You are presented with multiple paper abstracts, alongside the associated literature review and chapter titles. Given the provided information, you are going to do a literature review. Please generate a coherent literature review that synthesizes the content, draws connections between different works, highlights major findings, and identifies any gaps in the existing research.

Literature Review Title: A review on Harmonic Wavelets and their fractional extension 
Chapter Title:  Dilated and translated instances 
1. Abstract of Cited Paper (BIB001):  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. 
2. Abstract of Cited Paper (BIB002):  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.

Literature Review:
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.