A Review on Harmonic Wavelets and Their Fractional Extension

In this paper a review on harmonic wavelets and their fractional generalization, within the local fractional calculus, will be discussed. The main properties of harmonic wavelets and fractional harmonic wavelets will be given, by taking into account of their characteristic features in the Fourier domain. It will be shown that the local fractional derivatives of fractional wavelets have a very simple expression thus opening new frontiers in the solution of fractional di erential problems.


Introduction
Harmonic wavelets are some kind of complex wavelets [19] 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.[13,10,13,15,16,25,32,33]. In the search for numerical approximation of dierential problems, the main idea is to approximate the unknown so-lution 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.[5,9,24]) 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).
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.
Indeed the idea of fractional order derivative is deeply rooted in the history of mathematics, since already Cauchy was wondering about the possible generalization of ordinary dierential operators to fractional order dierential operators.The main advantage of fractional order derivative is to have an additional parameter (the order of derivative) to be use in the analysis of dierential problems.On the other hand the main drawback for the fractional dierential operators is that this derivative is not univocally dened (see e.g.[1922] and references therein).
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 [12,31,36,37].
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 [17,18] and the sinc-derivative [19,20,22] 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.

Wavelets
Harmonic wavelets also known as Newland wavelets [1,3,5,7,8] 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.[24]), but they also enjoy some more special features especially in the function approximation.

Harmonic scaling function
The harmonic scaling function is dened as that is there follow the real and imaginary part of the scaling function ( Plots of real ϕ r (x) and imaginary part , ϕ i (x)} of the scaling function in the real plane are shown in Fig. 1.The parametric plot {ϕ r (x), ϕ i (x)} of the complex scaling function ϕ(x) is shown in Fig. 2.
It can be easily seen that it is, in particular, Fig. 2: Plot of the scaling function in the complex plane (0 ≤ x ≤ 4).
The complex conjugate of the function ϕ(x) is the function

Fractional prolungation of the scaling function
The scaling function (1) is the power series, with complex coecients, Let us slightly modify the harmonic scaling function by using the Mittag-Leer function, instead of the exponential.So that we have x αk Γ(αk + 1) . ( the Mittag-Leer function.
When α = 1, namely we have where δ(x) is the Dirac delta By a direct computation we have the fractional scaling function x k ,

Scaling function in Fourier domain
The Fourier transform of the scaling function ( 1) is dened as So that, in the frequency domain, i.e. with respect to the variable ω the Fourier transform is a function with a compact support (i.e. with a bounded frequency) χ(ω) being the characteristic function dened as The scaling function in Fourier domain is boxfunction thus being dened in a sharp domain with slow decay in frequency.
The Fourier transform of the fractional scaling function ( 9) can be also computed so that we have at the rst approximation 2.4.

Harmonic wavelet function
Theorem 1.The harmonic (Newland) wavelet function is dened as [3,4,7,8] and its Fourier transform is Proof: Starting from ϕ(x) we have to dene a lter and to derive the corresponding wavelet function (see e.g.[7]).From (10) we have so that, In order to have a multiresolution analysis [3,5,7,24] the wavelet function must be dened as (see e.g.[24]) where the bar stands for complex conjugation.
With the lter H ω while with H ω 2 + 2π we obtain from where there follows (14).
c 2018 Journal of Advanced Engineering and Computation (JAEC) By the inverse Fourier transform of ( 14) we get we get the harmonic wavelet (13).
The real and imaginary parts of ( 13) are: In particular, according to (3), ( 4), ( 13) it is The complex conjugate of the function ψ(x) is the function 2.5.

Fractional prolungation of the harmonic wavelet
From Eqs. ( 13), (8) we can dene the fractional prolungation of the harmonic wavelet as and its Fourier transform is 2.6.

Dilated and translated instances
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), (13), so that there will be a family of functions depending on the scaling parameter n and on the translation paremater k.
For each function of the wavelet family Let us now compute the Fourier transform of the parameter depending instances (19), 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 ( 19), (see e.g.[3]):

Multiscale harmonic wavelet reconstruction of functions
In this section we give the inner product space structure to the family of harmonic wavelets (19) and the harmonic wavelet reconstruction of functions.

Hilbert space structure
Let f (x), g(x) be given two complex functions, the inner (or scalar or dot) product, of these functions is where we have used the Parseval identity for the equivalent inner product in the Fourier domain.
With respect to the family of the fundamental functions (19), it can be shown that Theorem 3. Harmonic wavelets are orthonormal functions, such that where δ nm (δ hk ) is the Kronecker symbol.
Proof: It is (for an alternative proof see also [7]) Moreover, according to (11), by the change of For h = k (and n = m), trivially one has: and since, according to (3), the proof easily follows.
Analogously it can be easily shown that Moreover, the fundamental functions (1), (13) fullls the basic (even-odd) properties of scaling and wavelet, that is and the following Theorem 4. The harmonic scaling function and the harmonic wavelets fulll the conditions Proof: According to (10)-( 22) one has where δ(ω) is the Dirac delta function.

Wavelet reconstruction
Let f (x) ∈ B, where B is the space of complex functions, such that for any value of the parameters n, k, the following integrals, which dene the wavelet coecients, exist and have nite values According to (21), (22), these coecients can be equivalently computed in the Fourier domain, thus being where the hat stands for the Fourier transform.

Harmonic wavelet series
Let f (x) ∈ B be a complex funtion with nite wavelet coecients (26), (27).By taking into account the orthonormality of the basis functions (23), (25) the function f (x) can be expressed as a wavelet (convergent) series (see e.g.[7]).In fact, if we put the wavelet coecients can be easily computed by using the orthogonality of the basis and its conjugate.
In [7] (see also [24]) it was shown that, under suitable and quite general hypotheses on the function f (x), the wavelet series (28) converges to f (x).
The conjugate of the reconstruction (28) it is The wavelet approximation is obtained by xing an upper limit in the series expansion (28), Since wavelets are localized, they can capture with few terms the main features of functions dened in a short range interval.

1) Examples of Harmonic wavelet reconstruction
Let us give a couple of examples to show the powerful approximation obtained by the harmonic wavelets.
Let us rst consider the reconstruction of the Gaussian function: The truncated wavelet series with so that if we compute the wavelet coecients α 0 , α * 0 , β 0 0 , β * 0 0 by using the Eqs. (26) (or ( 27)) we get being the error function dened as erf (x) There follows the zero order approximation of the Gaussian and since For instance, the second scale approximation N = 2, M = 0 for the Gaussian function e −(16x) 2 is (see Fig. 3) As expected, by increasing the scaling parameter N we will get a better approximation.
and, according to (21), and for a real function So that the wavelet coecient can be obtained by the fast Fourier transform.In [7] 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 ( 9), (17) in general are not orthogonal as can be checked by a direct computation of their inner product.However, they can be expressed, by the wavelet coecients with respect to the harmonic wavelet basis.By taking into account the simple form of the Fourier transform of the fractional functions ( 12), ( 18) 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 wavelet 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.[22]) 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 [22].

Fourier domain
It is clear from (27) that the reconstruction of a function f (x) it is impossible when its Fourier transform f (ω) is not dened.Moreover, the function (to be reconstructed) must be con- x 0 = 0, by using the so-called periodized harmonic wavelets [1,7,8]).
Among all functions f (x) some of them are constant under harmonic wavelet map (28).In fact, we have that, Theorem 5.For a non trivial function f (x) = 0 the corresponding wavelet coecients (27), in general, vanish when either In particular, it can be seen that the wavelet coecients (27) trivially vanish when Proof: For instance from (26) 1 , for cos(2kπx) from where by the change of variable 2πx = ξ and taking into account (20) there follows and, because of (11) χ(2π + 2πh) = 1, 0 < h < 1 so that ϕ 0 k (2πh) = 0, ∀h = 0.There follows that α h = 0, as well as the remaining wavelet coecients of cos(2kπx) (with k ∈ Z and k = 0) are trivially vanishing.Analogously, it can be shown that all wavelet coecients of cos(2kπx) (∀k ∈ Z) are zero.
As a consequence, a given function f (x), for which the coecients (26) are dened, admits the same wavelet coecients of or (by a simple tranformation) in terms of complex exponentials, so that the wavelet coecients of f (x) are dened unless an additional trigonometric series (the coecients A h , B h , C h being constant) as in (38).

Local fractional calculus
In order to get some advantages from the denition of the fractional harmonic wavelets we give in this section the denition of the local fraction derivative, and then we apply this operator to the fractional wavelets ( 9), (17).By taking into account that wavelets are localized functions, we need to dene a suitable local dierential operator as the ones proposed by Yang [3639]:

Local fractional derivative
Denition 1.The local fractional derivative of f (x) of order α at x = x 0 is the operator There follows that For any x in a suitable interval centered in x 0 , we can dene the local fractional derivative where we have ∆u j = u j+1 − u j , ∆u = max {∆u 0 , ∆u 1 , ∆u 2 , • • • } and [u j , u j+1 ] , u 0 = a, u N = b, is a partition of the interval [a, b].
For any x ∈ (a, b), we can also dene the integral operator a I (α) x f (x),
(47) d α dx α sin α (x α ) = cos α (x α ).The knowledge of the local fractional derivative of the fractional harmonic wavelets can be a fundamental tool in the search for numerical solution of fractional dierential equations.

Conclusion
In this paper the main properties of the complex harmonic wavelets are given.Moreover the fractional harmonic wavelets were dened and their local fractional derivatives explicitly computed.These fractional harmonic wavelets are the fundamental functions to build a model for the solution of fractional dierential problems.

c 2018 Fig. 1 :
Fig. 1: Plot of the scaling function in the complex plane
centrated around the origin (like a pulse) and should rapidly decay to zero.The reconstruction can be done also for periodic functions, or c 2018 Journal of Advanced Engineering and Computation (JAEC) functions localized in a point dierent from zero: α cos α (x α ) = − sin α (x α ).