A review on Harmonic Wavelets and their fractional extension <s> 2) Computation of the wavelet coecients in the 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> 2) Computation of the wavelet coecients in the Fourier domain <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> 2) Computation of the wavelet coecients in the Fourier domain <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> BIB003
According to  the wavelet coecient are obtained by Fourier transform.  and, according to  , n and for a real function So that the wavelet coecient can be obtained by the fast Fourier transform. In BIB001 it was given a simple algorithm for the computation of these coecients through the fast Fourier transform. 3) Harmonic wavelet coecients of the fractional harmonic scaling and wavelet The fractional harmonic scaling and wavelet functions BIB002 , (17)  we have for the scaling function ϕ α (x) Analogously for the fractional wavelet ψ α (x) So that according to (28) we get the fractional scaling as a wavelet series  and analogously for the fractional harmonic By taking into account Eqs. (1), (5), the basic functions on the right hand side can be simplied thus giving so that the fractional scaling is closely related to the sinc-fractional operator (see e.g. BIB003 ) and for the fractional wavelet, from (13), (16), analogously we get Also the fractional wavelet is closely related to the Shannon wavelet and the sinc-fractional wavelets BIB003 .