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Fractional Calculus and Shannon Wavelet
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.
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 [5–7] 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 [8–13].
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.
2.1. Properties of the Shannon Wavelet
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
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, 8–13].
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),
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
The connection coefficients fulfill some recursive formula as follows.
Theorem 3.1. The connection coefficients (2.26) are recursively given by
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:
Moreover, it can be shown also that
3.2. Taylor Series
By using the connection coefficients, it is easy to show the following theorem.
3.3. Functional Equations
The connection coefficients fulfill some identities as follows.
Theorem 3.4. For any and , it is or
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
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., [16–18]) 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
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
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