#### Abstract

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 Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.

#### 1. Introduction

Shannon wavelet theory [1, 2] is based on a family of orthogonal functions having many interesting properties. They enjoy the many advantages of wavelets [3, 4]; moreover, being analytical functions they are infinitely differentiable. Thus, enabling us to define the so-called connection coefficients  for any order derivative. Connection coefficients are an expedient tool for the projection of differential operators, useful for computing the wavelet solution of integrodifferential equations .

Wavelets are localized functions, in time and/or frequency, which are the basis for energy-bounded functions and in particular for -functions. So that localized pulse problems [14, 15] can be easily approached and analyzed. Moreover, wavelet allows the multiscale decomposition of problems, thus emphasizing the contribution of each scale. By defining a suitable inner product on the orthogonal family of scaling/wavelet functions, any -function can be approximated at a fixed scale, by a truncated series having, as basis, the scaling functions and the wavelet functions. The wavelet coefficients of these series represent the contribution of each scale.

Shannon wavelets are related to the harmonic wavelets [3, 5, 8], being the real part thereof, and to the well-known sinc function, which is the basic function in signal analysis. It should be also noticed that, as compared with other wavelet families, the main advantage of Shannon wavelets is that they are analytical functions, thus being infinitely differentiable. Moreover, they are sharply bounded in the frequency domain, so that, by taking into account the Parseval identity, any computation can be easily performed by their Fourier transforms.

The theory of connection coefficients was initially given [10, 13] for the compactly supported wavelet families, such as the Daubechies wavelets . The computation of these connection coefficients was based on the recursive equations of the wavelet theory and the explicit forms of these coefficients were given only up to the second order derivatives. The connection coefficients are the wavelet coefficients of the derivatives of the wavelet basis. These coefficients are a fundamental tool for the approximation of differential operators, with respect to the wavelet basis.

In some recent papers, the connection coefficients for Shannon wavelets have been explicitly computed up to any order derivative with a finite analytical form. This is due to the analytical form of Shannon wavelets and the discovery by Cattani of a suitable series expansion for the connection coefficients [2, 6, 7].

In the following, we will define the wavelet representation of fractional derivative, so that the fractional derivative of an -function can be easily computed by knowing the connection coefficients. The fractional derivatives of the Shannon scaling/wavelet basis are defined and the error of the approximation will be explicitly computed. Moreover, a comparison with the classical definition of Grünwald formula [16, 17] is given, by showing the major performance of wavelets, in terms of rate of convergence.

In particular, Section 2 gives some preliminary remarks, definitions, and properties about Shannon wavelets. Their corresponding connection coefficients are discussed in Section 3. This Section deals with some properties of connection coefficients, functional equalities, and error of approximation. Fractional derivatives of the Shannon scaling function and wavelets are given in Section 4. In this section, it is also shown that the fractional derivative is a semigroup. The error of the approximation is explicitly computed and compared with classical definitions of the fractional derivative, and in particular with the Grünwald formula.

#### 2. Preliminary Remarks

In this section, some remarks on Shannon wavelets and connection coefficients are given (see also ).

Shannon wavelet theory (see e.g. [1, 2, 6, 7, 9]) is based on the scaling function , also known as sinc function, and the wavelet function , respectively, defined as The corresponding families of translated and dilated instances wavelet [1, 2, 6, 7, 9], on which is based the multiscale analysis , are being, in particular, Let be the Fourier transform of the function , and its inverse transform, respectively. The Fourier transform of (2.1) give us  with

Analogously for the dilated and translated instances of scaling/wavelet function, in the frequency domain, it is

Both families of Shannon scaling and wavelet are -functions therefore, for each and , the inner product is defined as where the bar stands for the complex conjugate.

Shannon wavelets fulfill the following orthogonality properties (for the proof see e.g., [2, 7]): being the Kronecker symbols.

##### 2.1. Properties of the Shannon Wavelet

According to (2.2), Shannon wavelets can be easily computed at some special points, being in particular so that It is also 

In the following, we will be interested on the maximum values of these functions which can be easily computed. The maximum value of the scaling function can be found at the integers and the max values of are

Both families of scaling and wavelet functions belong to , thus having a bounded range and (slow) decay to zero

Let the set of functions in such that the integrals exist with finite values, then it can be shown [24, 7] that the series converges to .

According to (2.8), the coefficients can be also computed in the Fourier domain  so that In the frequency domain, (2.17) gives  When the upper bound for the series of (2.17) is finite, then we have the approximation The error of the approximation has been estimated in .

##### 2.2. Reconstruction of the Derivatives

In order to represent the differential operators in wavelet bases, we have to compute the wavelet decomposition of the derivatives. It can be shown [2, 7] that the derivatives of the Shannon wavelets are orthogonal functions: being the connection coefficients [2, 5, 6, 813].

The computation of connection coefficients can be easily performed in the Fourier domain, thanks to the equality (2.8)

In fact, in the Fourier domain, the -order derivative of the (scaling) wavelet functions are simply and, according to (2.7),

It has been shown [2, 6, 7] that the any order connection coefficients (2.22)1 of the Shannon scaling functions are or, by defining shortly as, when , and for ,

For the proof see .

Analogously for the connection coefficients (2.22)2 we have that the any order connection coefficients of the Shannon scaling wavelets are or, shortly for , and , respectively.

For the proof see .

#### 3. Remarks on Connection Coefficients

##### 3.1. Recursiveness

The connection coefficients fulfill some recursive formula as follows.

Theorem 3.1. The connection coefficients (2.26) are recursively given by

Proof. Let us show first when . From the definition (2.26), it is from where (3.1)2 follows. Analogously with simple computation we obtain (3.1)1.

Shorty and with some caution, (3.1) can be written as that is,

It is not so easy to find out a similar property also for the -coefficients as a function of however, there is a simple rule for the recursiveness of the scale (upper) indexes, as follows.

Theorem 3.2. The connection coefficients (2.30) are recursively given by the matrix at the lowest scale level:

Proof. As can be seen from (2.30) parameter appears only in the factor so that (3.5) follows from the identity

Moreover, it can be shown also that

##### 3.2. Taylor Series

By using the connection coefficients, it is easy to show the following theorem.

Theorem 3.3. If and the Taylor series of in is being and given by (2.16), (2.18) and the error.

Proof. From (2.17), the -order derivative () is so that by taking into account (3.5) the proof follows.

In particular, by a suitable choice of the initial point , (3.9) can be simplified. For instance, at the integers, , according to (2.10), (2.12) and (3.5), it is

##### 3.3. Functional Equations

The connection coefficients fulfill some identities as follows.

Theorem 3.4. For any and , it is or

Proof. From (2.21), by a Fourier transform of both sides and taking into account (2.24), we get from where the identity (3.12) follows.

In particular, by assuming, without restrictions, , we have the following (see Figure 1).

Corollary 3.5. For any it is

so that are the Fourier coefficients of the power .

Analogously, from (2.21)2, we have the following.

Theorem 3.6. For any and it is or

In particular, with , and taking into account (3.5), we have the following.

Corollary 3.7. For any it is

As a consequence of the previous theorems we have the following.

Theorem 3.8. For any it is

There we have the following.

Corollary 3.9. The Fourier transform of the derivatives of a function is

If we express as a Taylor series we have so that with is the solution of the functional equation

Moreover, the theorem of moments can be written as

##### 3.4. Error of the Approximation by Connection Coefficients

For a fixed scale of approximation in (2.21), it is possible to estimate the error as follows. It should be noticed that the approximation depends on a the upper bound of the limits in the sums.

Theorem 3.10 (error of the approximation of scaling functions derivatives). The error of the approximation in (2.21)1 is given by

Proof. The error of the approximation (2.21)1 is defined as Concerning the r.h.s, and according to (2.13), it is

Theorem 3.11 (error of the approximation of wavelet functions derivatives). The error of the approximation in (2.21)2 is given by

Proof. The error of the approximation is If , the r.h.s. according to (2.30) is zero; therefore, we assume that so that the last equation becomes

#### 4. Fractional Derivatives of the Wavelet Basis

The simplest way to define the fractional derivative is based on the assumption that the noninteger derivative of the exponential function formally coincides with the derivative with integer order so that For negative values of , this formula still holds true and it represents the integration.

It is known that the fractional derivative cannot be analytically computed except for some special functions, such as (see e.g., ) the following: From these, classical examples, we can see that the fractional derivative can be also interpreted as an interpolating function between derivatives with integer order, so that

More in general, let be a single-valued real function, then the Riemann-Liouville fractional order derivative is defined as  being the gamma function.

Other equivalent representations were given by Caputo (for a differentiable function) and by Grünwald (see e.g., [17, 18]) However, a drawback in the Grünwald definition, as well as in the Riemann-Liouville, is that it cannot be computed for negative values of the variable ().

##### 4.1. Fractional Derivative of the Shannon Scaling Function

Let us assume that the fractional order derivative is defined by a linear interpolation of the integer order derivatives, so that the fractional derivative of the scaling-wavelet basis with can be defined as

Let us show the following.

Theorem 4.1. The fractional derivative of the Shannon scaling functions is

Proof. From (4.9), by taking into account (2.21), it is and, when ,

With this definition, the fractional order derivative of the scaling functions is a commutative operator according to the following.

Theorem 4.2. The operator (4.10) is a semigroup, so that

Proof. Without loss of generality, let us show that According to (4.10)2, it is that is and, taking into account (2.26), by explicit computation we have By deriving, with respect to , we have that is, according to (2.26), From where, the proof follows due to the symmetry of the change .

It can be easily seen that together with (4.17) also the following equations hold: and, in general,

Moreover, when , then we can see that the definition (2.26) reduces to the ordinary derivative, according to the following.

Theorem 4.3. When , then

Proof. If we restrict to , according to the definition (2.26), it is and since we have

According to the definition (4.10), the fractional derivative is an interpolation between integer order derivative (see Figure 2).

##### 4.2. Error of the Approximation of (4.10)

In the definition (4.10), the fractional derivative depends on a fixed bound of the infinite series. In this section, it will be shown that the rate of convergence of the series, on the r.h.s of (4.10), is quite fast; already with low values of , the approximation is quite good (Figure 3).

###### 4.2.1. Rate of Convergence

If we compare the fractional derivative given by (4.10) with the Grünwald definition (4.6), we can see that the approximation by connection coefficients is good (see Figure 4), with a lower number of terms. Moreover, the definition based on connection coefficients can be extended also to negative values of the variable.

Since we have defined the fractional derivative on an infinite series , as well as the Grünwald formula, we can explicitly compute the error of the approximation as the difference between the approximated value at and the corresponding value of the infinite series at . For instance, with respect to (4.10), it is while for the Grünwald formula (4.6) we have Let us show the following.

Theorem 4.4. For , the approximation error of (4.10)2 is given by

Proof. By taking into account (4.22), it is

Analogously, the following can be shown.

Theorem 4.5. For , the approximation error of (4.6)2 is given by

Proof. At the integer , it is

##### 4.3. Fractional Derivative of the Shannon Wavelet

Analogously to (4.10), the following can be proved.

Theorem 4.6. The fractional derivative of the Shannon wavelet functions is

Proof. From (4.9), by taking into account (2.21)

Analogously to the fractional derivative of the scaling function, also for the wavelet function, the fractional order derivatives are enveloped by the integer order derivatives (Figure 5).

##### 4.4. Fractional Derivative of an Function

Let be a function such that (2.17) holds, then its fractional derivative can be computed as where the fractional derivatives of the scaling functions and wavelets are given by (4.10) and (4.32), respectively.

For instance, a good approximation of is (Figure 6)