A review on Harmonic Wavelets and their fractional extension <s> Wavelets <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> Wavelets <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
Harmonic wavelets also known as Newland wavelets  BIB002 5, BIB001 8] are complex orthonormal wavelets that are characterized by the sharply bounded frequency and slow decay in the space of variable. Like any other wavelet they depend both on the scale parameter n which dene the degree of renement, compression, or dilation and on a second parameter k which is related to the space localization. As we will see, harmonic wavelets fulll the fundamental axioms of multiresolution analysis (see e.g.  ), but they also enjoy some more special features especially in the function approximation.