A review on Harmonic Wavelets and their fractional extension <s> Dilated and translated instances <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> BIB001 </s> A review on Harmonic Wavelets and their fractional extension <s> Dilated and translated instances <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> BIB002 </s> A review on Harmonic Wavelets and their fractional extension <s> Dilated and translated instances <s> An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for any function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grunwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence. <s> BIB003
In order to have a family of (harmonic) wavelet functions we have to dene the dilated (compressed) and translated instances of the fundamental functions (1), BIB002 , so that there will be a family of functions depending on the scaling parameter n and on the translation paremater k. From Eqs. (1), (13)  with n, k ∈ Z. For each function of the wavelet family Let us now compute the Fourier transform of the parameter depending instances BIB003 , by using the properties of the Fourier transform. It so that we can easily obtain the dilated and translated instances of the Fourier transform of BIB003 , (see e.g. BIB001 ): 3.