supported wavelet of the derivative operator. This method makes easy and simple the computation of the coefficients of the matrix representing the operator. This allows us to solve the two equations which result from the splitting technique of the main Vlasov equation. Some numerical results are exposed using different numbers of wavelets. <s> BIB003
The Fourier transform of the scaling function (1) is dened as So that, in the frequency domain, i.e. with respect to the variable ω the Fourier transform is a function with a compact support (i.e. with a bounded frequency) χ(ω) being the characteristic function dened as The scaling function in Fourier domain is boxfunction thus being dened in a sharp domain with slow decay in frequency. The Fourier transform of the fractional scaling function (9) can be also computed so that we have at the rst approximation 2.4. Harmonic wavelet function Theorem 1. The harmonic (Newland) wavelet function is dened as BIB002  BIB001 8] ψ(x) def = e 4πix − e 2πix 2πix = e 2πix ϕ(x) BIB003 and its Fourier transform is Proof: Starting from ϕ(x) we have to dene a lter and to derive the corresponding wavelet function (see e.g. BIB001 ). From (10) we have so that, In order to have a multiresolution analysis BIB002 5, BIB001  ] the wavelet function must be dened as (see e.g.  ) where the bar stands for complex conjugation. With the lter H ω 2 − 2π = χ(ω) we have while with H ω 2 + 2π we obtain from where there follows  . By the inverse Fourier transform of (14) we we get the harmonic wavelet BIB003 . The real and imaginary parts of (13) are: In particular, according to (3), (4), (13) it is The complex conjugate of the function ψ(x) is the function 2πix . 2.5. Fractional prolungation of the harmonic wavelet From Eqs. (13), (8) we can dene the fractional prolungation of the harmonic wavelet as and its Fourier transform is 2.6.