This paper deals with Chebyshev wavelets. We analyze their properties computing their Fourier transform. Moreover, we discuss the differential properties of Chebyshev wavelets due to the connection coefficients. Uniform convergence of Chebyshev wavelets and their approximation error allow us to provide rigorous proofs. In particular, we expand the mother wavelet in Taylor series with an application both in fractional calculus and fractal geometry. Finally, we give two examples concerning the main properties proved.

1. Introduction

In the last four decades, wavelet analysis rose to the role of mathematical theory due to the introduction of multiresolution analysis [1]. In the current literature, 1909 is often recognised as the birth of wavelet analysis, when Haar introduced a complete orthonormal system for the space . Nowadays, wavelet analysis is a mathematical tool widely applied in different fields. Image compression, electromagnetism, and PDE image are just three examples where wavelet methods currently play a meaningful role (see, e.g., [25]). In particular, Mallat produced a fast wavelet decomposition and reconstruction algorithm [6]. Over the course of time, the Mallat algorithm became the base for many wavelet applications in pure and applied science. Quite recently, wavelet analysis was also used for several techniques in image fusion, where each algorithm leads to different image decompositions. Fusing two types of information (temporal and spectral), the discrete wavelet transform (DWT) enabled the development of many DWT-based techniques. An application of the wavelet analysis in image fusion is the discrete shapelet transform (DST), which estimates the degree of similarity between the signal under analysis and a prespecified shape. This discrete transform consists of a fractal-based criterion to redefine the original Daubechies’ DWTs, leading to a time-frequency-shape joint analysis. Replacing the fractal-based criterion with a correlation-based formulation, the DST can be improved significantly. More specifically, the DST of the second generation simplifies both the study of filter coefficients and the interpretation of the transformed signal [7].

The main advantage of wavelet analysis is their decomposition of mathematical entities (e.g., images and time series) into components at different scales. This property is a consequence of the multiresolution analysis. Approximation in Fourier basis can lead to unpleasant results, as in the case of Gibbs phenomenon. These approximation problems can occur in any other reconstruction, but wavelets. Likewise, fractal geometry allows us to describe irregular sets by the concepts of fractal dimension and lacunarity [8, 9]. As is well known, irregular sets provide a better representation of different natural phenomena than the classical Euclidean models. Thus, fractal-like sets are currently used for many real-world applications (e.g., antenna theory and dynamical systems). Quite recently, considerable attention has been paid to the application of hybrid methods based both on wavelet analysis and fractal geometry in nonlinear modelling. For a fuller and deeper treatment on fractal-wavelet analysis, we refer the reader to the results of Jorgensen [10, 11].

Chebyshev wavelets are generally used for numerical methods in integral equations and PDEs. In particular, Chebyshev wavelets allowed the introduction of these methods due to the operational matrices and defined in (12) and (13), respectively. In [12], Hyedari et al. introduced a numerical method based on Chebyshev wavelets for solution of PDEs with boundary conditions of the telegraph type. Biazar and Ebraimi proposed a method based on Chebyshev wavelets for solving nonlinear systems of Volterra integral equations [13]. Similarly, Singh and Saha Ray [14] dealt with the stochastic Itô-Volterra integral equations by Chebyshev wavelets of the second kind. Following the recent trends in nonlinear analysis, Chebyshev wavelets were also used for the numerical solutions of fractional differential equations (see, e.g., [15, 16]). Current literature showed that these methods depends on the different operational matrices in the sense of [1719]. Moreover, Chebyshev wavelets provided sharp estimates of functions in Hölder spaces of order [20].

In this paper, we give new results on Chebyshev wavelets. More precisely, we deal with the differentiability of Chebyshev wavelets and the possibility to use their derivatives to reconstruct a function. The differential properties of Chebyshev wavelets, expressed by the connection coefficients (also called refinable integrals), are given by finite series in terms of the Kronecker delta. Moreover, we treat the -order derivative of Chebyshev wavelets and compute its Fourier transform. In the same spirit, we expand in Taylor series a function by Chebyshev wavelets and connection coefficients. Accordingly, Taylor expansion of the mother wavelet allows us to define the local fractional derivative of Chebyshev wavelets. More precisely, the introduction of local fractional calculus in these wavelet bases enables us to extend the local fractional derivative to nonsmooth continuous functions (e.g., fractal sets or random signals).

The rest of the paper is divided into three sections. In Section 2 we give some remarks on wavelet analysis and, particularly, on Chebyshev wavelets. Section 3 is devoted to differential properties of Chebyshev wavelets by connection coefficients. In Section 4, we deal with the Taylor expansion of Chebyshev wavelets. Finally, Section 5 extends the sought results on Chebyshev wavelets to fractal-like sets by local fractional calculus.

2. Remarks on Wavelet Analysis

This section is to devoted to recall some basic definitions and properties of wavelet analysis, which will be used throughout the paper. From now on, we refer to the set of natural numbers, denoted by , as the set of strictly positive integer numbers, that is . Thus, . Moreover, we will use the notation to denote the th falling factorial of [21].

Definition 1. The th-order Chebyshev polynomials of the first kind are defined by so that

Thus, , , , and so on. Definition 1 refers to the trigonometric representation of these polynomials. In literature, Chebyshev polynomials are usually defined as solutions of some Sturm-Liouville differential equations (today called Chebyshev differential equations). In particular, the definition in terms of Sturm-Liouville form leads us to prove the orthogonality of the Chebyshev polynomials with regards to the weight function: that is,

where is the Kronecker delta. Furthermore, for any , Chebyshev polynomials can be written by the following general recurrence relation:

which for gives

Chebyshev polynomials of the second, third, and fourth kinds can be defined and handled in much the same way. Moreover, all four Chebyshev polynomials admits a matrix representation (see [13] for more details). For instance, (3) can be written in matrix form as follows: or where is the matrix of the coefficients in (3) while and are the left-hand side and the right-hand side vectors in (4), respectively.

The properties of Chebyshev polynomials mentioned above lay the foundation for introducing a corresponding wavelet bases, termed Chebyshev wavelets. To this scope and before going ahead, let us recall the definition of wavelet orthonormal basis on . Wavelets are a family of functions generated by dilation and translation of one single function (called mother wavelet). In literature, all other functions of this family are usually called daughter wavelets. Thus, a family of continuous wavelets is given by

where and correspond to the scale factor and time shift, respectively. In what follows, therefore, we can assume in (5) which is the most common value for . Clearly, (5) for dilation and translation parameters and gives the following family of discrete wavelets:

which for and yields

The family of functions (6) is a wavelet basis for which becomes orthonormal for and .

2.1. Chebyshev Wavelets

Multiresolution analysis shows that Chebyshev wavelets can be built as recursive wavelets for piecewise polynomial spaces on . For this construction, we refer the reader to [22, 23], in which the problem is widely discussed.

Definition 2. Let and with . Chebyshev wavelets are defined as follows: where

Remark 3. In Definition 2, Chebyshev wavelets depends on four parameters, that is, . Moreover, ; thus,

Remark 4. In view of (1), Definition 2 implies that Chebyshev wavelets are defined on the real interval . Note that the orthogonality of the Chebyshev polynomials on with regard to implies the orthogonality of the Chebyshev wavelets on with regard to the weight function with and as in Definition 2 (see [24] for more details).

A function can be expanded in terms of the wavelet basis as follows:

The coefficients , usually termed wavelet coefficients, are given by where denotes the inner product. The series representation in (8) is called a wavelet series. In the case of Chebyshev wavelets, the previous inner product is defined in , that is,

Thus, any function can be expanded in terms of Chebyshev wavelets as follows: where the wavelet coefficients are given by

2.2. Function Approximation and Operational Matrix

Convergence of series (9) on implies that can be approximated as follows:

where and are column vectors. For simplicity of notation and without loss of generality, we rewrite (11) as

where and . The index is given by , and thus,

Likewise, Chebyshev wavelets allow us to approximate every function of two variables defined over as follows:

where being

We may now integrate the vector , precisely given by

where is the operational matrix of integration. It is worth noticing that, due to introduction of Chebyshev wavelets, the matrix is sparse (see [17, 18] for more details). Furthermore, allows the -times integration of given by

Similarly, we can differentiate as follows: and so

where is the operational matrix of differentiation [12, 25].

Finally, we point out that the product of two Chebyshev wavelets can be approximated as [19] follows:

where is a column vector and is a matrix. In literature, is called the operational matrix of product. In particular, for , we get

where is a diagonal matrix. In recent years, approximation (14) has been applied for solving integral equations, PDEs, and boundary valued problems (see, e.g., [17, 18]).

3. Fourier Transform, Differentiability, and Connection Coefficients

In this section, we study the differentiability of Chebyshev wavelets. More precisely, we prove our results in the weighted function space by the introduction of connection coefficients.

3.1. Differentiability of Functions in Chebyshev Wavelet Bases

Theorem 5. ( convergence) A function with bounded second derivative on , i.e., for any , can be expanded as an infinite sum of Chebyshev wavelets and the series converges uniformly to the function , that is where .

Proof. We only sketch the proof. For a fuller treatment, we refer the reader to [24].
First, Now, if , the change of variable in (16) gives where Since , it follows that Similarly, if , (17) implies that Furthermore, for , the series in (15) converges. In fact, is an orthogonal system, which implies the convergence of . It follows that Accordingly, the series converges to uniformly, as desired.

Theorem 5 leads to computation of the approximation error of the wavelet expansion (30), as stated in the following proposition.

Proposition 6. (estimation) Under the same hypotheses as in Theorem 5, we have where ⟦⟧ is the Iverson bracket notation and

Proof. First, recall that the series in (30) can be approximated with the truncated series in (11). Throughout the proof, we write instead of to avoid confusion. Accordingly, Now, the change of variable in (20) and relabeling as gives Moreover, (2) implies that therefore From Definition 2 we see that , thus combining (18), (19) with (21) it follows The proof is complete.

Theorem 5 and Proposition 6 show uniform convergence and accuracy estimation of Chebyshev wavelets, which lay the foundation for their wide application to the theory of integral equations (see, e.g., [1215, 17, 18, 24]).

Our next goal is to rewrite Chebyshev wavelets as a power series. We recall [26] that

The change of variable in the previous series gives

where denotes the set of even numbers. Clearly, implies that , hence . Moreover, the previous change of variable entails that

thus, the lower index of summation in (22) is . For simplicity of notation, we set thus,

It is worth noticing that all contributions of the summation index in (24) are subject to the condition in (23), i.e., ; thus, half of them vanish. More precisely, we have that the lower index of summation is for and for .

We see from (24) that (7) can be rewritten as follows:

Since the parameters and give, respectively, a dilation and a translation of the wavelet basis (7), the wavelet mother is such that . As a consequence, the wavelet mother depends only on the its associated Chebyshev polynomial . According to the current symbology in wavelet analysis, we define the wavelet mother as follows: therefore,

3.2. Connection Coefficients

The differential operators can be represented in wavelet bases if we compute the wavelet decomposition of the derivatives. Let be a function with such that with bounded second derivative on . The wavelet reconstruction (30) allows us to compute the derivatives of as follows:

Thus, according to (8), the derivatives of up to order are uniquely determined by

On the other hand, the first derivative of Chebyshev wavelets are given by

We note that the first derivative in (27) depends on , i.e., Chebyshev polynomials of the second kind. More specifically, it can be written as a Chebyshev wavelet of the second kind (see [20]). The computation of the derivatives (26) is more complicated for . In particular, high-order derivatives in (26) cannot be easily derived. Therefore, according to (8), we next turn to the wavelet decomposition of the derivatives (26), that is, with

The coefficients (29) are called connection coefficients with an obvious intuitive meaning. Their computation can be obtained in the Fourier domain due to the Parseval–Plancherel identity:

where denotes the Fourier transform of defined as follows:


Let us now compute the Fourier transform of .

Lemma 7. The Fourier transform of Chebyshev polynomials is given by where

Proof. The proof falls naturally into two parts ( and ). For , it follows immediately that and Let us now turn to the case . We begin by recalling the differentiation property of Fourier transform which holds in the space of tempered distributions on the real line . Accordingly, from (32) for , we get where is the Dirac delta distribution. Therefore, Furthermore, since the desired result plainly follows.

On the one hand, in the proof of Lemma 7, we used the fact that . On the other hand, the principal significance of Lemma 7 is that the Fourier transform of Chebyshev polynomials is nothing but a sum of derivatives of the Dirac delta. Condition (31) on coefficients implies that if the Fourier transform of is the sum of all even order derivatives of the Dirac delta. Likewise, if , the Fourier transform of is the sum of all odd order derivatives of the Dirac delta. We note that for the Fourier transform always contains the Dirac delta . Moreover, the presence of the power in (31) implies that

where denotes the set of imaginary numbers. These results are shown in Table 1 for the .

Now, we are in a position to compute the Fourier transform in (30) and connection coefficients.

Theorem 8. Let and be defined as in (7). Moreover, let . The following statements hold: with where with as in (23), and

Proof. First, from (25), it follows that Combining the proof of Lemma 7 and binomial theorem gives With the same notation as in Lemma 7, we set hence, This proves (i). Furthermore, (ii) follows straightforwardly from (i) and (30).
Finally, we can prove (iii). From (29), we have By (7), Moreover, which follows from the hypothesis that . We proved in Appendix that with and . Thus, Now, we can proceed to compute . Note that, expanding the integrand in (33), we have The assumption implies that the integrand in (33) holds for . As a consequence, For simplicity of notation, we indicate the last four integrals with ,, , and , respectively. Thus, The change of variable in , and relabeling as , gives We note that the last computation follows from the binomial theorem. Moreover, Since in (35), the result above holds in the computation of . Obviously, (36) vanishes for odd. Therefore, and so Let us now pass to the second integral . We can proceed analogously to the computation of . In fact, (35) implies that Furthermore, As in computation of , the result above holds because . We note that (37) vanishes for odd. Thus, and so Similarly, we can compute the integral and . In fact, the same change of variable as in gives Thus, in (34). This completes the proof.

Remark 9. In the proof of Theorem 8, the hypothesis played a fundamental role in the computation of connection coefficients . Indeed, for , the integrand in (33) holds for . Thus, we get a similar computation as in the proof of Theorem 8 but the integral and will be defined on . The details are left to the reader.

Remark 10. Theorem 8 allows us to define the connection coefficients of Chebyshev wavelets. Moreover, we note that the proof of Theorem 5 gives an upper bound on connection coefficients , i.e., Theorems 5 and 8 allow us the reconstruction for functions together with their derivative. Moreover, Theorem 8 gives the Fourier transform for any order derivative of Chebyshev wavelets, as the next example shows.

Example 1. Let us consider Chebyshev wavelets for re and : For the sake of simplicity and without loss of generality, we consider only . Thus, we leave it to the reader to deal with .

From Theorem 8, we have

where we used that , , and . Finally, we conclude that the sixth-order derivative of has the following Fourier transform:

Figures 1 and 2 show the graph of and . We note that the Fourier transform of exbibits an impulsive behaviour. In particular, the imaginary part of depends also on the distributional derivative of the Dirac delta, i.e.,

We may approximate the Dirac delta as [27] follows: thus,

Approximation (39) allows us to draw . In Figure 1, we show the imaginary part of with . Precisely, it is worth noticing that the Dirac delta in Figure 1 was multiplied by a factor of which takes into account the approximation error introduced by (39).

4. Taylor Series and Chebyshev Wavelets

In Section 3, we introduced the connection coefficients (29) that allows us to prove the following statement.

Theorem 11. Let be a function such that with bounded second derivative on . Then, the Taylor series of in is given by with as in (10).

Proof. The wavelet expansion (30) entails that the th-order derivative of (with ) can be expanded as follows: