A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> Abstract A multilevel wavelet collection method for the solution of partial differential equations is developed. Two different approaches of treating general boundary conditions are suggested. Both are based on the wavelet interpolation technique developed in the present research. The first approach uses wavelets as a basis and results in a differential-algebraic system of equations, where the algebraic part arises from boundary conditions. The second approach utilizes extended wavelets, which satisfy boundary conditions exactly. This approach results in a system of coupled ordinary differential equations. The method is tested on the one-dimensional Burgers equation with small viscosity. The solutions are compared with those resulting from the use of other numerical algorithms. The present results indicate that the method is competitive with well-established numerical algorithms. <s> BIB001 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> 1. A small sample 2. General constructions 3. Some important wavelets 4. Compactly supported wavelets 5. Multivariable wavelets 6. Function spaces 7. Unconditional convergence 8. Wavelet bases in Lp and H1 9. Wavelets and smoothness of functions. <s> BIB002 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> A simply supported boundary value problem of biharmonic equations in the unit disk is reduced into an equivalent second kind natural boundary integral equation (NBIE) with hypersingular kernel (in the sense of Hadamard finite part). Trigonometric Hermite interpolatory wavelets introduced by Quak [Math. Comput. 65 (1996) 683-722] as trial functions are used to its Galerkin discretization with 2^J^+^1 degrees of freedom on the boundary. It is proved that the stiffness matrix is a block diagonal matrix and its diagonal elements are some symmetric and block circulant submatrices. The simple computational formulae of the entries in stiffness matrix are obtained. These show that we only need to compute 2(2^J+J+1) elements of a 2^J^+^2x2^J^+^2 stiffness matrix. The error estimates for the approximation solutions are established. Finally, numerical examples are given. <s> BIB003 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> Abstract The multiscale solution of the Klein‐Gordon equations in the linear theory of (two‐phase) materials with microstructure is defined by using a family of wavelets based on the harmonic wavelets. The connection coefficients are explicitly computed and characterized by a set of differential equations. Thus the propagation is considered as a superposition of wavelets at different scale of approximation, depending both on the physical parameters and on the connection coefficients of each scale. The coarse level concerns with the basic harmonic trend while the small details, arising at more refined levels, describe small oscillations around the harmonic zero‐scale approximation. <s> BIB004 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <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> BIB005 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of functions. The differential properties ::: of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the -functions. <s> BIB006 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <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> BIB007 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the MittagLeffler function. <s> BIB008 </s> A review on Harmonic Wavelets and their fractional extension <s> Introduction <s> In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator. <s> BIB009
Harmonic wavelets are some kind of complex wavelets BIB007 which are analitically dened, innitely dierentiable, and band-limited in the Fourier domain. Although the slow decay in the space domain, their sharp localization in frequency, is a good property especially for the analysis of wave evolution problems (see e.g. BIB005  BIB005 BIB004 16,  32,  . In the search for numerical approximation of dierential problems, the main idea is to approximate the unknown solution by some wavelet series and then by computing the integrals (or derivatives) of the basic wavelet functions, to convert the starting dierential problem into an algebraic system for the wavelet coecients (see e.g. [2630] ). Wavelets are some special functions (see e.g. [5, BIB002  ) which depend on two parameters, the scale parameter (also called renement, compression, or dilation parameter) and a the localization (translation) parameter. These functions fulll the fundamental axioms of multiresolution analysis so that by a suitable choice of the scale and translation parameter one is able to easily and quickly approximate (almost) all functions (even tabular) with decay to innity. Therefore wavelets seems to be the more expedient tool for studying dierential problems which are localized (in time or in frequency). There exists a very large literature devoted to wavelet solution of partial dierential and integral equations (see e.g. the pioneristic works  BIB005  BIB001 ) integral equations (see e.g.  BIB003  and more general integro-dierential equations and operators (see e.g. [2630] ). By using the derivatives (or integrals) of the wavelet basis the PDE equation can be transformed into an innite dimensional system of ordinary dierential equations. By xing the scale of approximation, the projection correspond to the choice of a nite set of wavelet spaces, thus obtaining the numerical (wavelet) approximation. By using the orthogonality of the wavelet basis and the computation of the inner product of the basis functions with their derivatives or integrals (operational matrix, also called connection coecients), we can convert the dierential problem into an algebraic system and thus we can easily derive the wavelet approximate solution. The approximation depends on the xed scale (of approximation) and on the number of dilated and translated instances of the wavelets. However, due to their localization property just a few instances are able to capture the main feature of the signal, and for this reason it is enough to compute a few number of wavelet coecients to quickly get a quite good approximation of the solution. In recent years there has been a fast rising interest for the fractional dierential problems. We will not go deeply into this subject, since we will focus only on a special fractional operator, the so-called local fractional derivative, as dened by Yang  BIB008   . The local fractional derivative when applied to the most popular functions give a natural generalization of known results and fullls the basica axioms of the fractional calculus. In the following after reviewing on the classical Harmonic wavelet, the fractional harmonic wavelets will be dened. Moreover their local fractional derivatives will be explicitly computed. It will be shown that these fractional derivatives, are some kind of generalization already obtained for the so called Shannon wavelets BIB006  and the sinc-derivative BIB007  BIB009 The paper is organized as follows: in section 2 some preliminary denitions about harmonic (complex wavelets) together with their fractional counterparts are given. The harmonic wavelet reconstruction of functions is described in section 3. In the same section, the harmonic wavelet representation of the fractional harmonic functions will be also given. Section 4 shows some characteristic features of harmonic wavelets. In section 5 the basic denitions and properties of local fractional derivatives are given and the local fractional derivatives of the fractional harmonic wavelets will be explicitly computed.