# Chebyshev Wavelet Analysis

Authors: Emanuel Guariglia; Rodrigo Capobianco Guido
Journal: Journal of Function Spaces
 (2022)

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## Abstract

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.


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## Body

## 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 L20,1. 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., [2–5]). 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 matricesP and D 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 [17–19]. Moreover, Chebyshev wavelets provided sharp estimates of functions in Hölder spaces of order α [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 thep-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 Section2 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 byℕ, as the set of strictly positive integer numbers, that is ℕ=1,2,3,⋯. Thus, ℕ0=ℕ∪0. Moreover, we will use the notation xn¯ to denote the nth falling factorial of x [21].Definition 1.
Thenth-order Chebyshev polynomials of the first kind are defined by
(1)Tnx=cosnarccosx,x∈−1,1,n∈ℕ0,so that
Tncosθ=cosnθ,θ∈0,π,n∈ℕ0.Thus,T0x=1, T1x=x, T2x=2x2−1, 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:
wx=11−x2,x∈−1,1,that is,
(2)∫−11TmxTnx1−x2dx=π2δm,n,m≠0,n≠0,π,m=n=0,whereδm,n is the Kronecker delta. Furthermore, for any n,m∈ℕ0, Chebyshev polynomials can be written by the following general recurrence relation:
Tn+mx=2TnxTmx−Tn−mx,which form=1 gives
(3)Tn+1x=2xTnx−Tn−1x,n≥1.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:
(4)1000⋯0−2x100⋯01−2x10⋯001−2x1⋯0⋮⋱⋱⋱⋱⋮0⋯01−2x1T0xT1xT2xT3x⋮Tnx=1−x00⋮0,or
At=c,where A is the n+1×n+1 matrix of the coefficients in (3) while t and c 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 onℝ. Wavelets are a family of functions generated by dilation and translation of one single function ψ (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
(5)ψa,bx=a−1/pψx−ba,p>0,a,b∈ℝ,a≠0,wherea and b correspond to the scale factor and time shift, respectively. In what follows, therefore, we can assume p=2 in (5) which is the most common value for p. Clearly, (5) for dilation and translation parameters a−k and nba−k gives the following family of discrete wavelets:
(6)ψnkx=ak/2ψakx−nb,a>1,b>0,k,n∈ℤ,which fora=2 and b=1 yields
ψnkx=2k/2ψ2kx−n,k,n∈ℤ.The family of functions (6) is a wavelet basis for L2ℝ which becomes orthonormal for a=2 and b=1.


### 2.1. Chebyshev Wavelets

Multiresolution analysis shows that Chebyshev wavelets can be built as recursive wavelets for piecewise polynomial spaces on0,1. For this construction, we refer the reader to [22, 23], in which the problem is widely discussed.Definition 2.
Letn=1,2,⋯,2k−1 and m=0,1,⋯,M−1 with k,M∈ℕ2. Chebyshev wavelets are defined as follows:
(7)ψn,mkx=2k/2T~m2kx−2n+1,n−12k−1≤x<n2k−1,0,otherwise,where
T~mx=1πδm+2π1−δmTmx.Remark 3.
In Definition2, Chebyshev wavelets depends on four parameters, that is, ψn,mkx=ψk,n,m,x. Moreover, m∈ℕ0; thus,
T~mx=1π,m=0,2πTmx,m>0.Remark 4.
In view of (1), Definition 2 implies that Chebyshev wavelets are defined on the real interval 0,1. Note that the orthogonality of the Chebyshev polynomials on −1,1 with regard to w implies the orthogonality of the Chebyshev wavelets on 0,1 with regard to the weight function wkx=w2k−1x−n+1 with n and x as in Definition 2 (see [24] for more details).A functionf∈L2ℝ can be expanded in terms of the wavelet basis ψnkk,n∈ℤ as follows:
(8)fx=∑n=−∞∞∑k=−∞∞βnkψnkx.The coefficientsψnk, usually termed wavelet coefficients, are given by βnk=fx,ψnkx 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 Lw20,1, that is,
fx,gxw≔∫01fxgx¯wxdx,f,g∈Lw20,1.Thus, any functionf∈Lw20,1 can be expanded in terms of Chebyshev wavelets as follows:
(9)fx=∑n=1∞∑m=0∞βn,mkψn,mkx,where the wavelet coefficients are given by
(10)βn,mk=fx,ψn,mkxwk.


### 2.2. Function Approximation and Operational Matrix

Convergence of series (9) on Lw20,1 implies that f can be approximated as follows:
(11)fx≃∑n=12k−1∑m=0M−1βn,mkψn,mkx=BTΨx,whereB and Ψ are m~=2k−1M column vectors. For simplicity of notation and without loss of generality, we rewrite (11) as
fx≃∑i=1m~βiψix=BTΨx,whereβi=βn,mk and ψi=ψn,mk. The index i is given by i=Mn−1+m+1, and thus,
B≔β1,β2,⋯,βm~T,Ψx≔ψ1,ψ2,⋯,ψm~T.Likewise, Chebyshev wavelets allow us to approximate every function of two variablesu=ux,y defined over 0,1×0,1 as follows:
ux,y≃∑i=1m~∑j=1m~uijψixψiy=ΨTxUΨy,whereU=uij being
uij=ψix,ux,y,ψjywkywkx,i,j=1,2,⋯,m~.We may now integrate the vectorΨx, precisely given by
(12)∫0xΨtdt=PΨx,whereP is the m~×m~ operational matrix of integration. It is worth noticing that, due to introduction of Chebyshev wavelets, the matrix P is sparse (see [17, 18] for more details). Furthermore, Pn allows the n-times integration of Ψx given by
∫0x⋯∫0x⏟n−timesΨtdt1⋯dtn=PnΨx.Similarly, we can differentiateΨx as follows:
dΨxdx=DΨx,and so
(13)dnΨxdxn=DnΨx,whereD is the m~×m~ operational matrix of differentiation [12, 25].Finally, we point out that the product of two Chebyshev wavelets can be approximated as [19] follows:
ΨxΨTxX≃X~Ψx,whereX is a m~ column vector and X~ is a m~×m~ matrix. In literature, X~ is called the operational matrix of product. In particular, for X=B, we get
(14)ΨxΨTxB≃B~Ψx,whereB~ is a diagonal m~×m~ matrix. In recent years, approximation (14) has been applied for solving integral equations, PDEs, and boundary valued problems (see, e.g., [17, 18]).


## 2.1. Chebyshev Wavelets

Multiresolution analysis shows that Chebyshev wavelets can be built as recursive wavelets for piecewise polynomial spaces on0,1. For this construction, we refer the reader to [22, 23], in which the problem is widely discussed.Definition 2.
Letn=1,2,⋯,2k−1 and m=0,1,⋯,M−1 with k,M∈ℕ2. Chebyshev wavelets are defined as follows:
(7)ψn,mkx=2k/2T~m2kx−2n+1,n−12k−1≤x<n2k−1,0,otherwise,where
T~mx=1πδm+2π1−δmTmx.Remark 3.
In Definition2, Chebyshev wavelets depends on four parameters, that is, ψn,mkx=ψk,n,m,x. Moreover, m∈ℕ0; thus,
T~mx=1π,m=0,2πTmx,m>0.Remark 4.
In view of (1), Definition 2 implies that Chebyshev wavelets are defined on the real interval 0,1. Note that the orthogonality of the Chebyshev polynomials on −1,1 with regard to w implies the orthogonality of the Chebyshev wavelets on 0,1 with regard to the weight function wkx=w2k−1x−n+1 with n and x as in Definition 2 (see [24] for more details).A functionf∈L2ℝ can be expanded in terms of the wavelet basis ψnkk,n∈ℤ as follows:
(8)fx=∑n=−∞∞∑k=−∞∞βnkψnkx.The coefficientsψnk, usually termed wavelet coefficients, are given by βnk=fx,ψnkx 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 Lw20,1, that is,
fx,gxw≔∫01fxgx¯wxdx,f,g∈Lw20,1.Thus, any functionf∈Lw20,1 can be expanded in terms of Chebyshev wavelets as follows:
(9)fx=∑n=1∞∑m=0∞βn,mkψn,mkx,where the wavelet coefficients are given by
(10)βn,mk=fx,ψn,mkxwk.


## 2.2. Function Approximation and Operational Matrix

Convergence of series (9) on Lw20,1 implies that f can be approximated as follows:
(11)fx≃∑n=12k−1∑m=0M−1βn,mkψn,mkx=BTΨx,whereB and Ψ are m~=2k−1M column vectors. For simplicity of notation and without loss of generality, we rewrite (11) as
fx≃∑i=1m~βiψix=BTΨx,whereβi=βn,mk and ψi=ψn,mk. The index i is given by i=Mn−1+m+1, and thus,
B≔β1,β2,⋯,βm~T,Ψx≔ψ1,ψ2,⋯,ψm~T.Likewise, Chebyshev wavelets allow us to approximate every function of two variablesu=ux,y defined over 0,1×0,1 as follows:
ux,y≃∑i=1m~∑j=1m~uijψixψiy=ΨTxUΨy,whereU=uij being
uij=ψix,ux,y,ψjywkywkx,i,j=1,2,⋯,m~.We may now integrate the vectorΨx, precisely given by
(12)∫0xΨtdt=PΨx,whereP is the m~×m~ operational matrix of integration. It is worth noticing that, due to introduction of Chebyshev wavelets, the matrix P is sparse (see [17, 18] for more details). Furthermore, Pn allows the n-times integration of Ψx given by
∫0x⋯∫0x⏟n−timesΨtdt1⋯dtn=PnΨx.Similarly, we can differentiateΨx as follows:
dΨxdx=DΨx,and so
(13)dnΨxdxn=DnΨx,whereD is the m~×m~ operational matrix of differentiation [12, 25].Finally, we point out that the product of two Chebyshev wavelets can be approximated as [19] follows:
ΨxΨTxX≃X~Ψx,whereX is a m~ column vector and X~ is a m~×m~ matrix. In literature, X~ is called the operational matrix of product. In particular, for X=B, we get
(14)ΨxΨTxB≃B~Ψx,whereB~ is a diagonal m~×m~ 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 spaceLw20,1 by the introduction of connection coefficients.


### 3.1. Differentiability ofLw2 Functions in Chebyshev Wavelet Bases

Theorem 5.
(Lw2 convergence) A function f∈Lw20,1 with bounded second derivative on 0,1, i.e., f″x≤A for any x∈0,1, can be expanded as an infinite sum of Chebyshev wavelets and the series converges uniformly to the function f, that is
(15)fx=∑n=1∞∑m=0∞βn,mkψn,mkx,
whereβn,mk=fx,ψn,mkxwk.Proof.
We only sketch the proof. For a fuller treatment, we refer the reader to [24].
First,(16)βn,mk=∫01fxψn,mkxwkxdx=∫n−1/2k−1n/2k−1fxT~m2kx−2n+11−2kx−2n+12dx.
Now, ifm>1, the change of variable 2kx−2n+1=cosθ in (16) gives
(17)βn,mk=223k/2mπ∫0πf′cosθ+2n−12ksinmθsinθdθ=125k/2m2π∫0πf″cosθ+2n−12khmθdθ,where
hmθ=sinθsinm−1θm−1−sinm+1θm+1.
Sincen≤2k−1, it follows that
(18)βn,mk≤2πA2n5/2m2−1.
Similarly, ifm=1, (17) implies that
(19)βn,1k≤2π2n3/2max0≤x≤1f′x.
Furthermore, form=0, the series in (15) converges. In fact, ψn,0kn=1∞ is an orthogonal system, which implies the convergence of ∑n=1∞βn,0kψn,0kx. It follows that
∑n=1∞∑m=0∞βn,mkψn,mkx≤∑n=1∞βn,0kψn,0kx+∑n=1∞∑m=1∞βn,mk<∞.
Accordingly, the series∑n=1∞∑m=0∞βn,mkψn,mkx converges to fx uniformly, as desired.Theorem5 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 Theorem5, we have
σk,M≤π2∑n=2k−1+1∞1n3max20≤x≤1f′xM=1+A24n2∑m=M∞1m2−12M≠11/2,
where ⟦⟧ is the Iverson bracket notation andσk,M≔∫01fx−∑n=12k−1∑m=0M−1βn,mkψn,mkx2wkxdx1/2.Proof.
First, recall that the series in (30) can be approximated with the truncated series in (11). Throughout the proof, we write βn,m instead of βn,mk to avoid confusion. Accordingly,
(20)σk,M2=∫01fx−∑n=12k−1∑m=0M−1βn,mψn,mkx2wkxdx=∫01∑n=2k−1+1∞∑m=M∞βn,m2ψn,mkx2wkxdx=∑n=2k−1+1∞∑m=M∞βn,m22k+1π∫n−1/2k−1n/2k−1Tm22kx−2n+11−2kx−2n+12dx.
Now, the change of variable2kx−2n+1=x′ in (20) and relabeling x′ as x gives
σk,M2=∑n=2k−1+1∞∑m=M∞βn,m22π∫−11Tm2x1−x2dx.
Moreover, (2) implies that
∫−11Tm2x1−x2dx=π2,m≥1,therefore
(21)σk,M2=∑n=2k−1+1∞∑m=M∞βn,m2.
From Definition2 we see that M∈ℕ, thus combining (18), (19) with (21) it follows
σk,M2=∑n=2k−1+1∞βn,12M=1+∑m=M∞βn,m2M≠1≤π4∑n=2k−1+1∞1n3max20≤x≤1f′xM=1+A24n2∑m=M∞1m2−12M≠1.
The proof is complete.Theorem5 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., [12–15, 17, 18, 24]).Our next goal is to rewrite Chebyshev wavelets as a power series. We recall [26] that
Tmx=m2∑r=0m/2−1rm−rm−rr2xm−2r,m>0.The change of variabler′=m−2r in the previous series gives
(22)Tmx=m∑r=0m−1m−r/2m+rm+r2m−r22xr=∑r=0m2rm−1ωm+rm+r2m−r2xr,m+r∈E,ω=m−r2,whereE denotes the set of even numbers. Clearly, m+r∈E implies that m−r∈E, hence m±r∈E. Moreover, the previous change of variable entails that
m−2m/2=0,m∈E,1,m∈O,thus, the lower index of summation in (22) is r=0. For simplicity of notation, we set
(23)αrm≔2rm−1ωm+rm+r2m−r2,m+r∈E,ω=m−r2,thus,
(24)Tmx=∑r=0mαrmxr,m>0.It is worth noticing that all contributions of the summation indexr in (24) are subject to the condition m+r∈E in (23), i.e., m±r∈E; thus, half of them vanish. More precisely, we have that the lower index of summation is r=0 for m∈E\0 and r=1 for m∈E.We see from (24) that (7) can be rewritten as follows:
(25)ψn,mkx=2k/21πδm+2π1−δm∑r=0mαrm2kx−2n+1r,n−12k−1≤x<n2k−1,0,otherwise.Since the parametersn and k give, respectively, a dilation and a translation of the wavelet basis (7), the wavelet mother ψ is such that ψ=ψm,x. As a consequence, the wavelet mother ψ depends only on the its associated Chebyshev polynomial Tmx. According to the current symbology in wavelet analysis, we define the wavelet mother ψ as follows:
ψx≔ψ1,m1x,therefore,
ψx=1πδm+2π1−δm∑r=0mαrmx−1r,0≤x<1,0,otherwise.


### 3.2. Connection Coefficients

The differential operators can be represented in wavelet bases if we compute the wavelet decomposition of the derivatives. Letf be a Cp function with p>0 such that f∈Lw20,1 with bounded second derivative on 0,1. The wavelet reconstruction (30) allows us to compute the derivatives of f as follows:
dpdxpfx=∑n=1∞∑m=0∞βn,mkdpdxpψn,mkx.Thus, according to (8), the derivatives of f up to order p are uniquely determined by
(26)dpdxpψn,mkx.On the other hand, the first derivative of Chebyshev wavelets are given by(27)ddxψn,mkx=m2π23k/2Um−12kx−2n+1,n−12k−1≤x<n2k−1,0,otherwise.We note that the first derivative in (27) depends on Um−1x, 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 p>1. 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,
(28)dpdxpψn,mkx=∑l=1∞∑q=0∞γnlmqpkhψl,qhx,with
(29)γnlmqpkh=dpdxpψn,mkx,ψl,qhxwh.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:
f,g≔∫−∞∞fxgx¯dx=∫−∞∞f^ξg^ξ¯dξ=f^,g^,f,g∈L1ℝ∩L2ℝ,wheref^ denotes the Fourier transform of f defined as follows:
f^ξ≔∫−∞∞fxe−2πiξxdx,f∈L1ℝ.Therefore,(30)dpdxpψn,mkx∧=2πiξpψ∧n,mkξ.Let us now compute the Fourier transform ofTmx.Lemma 7.
The Fourier transform of Chebyshev polynomialsTmx is given by
T∧mξ=∑r=0mcrmδrξ,where
(31)crm=imπrmm+rm+r2m−r2,m+r∈E.Proof.
The proof falls naturally into two parts (m=0 and m>0). For m=0, it follows immediately that c00=1and
T0x=1↔Fδξ.
Let us now turn to the casem>0. We begin by recalling the differentiation property of Fourier transform
(32)xnfx↔Fi2πndndξnf∧ξ,n∈ℕ0,
which holds in the space of tempered distributions on the real lineS′ℝ. Accordingly, from (32) for fx=δx, we get
xn↔Fi2πnδnξ,n∈ℕ0,
whereδ is the Dirac delta distribution. Therefore,
Tmξ=24∑r=0mαrmFxr=∑r=0mαrmi2πrδrξ,m>0.
Furthermore, sinceαrmi2πr=crm,m>0,the desired result plainly follows.On the one hand, in the proof of Lemma7, we used the fact that T^0ξ=δξ. On the other hand, the principal significance of Lemma 7 is that the Fourier transform of Chebyshev polynomials Tmx is nothing but a sum of derivatives of the Dirac delta. Condition (31) on coefficients crm implies that if m∈E the Fourier transform of Tmx is the sum of all even order derivatives of the Dirac delta. Likewise, if m∈O, the Fourier transform of Tmx is the sum of all odd order derivatives of the Dirac delta. We note that for m∈E the Fourier transform T^mξ always contains the Dirac delta δξ. Moreover, the presence of the power im in (31) implies that
T^mξ∈ℝ,m∈E,I,m∈0,whereI denotes the set of imaginary numbers. These results are shown in Table 1 for the m=1,2,⋯,10.Table 1
Fourier transform of Chebyshev polynomialsTmx for m=1,2,⋯,10.
mTmxT∧mξ1xiδ′ξ/2π22x2−1−δξ−δ′′ξ/2π234x3−3x3π2δ′ξ+δ3ξ/2π3i48x4−8x2+1δξ+4π2δ′′ξ+δ4ξ/2π4516x5−20x3+5xi5π4δ′ξ+5π2δ3ξ+δ5ξ/2π5632x6−48x4+18x2−1−δξ−9π4δ′′ξ+6π2δ4ξ+δ6ξ/2π6764x7−112x5+56x3−7x7π6δ′ξ+14π4δ3ξ+14π2δ5ξ+δ7ξ/2π7i8128x8−256x6+160x4−32x2+1δξ+16π6δ′′ξ+20π4δ4ξ+8π2δ6ξ+δ8ξ/2π89259x9−576x7+432x5−120x3+9xi9π8δ′ξ+30π6δ3ξ+27π4δ5ξ+9π2δ7ξ+δ9ξ/2π910512x10−1280x8+1120x6−400x4+50x2−1−δξ−25π8δ′′ξ+50π6δ4ξ+35π4δ6ξ+10π2δ8ξ+δ10ξ/2π10Now, we are in a position to compute the Fourier transform in (30) and connection coefficients.Theorem 8.
Letn and k be defined as in (7). Moreover, let m,p∈ℕ. The following statements hold:
ψ∧n,mkξ=2k/21πδmδξ+2π1−δm∑r=0m∑t=0rcr,tm,nδtξ,
withcr,tm,n≔2k−1t+rim−r+t/πtm/m+rm+r/2m−r/2rt1−2nr−t,m+r∈E,dpdxpψn,mkx∧=2k/22πiξp1πδmδξ+2π1−δm∑r=0m∑t=0rcr,tm,nδtξ,γnlmqpkh=21+p+k−hk−h/2πm∑0≤r≤m−p:m−p−r∈Edm,rp2δqλδq+1−δqλ1−δq−2π4dmpm−p∈E,l≤n,where
dm,rp≔m+p+r2−1p−1¯m+p−r2−1p−1,λ1−δq≔λδq1+∑ν=1qανqs+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,with ανq as in (23), dmp=dm,0p and
λδq≔∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−ss+1/2!s+1s/2!,s∈E.Proof.
First, from (25), it follows that
ψ^n,mkξ=2k/21πδmδξ+2π1−δmF∑r=0mαrm2kx−2n+1r.
Combining the proof of Lemma7 and binomial theorem gives
F∑r=0mαrm2kx−2n+1r=∑r=0m∑t=0rαrm2krrt−2n+12kr−ti2πtδtξ=23∑r=0m∑t=0r2k−1t+rim−r+tπtmm+rm+r2m−r2rt1−2nr−tδtξ,m+r∈E.
With the same notation as in Lemma7, we set
cr,tm,n=2k−1t+rim−r+tπtmm+rm+r2m−r2rt1−2nr−t,m+r∈E,hence,
F∑r=0mαrm2kx−2n+1r=∑r=0m∑t=0rcr,tm,nδtξ.
This proves (i). Furthermore, (ii) follows straightforwardly from (i) and (30).
Finally, we can prove (iii). From (29), we have
(33)γnlmqpkh=∫01dpdxpψn,mkxψl,qhxwhxdx.
By (7),
dpdxpψn,mkx=2k/2dpdxp1πδm+2π1−δmTm2kx−2n+1=2k/22π1−δmdpdxpTm2kx−2n+1,n−12k−1≤x<n2k−1.
Moreover,dpdxpψn,mkxψl,qhx=2h+k/22π1−δmδqdpdxpTm2kx−2n+1+2h+k/22π1−δm1−δqdpdxpTm2kx−2n+1Tq2hx−2l+1=2h+k/2dpdxpTm2kx−2n+12πδq+2π1−δqTq2hx−2l+1,
which follows from the hypothesis thatm≥1. We proved in Appendix that
dpdxpTm2kx−2n+1=2pm∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−2p−1mdmpm−p∈E,with
dm,rp≔m+p+r2−1p−1¯m+p−r2−1p−1,
anddmp=dm,0p. Thus,
dpdxpψn,mkxψl,qhx=2p+h+k/2m2πδq+2π1−δqTq2hx−2l+1∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−dmp2m−p∈E,
Now, we can proceed to computeγnlmqpkh. Note that, expanding the integrand in (33), we have
(34)n−12k−1≤x<n2k−1,n=1,2,⋯,2k−1,l−12h−1≤x<l2h−1,l=1,2,⋯,2h−1.
The assumptionl≤n implies that the integrand in (33) holds for 342. As a consequence,
γnlmqpkh=∫l−1/2h−1l/2h−12p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−dmp2m−p∈E·2πδq+2π1−δqTq2hx−2l+111−2hx−2l+12dx=2p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rp2πδq∫l−1/2h−1l/2h−1Tr2kx−2n+111−2hx−2l+12dx+2π1−δq∫l−1/2h−1l/2h−1Tr2kx−2n+1Tq2hx−2l+111−2hx−2l+12dx−2p+h+kh+k/2mdmp2m−p∈E2πδq∫l−1/2h−1l/2h−111−2hx−2l+12dx+2π1−δq·∫l−1/2h−1l/2h−1Tq2hx−2l+111−2hx−2l+12dx.
For simplicity of notation, we indicate the last four integrals withI1,I2, I3, and I4, respectively. Thus,
(35)γnlmqpkh=2p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rp2πδqI1+2π1−δqI2−dmp22πδqI3+2π1−δqI4m−p∈E.
The change of variablex′=2hx−2l+1 in I1, and relabeling x′ as x, gives
(36)I1=12h∫−11Tr2k−hx+2l−1−2n+111−x2dx=2412h∫−11∑t=0rαtr2k−hx+2l−1−2n+1t11−x2dx=12h∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−s∫−11xs1−x2dx.
We note that the last computation follows from the binomial theorem. Moreover,(37)∫−11xs1−x2dx=π−1s+1Γs+1/2sΓs/2,Res>−1,
Sinces≥0 in (35), the result above holds in the computation of I1. Obviously, (36) vanishes for s odd. Therefore,
∫−11xs1−x2dx=2πΓs+1/2sΓs/2=2πs+1/2!s+1s/2!,s∈E, and so
I1=π2h−1∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−ss+1/2!s+1s/2!,s∈E.
Let us now pass to the second integralI2. We can proceed analogously to the computation of I1. In fact, (35) implies that
I2=12h∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−s∫−11xs1−x2Tqxdx=2412h∑t=0r∑j=0t∑s=0j∑ν=0qαtrανqtjjs2k−hj1−2nt−j2l−1j−s∫−11xs+ν1−x2dx.
Furthermore,(38)∫−11xs+ν1−x2dx=π−1s+ν+1Γs+ν+1/22Γs+ν+2/2,Res+ν>−1.
As in computation ofI1, the result above holds because s+ν≥0. We note that (37) vanishes for s+ν odd. Thus,
∫−11xs+ν1−x2dx=πΓs+ν+1/2Γs+ν+2/2=πs+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,s+ν∈E,and so
I2=π2h∑t=0r∑j=0t∑s=0j∑ν=0qαtrανqtjjs2k−hj1−2nt−j2l−1j−ss+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,s+ν∈E.
Similarly, we can compute the integralI3 and I4. In fact, the same change of variable as in I1gives
I3=12h∫−1111−x2dx=π2h,I4=12h∫−11Tqx11−x2dx=2π2hδq.
Thus,1−δqI4=0 in (34). This completes the proof.Remark 9.
In the proof of Theorem8, the hypothesis l≤n played a fundamental role in the computation of connection coefficients γnlmqpkh. Indeed, for n<l, the integrand in (33) holds for 341. Thus, we get a similar computation as in the proof of Theorem 8 but the integral I1 and I2 will be defined on n−1/2k−1,n/2k−1. The details are left to the reader.Remark 10.
Theorem8 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 γnlmqpkh, i.e.,
(39)γnlmqpkh≤21+p+k−h/2πm∑0≤r≤m−p:m−p−r∈Edm,rp2δqλδq+1−δqλ1−δq,l≤n.
Theorems5 and 8 allow us the reconstruction for Lw20,1 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 fork=2 re and m=1:
ψ1,12x=22π4x−1,0≤x<12,0,otherwise,ψ2,12x=22π4x−3,12≤x<1,0,otherwise.
For the sake of simplicity and without loss of generality, we consider onlyψ∧1,12. Thus, we leave it to the reader to deal with ψ∧2,12.From Theorem8, we have
ψ^1,12ξ=2π∑r=01∑t=0rcr,t1,1δtξ=2πc0,01,1+c1,01,1δξ+c1,01,1δ′ξ=2π−1+iδξ+2iπδ′ξ,where we used thatc0,01,1=i, c1,01,1=−1, and c1,11,1=2i/π. Finally, we conclude that the sixth-order derivative of ψ1,12 has the following Fourier transform:
d6dx6ψ1,12x^=2πiξ62π−1+iδξ+2iπδ′ξ=64π52πξ6π1−iδξ−2iδ′ξ.Figures1 and 2 show the graph of ψ1,12 and ψ^1,12. We note that the Fourier transform of ψ1,12 exbibits an impulsive behaviour. In particular, the imaginary part of ψ^1,12 depends also on the distributional derivative of the Dirac delta, i.e.,
Reψ^1,12=Reψ^1,12δ,Imψ^1,12=Imψ^1,12δ,δ′.Figure 1
Chebyshev waveletsψn∗,12 with n∗=1,2.Figure 2
The Fourier transform ofψ1,12. We drew the imaginary part of ψ^1,12 by (101) with ε=0.01.We may approximate the Dirac delta as [27] follows:
δx≈1επe−x2/ε,ε≪1,thus,
(40)δ′x=−2xεεπe−x2/ε,ε≪1.Approximation (39) allows us to draw ψ∧1,12. In Figure 1, we show the imaginary part of ψ∧1,12 with ε=0.01. Precisely, it is worth noticing that the Dirac delta δξ in Figure 1 was multiplied by a factor of 51 which takes into account the approximation error introduced by (39).


## 3.1. Differentiability ofLw2 Functions in Chebyshev Wavelet Bases

Theorem 5.
(Lw2 convergence) A function f∈Lw20,1 with bounded second derivative on 0,1, i.e., f″x≤A for any x∈0,1, can be expanded as an infinite sum of Chebyshev wavelets and the series converges uniformly to the function f, that is
(15)fx=∑n=1∞∑m=0∞βn,mkψn,mkx,
whereβn,mk=fx,ψn,mkxwk.Proof.
We only sketch the proof. For a fuller treatment, we refer the reader to [24].
First,(16)βn,mk=∫01fxψn,mkxwkxdx=∫n−1/2k−1n/2k−1fxT~m2kx−2n+11−2kx−2n+12dx.
Now, ifm>1, the change of variable 2kx−2n+1=cosθ in (16) gives
(17)βn,mk=223k/2mπ∫0πf′cosθ+2n−12ksinmθsinθdθ=125k/2m2π∫0πf″cosθ+2n−12khmθdθ,where
hmθ=sinθsinm−1θm−1−sinm+1θm+1.
Sincen≤2k−1, it follows that
(18)βn,mk≤2πA2n5/2m2−1.
Similarly, ifm=1, (17) implies that
(19)βn,1k≤2π2n3/2max0≤x≤1f′x.
Furthermore, form=0, the series in (15) converges. In fact, ψn,0kn=1∞ is an orthogonal system, which implies the convergence of ∑n=1∞βn,0kψn,0kx. It follows that
∑n=1∞∑m=0∞βn,mkψn,mkx≤∑n=1∞βn,0kψn,0kx+∑n=1∞∑m=1∞βn,mk<∞.
Accordingly, the series∑n=1∞∑m=0∞βn,mkψn,mkx converges to fx uniformly, as desired.Theorem5 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 Theorem5, we have
σk,M≤π2∑n=2k−1+1∞1n3max20≤x≤1f′xM=1+A24n2∑m=M∞1m2−12M≠11/2,
where ⟦⟧ is the Iverson bracket notation andσk,M≔∫01fx−∑n=12k−1∑m=0M−1βn,mkψn,mkx2wkxdx1/2.Proof.
First, recall that the series in (30) can be approximated with the truncated series in (11). Throughout the proof, we write βn,m instead of βn,mk to avoid confusion. Accordingly,
(20)σk,M2=∫01fx−∑n=12k−1∑m=0M−1βn,mψn,mkx2wkxdx=∫01∑n=2k−1+1∞∑m=M∞βn,m2ψn,mkx2wkxdx=∑n=2k−1+1∞∑m=M∞βn,m22k+1π∫n−1/2k−1n/2k−1Tm22kx−2n+11−2kx−2n+12dx.
Now, the change of variable2kx−2n+1=x′ in (20) and relabeling x′ as x gives
σk,M2=∑n=2k−1+1∞∑m=M∞βn,m22π∫−11Tm2x1−x2dx.
Moreover, (2) implies that
∫−11Tm2x1−x2dx=π2,m≥1,therefore
(21)σk,M2=∑n=2k−1+1∞∑m=M∞βn,m2.
From Definition2 we see that M∈ℕ, thus combining (18), (19) with (21) it follows
σk,M2=∑n=2k−1+1∞βn,12M=1+∑m=M∞βn,m2M≠1≤π4∑n=2k−1+1∞1n3max20≤x≤1f′xM=1+A24n2∑m=M∞1m2−12M≠1.
The proof is complete.Theorem5 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., [12–15, 17, 18, 24]).Our next goal is to rewrite Chebyshev wavelets as a power series. We recall [26] that
Tmx=m2∑r=0m/2−1rm−rm−rr2xm−2r,m>0.The change of variabler′=m−2r in the previous series gives
(22)Tmx=m∑r=0m−1m−r/2m+rm+r2m−r22xr=∑r=0m2rm−1ωm+rm+r2m−r2xr,m+r∈E,ω=m−r2,whereE denotes the set of even numbers. Clearly, m+r∈E implies that m−r∈E, hence m±r∈E. Moreover, the previous change of variable entails that
m−2m/2=0,m∈E,1,m∈O,thus, the lower index of summation in (22) is r=0. For simplicity of notation, we set
(23)αrm≔2rm−1ωm+rm+r2m−r2,m+r∈E,ω=m−r2,thus,
(24)Tmx=∑r=0mαrmxr,m>0.It is worth noticing that all contributions of the summation indexr in (24) are subject to the condition m+r∈E in (23), i.e., m±r∈E; thus, half of them vanish. More precisely, we have that the lower index of summation is r=0 for m∈E\0 and r=1 for m∈E.We see from (24) that (7) can be rewritten as follows:
(25)ψn,mkx=2k/21πδm+2π1−δm∑r=0mαrm2kx−2n+1r,n−12k−1≤x<n2k−1,0,otherwise.Since the parametersn and k give, respectively, a dilation and a translation of the wavelet basis (7), the wavelet mother ψ is such that ψ=ψm,x. As a consequence, the wavelet mother ψ depends only on the its associated Chebyshev polynomial Tmx. According to the current symbology in wavelet analysis, we define the wavelet mother ψ as follows:
ψx≔ψ1,m1x,therefore,
ψx=1πδm+2π1−δm∑r=0mαrmx−1r,0≤x<1,0,otherwise.


## 3.2. Connection Coefficients

The differential operators can be represented in wavelet bases if we compute the wavelet decomposition of the derivatives. Letf be a Cp function with p>0 such that f∈Lw20,1 with bounded second derivative on 0,1. The wavelet reconstruction (30) allows us to compute the derivatives of f as follows:
dpdxpfx=∑n=1∞∑m=0∞βn,mkdpdxpψn,mkx.Thus, according to (8), the derivatives of f up to order p are uniquely determined by
(26)dpdxpψn,mkx.On the other hand, the first derivative of Chebyshev wavelets are given by(27)ddxψn,mkx=m2π23k/2Um−12kx−2n+1,n−12k−1≤x<n2k−1,0,otherwise.We note that the first derivative in (27) depends on Um−1x, 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 p>1. 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,
(28)dpdxpψn,mkx=∑l=1∞∑q=0∞γnlmqpkhψl,qhx,with
(29)γnlmqpkh=dpdxpψn,mkx,ψl,qhxwh.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:
f,g≔∫−∞∞fxgx¯dx=∫−∞∞f^ξg^ξ¯dξ=f^,g^,f,g∈L1ℝ∩L2ℝ,wheref^ denotes the Fourier transform of f defined as follows:
f^ξ≔∫−∞∞fxe−2πiξxdx,f∈L1ℝ.Therefore,(30)dpdxpψn,mkx∧=2πiξpψ∧n,mkξ.Let us now compute the Fourier transform ofTmx.Lemma 7.
The Fourier transform of Chebyshev polynomialsTmx is given by
T∧mξ=∑r=0mcrmδrξ,where
(31)crm=imπrmm+rm+r2m−r2,m+r∈E.Proof.
The proof falls naturally into two parts (m=0 and m>0). For m=0, it follows immediately that c00=1and
T0x=1↔Fδξ.
Let us now turn to the casem>0. We begin by recalling the differentiation property of Fourier transform
(32)xnfx↔Fi2πndndξnf∧ξ,n∈ℕ0,
which holds in the space of tempered distributions on the real lineS′ℝ. Accordingly, from (32) for fx=δx, we get
xn↔Fi2πnδnξ,n∈ℕ0,
whereδ is the Dirac delta distribution. Therefore,
Tmξ=24∑r=0mαrmFxr=∑r=0mαrmi2πrδrξ,m>0.
Furthermore, sinceαrmi2πr=crm,m>0,the desired result plainly follows.On the one hand, in the proof of Lemma7, we used the fact that T^0ξ=δξ. On the other hand, the principal significance of Lemma 7 is that the Fourier transform of Chebyshev polynomials Tmx is nothing but a sum of derivatives of the Dirac delta. Condition (31) on coefficients crm implies that if m∈E the Fourier transform of Tmx is the sum of all even order derivatives of the Dirac delta. Likewise, if m∈O, the Fourier transform of Tmx is the sum of all odd order derivatives of the Dirac delta. We note that for m∈E the Fourier transform T^mξ always contains the Dirac delta δξ. Moreover, the presence of the power im in (31) implies that
T^mξ∈ℝ,m∈E,I,m∈0,whereI denotes the set of imaginary numbers. These results are shown in Table 1 for the m=1,2,⋯,10.Table 1
Fourier transform of Chebyshev polynomialsTmx for m=1,2,⋯,10.
mTmxT∧mξ1xiδ′ξ/2π22x2−1−δξ−δ′′ξ/2π234x3−3x3π2δ′ξ+δ3ξ/2π3i48x4−8x2+1δξ+4π2δ′′ξ+δ4ξ/2π4516x5−20x3+5xi5π4δ′ξ+5π2δ3ξ+δ5ξ/2π5632x6−48x4+18x2−1−δξ−9π4δ′′ξ+6π2δ4ξ+δ6ξ/2π6764x7−112x5+56x3−7x7π6δ′ξ+14π4δ3ξ+14π2δ5ξ+δ7ξ/2π7i8128x8−256x6+160x4−32x2+1δξ+16π6δ′′ξ+20π4δ4ξ+8π2δ6ξ+δ8ξ/2π89259x9−576x7+432x5−120x3+9xi9π8δ′ξ+30π6δ3ξ+27π4δ5ξ+9π2δ7ξ+δ9ξ/2π910512x10−1280x8+1120x6−400x4+50x2−1−δξ−25π8δ′′ξ+50π6δ4ξ+35π4δ6ξ+10π2δ8ξ+δ10ξ/2π10Now, we are in a position to compute the Fourier transform in (30) and connection coefficients.Theorem 8.
Letn and k be defined as in (7). Moreover, let m,p∈ℕ. The following statements hold:
ψ∧n,mkξ=2k/21πδmδξ+2π1−δm∑r=0m∑t=0rcr,tm,nδtξ,
withcr,tm,n≔2k−1t+rim−r+t/πtm/m+rm+r/2m−r/2rt1−2nr−t,m+r∈E,dpdxpψn,mkx∧=2k/22πiξp1πδmδξ+2π1−δm∑r=0m∑t=0rcr,tm,nδtξ,γnlmqpkh=21+p+k−hk−h/2πm∑0≤r≤m−p:m−p−r∈Edm,rp2δqλδq+1−δqλ1−δq−2π4dmpm−p∈E,l≤n,where
dm,rp≔m+p+r2−1p−1¯m+p−r2−1p−1,λ1−δq≔λδq1+∑ν=1qανqs+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,with ανq as in (23), dmp=dm,0p and
λδq≔∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−ss+1/2!s+1s/2!,s∈E.Proof.
First, from (25), it follows that
ψ^n,mkξ=2k/21πδmδξ+2π1−δmF∑r=0mαrm2kx−2n+1r.
Combining the proof of Lemma7 and binomial theorem gives
F∑r=0mαrm2kx−2n+1r=∑r=0m∑t=0rαrm2krrt−2n+12kr−ti2πtδtξ=23∑r=0m∑t=0r2k−1t+rim−r+tπtmm+rm+r2m−r2rt1−2nr−tδtξ,m+r∈E.
With the same notation as in Lemma7, we set
cr,tm,n=2k−1t+rim−r+tπtmm+rm+r2m−r2rt1−2nr−t,m+r∈E,hence,
F∑r=0mαrm2kx−2n+1r=∑r=0m∑t=0rcr,tm,nδtξ.
This proves (i). Furthermore, (ii) follows straightforwardly from (i) and (30).
Finally, we can prove (iii). From (29), we have
(33)γnlmqpkh=∫01dpdxpψn,mkxψl,qhxwhxdx.
By (7),
dpdxpψn,mkx=2k/2dpdxp1πδm+2π1−δmTm2kx−2n+1=2k/22π1−δmdpdxpTm2kx−2n+1,n−12k−1≤x<n2k−1.
Moreover,dpdxpψn,mkxψl,qhx=2h+k/22π1−δmδqdpdxpTm2kx−2n+1+2h+k/22π1−δm1−δqdpdxpTm2kx−2n+1Tq2hx−2l+1=2h+k/2dpdxpTm2kx−2n+12πδq+2π1−δqTq2hx−2l+1,
which follows from the hypothesis thatm≥1. We proved in Appendix that
dpdxpTm2kx−2n+1=2pm∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−2p−1mdmpm−p∈E,with
dm,rp≔m+p+r2−1p−1¯m+p−r2−1p−1,
anddmp=dm,0p. Thus,
dpdxpψn,mkxψl,qhx=2p+h+k/2m2πδq+2π1−δqTq2hx−2l+1∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−dmp2m−p∈E,
Now, we can proceed to computeγnlmqpkh. Note that, expanding the integrand in (33), we have
(34)n−12k−1≤x<n2k−1,n=1,2,⋯,2k−1,l−12h−1≤x<l2h−1,l=1,2,⋯,2h−1.
The assumptionl≤n implies that the integrand in (33) holds for 342. As a consequence,
γnlmqpkh=∫l−1/2h−1l/2h−12p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rpTr2kx−2n+1−dmp2m−p∈E·2πδq+2π1−δqTq2hx−2l+111−2hx−2l+12dx=2p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rp2πδq∫l−1/2h−1l/2h−1Tr2kx−2n+111−2hx−2l+12dx+2π1−δq∫l−1/2h−1l/2h−1Tr2kx−2n+1Tq2hx−2l+111−2hx−2l+12dx−2p+h+kh+k/2mdmp2m−p∈E2πδq∫l−1/2h−1l/2h−111−2hx−2l+12dx+2π1−δq·∫l−1/2h−1l/2h−1Tq2hx−2l+111−2hx−2l+12dx.
For simplicity of notation, we indicate the last four integrals withI1,I2, I3, and I4, respectively. Thus,
(35)γnlmqpkh=2p+h+k/2m∑0≤r≤m−p:m−p−r∈Edm,rp2πδqI1+2π1−δqI2−dmp22πδqI3+2π1−δqI4m−p∈E.
The change of variablex′=2hx−2l+1 in I1, and relabeling x′ as x, gives
(36)I1=12h∫−11Tr2k−hx+2l−1−2n+111−x2dx=2412h∫−11∑t=0rαtr2k−hx+2l−1−2n+1t11−x2dx=12h∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−s∫−11xs1−x2dx.
We note that the last computation follows from the binomial theorem. Moreover,(37)∫−11xs1−x2dx=π−1s+1Γs+1/2sΓs/2,Res>−1,
Sinces≥0 in (35), the result above holds in the computation of I1. Obviously, (36) vanishes for s odd. Therefore,
∫−11xs1−x2dx=2πΓs+1/2sΓs/2=2πs+1/2!s+1s/2!,s∈E, and so
I1=π2h−1∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−ss+1/2!s+1s/2!,s∈E.
Let us now pass to the second integralI2. We can proceed analogously to the computation of I1. In fact, (35) implies that
I2=12h∑t=0r∑j=0t∑s=0jαtrtjjs2k−hj1−2nt−j2l−1j−s∫−11xs1−x2Tqxdx=2412h∑t=0r∑j=0t∑s=0j∑ν=0qαtrανqtjjs2k−hj1−2nt−j2l−1j−s∫−11xs+ν1−x2dx.
Furthermore,(38)∫−11xs+ν1−x2dx=π−1s+ν+1Γs+ν+1/22Γs+ν+2/2,Res+ν>−1.
As in computation ofI1, the result above holds because s+ν≥0. We note that (37) vanishes for s+ν odd. Thus,
∫−11xs+ν1−x2dx=πΓs+ν+1/2Γs+ν+2/2=πs+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,s+ν∈E,and so
I2=π2h∑t=0r∑j=0t∑s=0j∑ν=0qαtrανqtjjs2k−hj1−2nt−j2l−1j−ss+ν+2s+ν+1/2!s+ν+1s+ν+2/2!,s+ν∈E.
Similarly, we can compute the integralI3 and I4. In fact, the same change of variable as in I1gives
I3=12h∫−1111−x2dx=π2h,I4=12h∫−11Tqx11−x2dx=2π2hδq.
Thus,1−δqI4=0 in (34). This completes the proof.Remark 9.
In the proof of Theorem8, the hypothesis l≤n played a fundamental role in the computation of connection coefficients γnlmqpkh. Indeed, for n<l, the integrand in (33) holds for 341. Thus, we get a similar computation as in the proof of Theorem 8 but the integral I1 and I2 will be defined on n−1/2k−1,n/2k−1. The details are left to the reader.Remark 10.
Theorem8 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 γnlmqpkh, i.e.,
(39)γnlmqpkh≤21+p+k−h/2πm∑0≤r≤m−p:m−p−r∈Edm,rp2δqλδq+1−δqλ1−δq,l≤n.
Theorems5 and 8 allow us the reconstruction for Lw20,1 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 fork=2 re and m=1:
ψ1,12x=22π4x−1,0≤x<12,0,otherwise,ψ2,12x=22π4x−3,12≤x<1,0,otherwise.
For the sake of simplicity and without loss of generality, we consider onlyψ∧1,12. Thus, we leave it to the reader to deal with ψ∧2,12.From Theorem8, we have
ψ^1,12ξ=2π∑r=01∑t=0rcr,t1,1δtξ=2πc0,01,1+c1,01,1δξ+c1,01,1δ′ξ=2π−1+iδξ+2iπδ′ξ,where we used thatc0,01,1=i, c1,01,1=−1, and c1,11,1=2i/π. Finally, we conclude that the sixth-order derivative of ψ1,12 has the following Fourier transform:
d6dx6ψ1,12x^=2πiξ62π−1+iδξ+2iπδ′ξ=64π52πξ6π1−iδξ−2iδ′ξ.Figures1 and 2 show the graph of ψ1,12 and ψ^1,12. We note that the Fourier transform of ψ1,12 exbibits an impulsive behaviour. In particular, the imaginary part of ψ^1,12 depends also on the distributional derivative of the Dirac delta, i.e.,
Reψ^1,12=Reψ^1,12δ,Imψ^1,12=Imψ^1,12δ,δ′.Figure 1
Chebyshev waveletsψn∗,12 with n∗=1,2.Figure 2
The Fourier transform ofψ1,12. We drew the imaginary part of ψ^1,12 by (101) with ε=0.01.We may approximate the Dirac delta as [27] follows:
δx≈1επe−x2/ε,ε≪1,thus,
(40)δ′x=−2xεεπe−x2/ε,ε≪1.Approximation (39) allows us to draw ψ∧1,12. In Figure 1, we show the imaginary part of ψ∧1,12 with ε=0.01. Precisely, it is worth noticing that the Dirac delta δξ in Figure 1 was multiplied by a factor of 51 which takes into account the approximation error introduced by (39).


## 4. Taylor Series and Chebyshev Wavelets

In Section3, we introduced the connection coefficients (29) that allows us to prove the following statement.Theorem 11.
Letf be a Cq function such that f∈Lw20,1 with bounded second derivative on 0,1. Then, the Taylor series of f in x=x0 is given by
(41)fx=fx0+∑t=1∞∑n,l=1∞∑m,q=0∞βn,mkγnlmqtkhψl,qhx0x−x0tt!,
withβn,mk as in (10).Proof.
The wavelet expansion (30) entails that the pth-order derivative of f (with p≤q) can be expanded as follows:
fpx=∑n=1∞∑m=0∞βn,mkdpdxpψn,mkx=28∑n=1∞∑m=0∞βn,mk∑l=1∞∑q=0∞γnlmqpkhψl,qhx=∑n,l=1∞∑m,q=0∞βn,mkγnlmqpkhψl,qhx,as desired.
It is worthy noticing that a suitable choice of the initial pointx0 allows us to simplify the Taylor expansion (40). In particular, for x0=0, we have that
fx=41f0+∑t=1∞∑n,l=1∞∑m,q=0∞βn,mkγnlmqtkhψl,qh0xtt!.
Moreover, Definition2 gives that x0=0⟹l=h=1; thus,
ψl,qh0=ψ1,q10=21/2T~q−1=21πδq+2π−1q1−δq,γnlmqtkh=γn1mqtk1.
Therefore,(42)fx=f0+2π∑t=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δqxtt!.
Explicitly, this means that any functionf∈Lw20,1 can be expressed as a power series when the wavelet coefficients βn,mk are finite. Accordingly, the Taylor series for the wavelet mother ψx=ψ1,m1x is given by
(43)ψx=2πδm+2π1−δm−1m+2π∑t=1∞∑m,q=0∞β1,m1γ11mqt11δq+2−1q1−δqxtt!,
sinceψ0=2/πδm+2/π1−δm−1m. We note that in the last computation the sum on the index n reduces to one term because for the wavelet mother n=1.
Now, we are in a position to estimate the approximation error in (28) for a fixed scale of approximation. It is worth pointing out that the approximation depends on the upper bound in the sums.Theorem 12 (Approximation error of wavelet derivatives).
Letp∈ℕ and q∈ℕ0 be such that p≤q+1. Under the same hypotheses as in Definition 2, the approximation error in (28)
εk,M≔∫01dpdxpψl,qhx−∑n=12k−1∑m=0M−1γlnqmphkψn,mkx2wkxdx1/2,is bounded by
εk,M≤21+p+h−k/2πqq−p+1∑n=2k−1+1∞∑m=M∞dm,q,p,r∗21/2,n≤l,
where is the Iverson bracket notation,dq,p stands for dqp defined as in Theorem 8 and
dm,q,p,r∗≔max0≤r≤q−p:q−p−r∈Edq,rp·2δmλδm+1−δmλ1−δm,
withdq,rp as in Theorem 8.Proof.
The proof can be handled in a similar way as in the proof of Proposition6. In order to avoid confusion, we rewrite (28) as follows:
dpdxpψl,qhx=∑n=1∞∑m=0∞γlnqmphkψn,mkx,with
γlnqmphk=dpdxpψl,qhx,ψn,mhxwh.
Moreover, throughout the proof, we denoteγlnqmphk briefly by γlnqm. Therefore,
εk,M2=∫01dpdxpψn,mkx−∑n=12k−1∑m=0M−1γlnqmψn,mkx2wkxdx=∫01∑n=2k−1+1∞∑m=M∞γlnqm2ψn,mkx2wkxdx=∑n=2k−1+1∞∑m=M∞γlnqm22k+1π∫n−1/2k−1n/2k−1Tm22kx−2n+11−2kx−2n+12dx.
Now, the proof of Proposition6 gives
εk,M2=∑n=2k−1+1∞∑m=M∞γlnqm2.
We note that the coefficientsdq,rp defined as in Theorem 8 are nonnegative. Thus, (38) implies that
γlnqm≤21+p+h−k/2πq∑0≤r≤q−p:q−p−r∈Edq,rp2δmλδm+1−δmλ1−δm≤21+p+h−k/2πqq−p+1dm,q,p,r∗,n≤l.
Accordingly,γlnqm2≤22+2p+h−kπq2q−p+12dm,q,p,r∗2,n≤l.
We conclude thatεk,M2≤4π22p+h−kq2q−p+12∑n=2k−1+1∞∑m=M∞dm,q,p,r∗2,n≤l,which gives the desired estimate.Remark 13.
We note that the termdm,q,p,r∗2 depends on m and n by the definition of dp,rp in Theorem 8. Moreover, the hypothesis p≤q+1 assures us that the derivative
dpdxpψl,qhx,does not identically vanish on its domain, i.e., we exclude trivial cases in the thesis of Theorem 12.


## 5. Fractional Calculus of Chebyshev Wavelets

This section is devoted to the local fractional calculus over Chebyshev wavelets. Our approach follows new trends in fractional calculus. Indeed, many problems in fractional calculus are nowadays dealt with hybrid methods (e.g., [28]). In particular, we show how the local fractional calculus extends the fractional derivative to nonsmooth continuous functions (e.g., fractal functions). In fact, the introduction of a local fractional derivative on Chebyshev wavelets by connection coefficients allows us to compute the local fractional derivative of non-smooth functions.


### 5.1. Chebyshev Wavelets on Fractal Sets of Dimensionν

According to the recent results on fractional calculus of wavelet bases (see, e.g., [29]), we can define the Chebyshev wavelets on a fractal set of dimension ν (with 0<ν≤1) as follows:
(44)ψνx=432πδm+2π1−δm−1m+2π∑p=1∞∑m,q=0∞β1,m1γ11mqt11δq+2−1q1−δqxνpΓνp+1.Thus, if we setγmp≔2πδm+2π1−δm−1m,p=0,2π∑m,q=0∞β1,m1γ11mqt11δq+2−1q1−δqxνpΓνp+1,p>0,we have that
(45)ψνxν=∑p=0∞γmpΓνp+1xνp.We note that power series expansion (44) for ν=1 reduces to Chebyshev wavelets on a regular domain:
ψ1x=ψx=∑p=0∞γmpp!xp.In addition to this, series expansion (44) for ν<1 gives Chebyshev wavelets in a continuous nonregular domain, which are functions of the variable xν. We point out that the variable xν can be defined in nonregular domains, such as the Cantor set shown (Figure 3). For more details, we refer the reader to [29].Figure 3
The middle third Cantor set.In the same spirit as (42), any function f∈Lw20,1 can be expanded in Taylor series on a fractal set of dimension ν as follows:
(46)fx=f0+2π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δqxνpΓνp+1.Obviously, Taylor expansion (45) generalizes (41) on a fractal set of dimension ν. In order to clarify the generalization of power series (44) for nonregular domain, we consider one of the simplest case of nonregular set: the Cantor set. For further considerations on the Cantor set, we refer the reader to [30, 31]. It suffices for our purposes to recall that the Cantor function (or Devil’s staircase) f is defined as follows:
fx=ulimn⟶∞fnx,where=u denotes uniform convergence, fn0,1⟶ℝ such that f0x=x and
fn+1x=12fn3x0≤x≤13,1,13≤x≤23,1+fn3x−2,23≤x≤1.Of course, the Cantor function maps the Cantor set C onto0,1. The family of functions fnn∈ℕ is a set of polygonal approximations of the Cantor function f. We note that the previous definition of fnn∈ℕ is equivalent to the following condition:
fn+1x−fnx≤2−n−1,x∈0,1,n∈ℕ.Moreover, Chalice [32] proved that the Cantor function f is the unique monotonic, real-valued function on 0,1 such that label=f0=0,fx3=fx2,f1−x=1−fx.This characterization gives us an easier way to draw the Cantor functionf (Figure 4). For more details on the Cantor function, we refer the reader to [33]. We note that the method of Chalice can be used to draw the graphics of the functions ψνxν both on the compact interval 0,1 and on the Cantor set.Figure 4
The Cantor function.


### 5.2. Local fractional Calculus

We recall that if the real functionfx is such that
(47)fx−fx0<εν,withx−x0<δ being ε,δ>0, then fx is called fractional continuous at x=x0. As in the classical case, we denote this property by
limx⟶x0fx=fx0.In the same spirit, the functionfx is called local fractional continuous on the interval a,b and denoted by f∈Cνa,b, if it satisfies the condition (46) for x∈a,b. Now, if f∈Cνa,b, the local fractional derivative of the real function fx of order ν at x=x0 is given [34] by
(48)fνx0=dνdxνfxx=x0=Γ1+νlimx⟶x0fx−fx0x−x0ν,0<ν≤1.The functionf is called ν-differentiable (or differentiable of order ν) at x=x0 if the limit in (47) exists and is finite. It is worth noticing that the local fractional derivative (47) does not satisfy the generalized Leibniz rule [21]. Nevertheless, if f and g are both ν-differentiable in the interval a,b in the sense of (47), we have that
(49)fgν=fνg+fgν.In fact, (47) implies that
dνdxνfxgx=Γ1+νlimh⟶0fx+hgx+h−fxgxhα=Γ1+νlimh⟶0fx+h−fxhαgx+fx+hgx+h−gxhα=fνxgx+fxgνx.Thus, theν-derivative of the product of two functions is the fractional equivalent of the product rule for integer derivatives.We note that the definition (47) works for almost all the rules of fractional calculus (except, of course, the generalized Leibniz rule). For instance, we have [35] that
dνdxνsinνxν=cosνxν,dνdxνcosνxν=−sinνxν.Moreover,(50)dνdxνxνk=Γ1+kνΓ1+k−1νxk−1ν,and thus, fork=1, it follows that
(51)dνxν=Γ1+νdxν⇒dxν=Γ1+νdxν.In recent years, the differential operator (50) was used to define integral transformations on Cantor sets (see, e.g., [36, 37]). The results of Section 4 allow us to obtain a series expansion of the local fractional derivative (47) as stated in the following theorem.Theorem 14.
Letf∈Lw20,1 such that the wavelet coefficients (10) exist and are finite. Then, the local fractional derivative of f in the neighborhood of x=0 is given by
dνdxνfx=∑p=1∞ApβΓ1+p−1νxp−1ν,where
Apβ≔2π∑n=1∞∑m,q=0∞βn,mkγn1mqpk1δq+2−1q1−δq.Proof.
The proof follows directly by (45) and (49). In fact, we have
dνdxνfx=2π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δq1Γνp+1dνdxνxνp=502π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δq1Γ1+p−1νxp−1ν.Example 2.
Let us consider the functiong:x∈ℝ⟶3x2e2xsin−2πx∈ℝ. This example shows the efficiency of the results proved in this paper. First, we note that g∈Lw20,1. In fact,
∫013x2e2xsin−2πx1−x2dx≈5.0275.
Moreover,g″ is bounded on 0,1. Accordingly, we can expand the function g in Chebyshev wavelets. Now, (11) implies that g can be approximated in terms of Chebyshev wavelets as follows:
(52)3x2e2xsin−2πx≃∑n=12k∗−1∑m=0M∗−1βn,mkψn,mkx,
where increasing the values of bothk∗ and M∗ provides a better approximation in (51). Obviously, wavelet coefficients βn,mk are computed by (10). The approximation error of g by Chebyshev wavelets is clearly shown in Figure 5.Figure 5
Comparison between the functiongx=3x2e2xsin−2πx and its approximation (dashed line) by Chebyshev wavelets with k∗=6∧M∗=21 (top) and k∗=9∧M∗=31 (bottom).On the one hand, Theorem14 implies that
(53)dνdxν3x2e2xsin−2πx≃∑p=1∞ApTβΓ1+p−1νxp−1ν,where the coefficientsApTβ is the truncated version of Apβ, i.e.,
ApTβ=2π∑n=12k∗−1∑m=0M∗−1∑q=0Q∗−1βn,mkγn1mqpk1δq+2−1q1−δq.On the other hand, (48) gives
(54)dνdxνgx=−3dνdxνx2e2xsin2πx=−32πνx2e2xΓ3Γ3−ν12πxν+1πνsin2πx+sin2πx+π2ν,where we used the fact [35] that
dνdxνeax=aνeax,dνdxνsinax=aνsinax+π2ν,a∈ℝ,dνdxνxn=Γn+1Γn+1−νxn−ν,n≥0.Comparison between (52) and (53) shows the efficiency of this method even with rough wavelet approximations (Figure 6).Figure 6
Comparison between the local fractional derivative ofgx=3x2e2xsin−2πx for ν=1/2 and its approximation (dashed line) by Chebyshev wavelets with k∗=9∧M∗=31.


## 5.1. Chebyshev Wavelets on Fractal Sets of Dimensionν

According to the recent results on fractional calculus of wavelet bases (see, e.g., [29]), we can define the Chebyshev wavelets on a fractal set of dimension ν (with 0<ν≤1) as follows:
(44)ψνx=432πδm+2π1−δm−1m+2π∑p=1∞∑m,q=0∞β1,m1γ11mqt11δq+2−1q1−δqxνpΓνp+1.Thus, if we setγmp≔2πδm+2π1−δm−1m,p=0,2π∑m,q=0∞β1,m1γ11mqt11δq+2−1q1−δqxνpΓνp+1,p>0,we have that
(45)ψνxν=∑p=0∞γmpΓνp+1xνp.We note that power series expansion (44) for ν=1 reduces to Chebyshev wavelets on a regular domain:
ψ1x=ψx=∑p=0∞γmpp!xp.In addition to this, series expansion (44) for ν<1 gives Chebyshev wavelets in a continuous nonregular domain, which are functions of the variable xν. We point out that the variable xν can be defined in nonregular domains, such as the Cantor set shown (Figure 3). For more details, we refer the reader to [29].Figure 3
The middle third Cantor set.In the same spirit as (42), any function f∈Lw20,1 can be expanded in Taylor series on a fractal set of dimension ν as follows:
(46)fx=f0+2π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δqxνpΓνp+1.Obviously, Taylor expansion (45) generalizes (41) on a fractal set of dimension ν. In order to clarify the generalization of power series (44) for nonregular domain, we consider one of the simplest case of nonregular set: the Cantor set. For further considerations on the Cantor set, we refer the reader to [30, 31]. It suffices for our purposes to recall that the Cantor function (or Devil’s staircase) f is defined as follows:
fx=ulimn⟶∞fnx,where=u denotes uniform convergence, fn0,1⟶ℝ such that f0x=x and
fn+1x=12fn3x0≤x≤13,1,13≤x≤23,1+fn3x−2,23≤x≤1.Of course, the Cantor function maps the Cantor set C onto0,1. The family of functions fnn∈ℕ is a set of polygonal approximations of the Cantor function f. We note that the previous definition of fnn∈ℕ is equivalent to the following condition:
fn+1x−fnx≤2−n−1,x∈0,1,n∈ℕ.Moreover, Chalice [32] proved that the Cantor function f is the unique monotonic, real-valued function on 0,1 such that label=f0=0,fx3=fx2,f1−x=1−fx.This characterization gives us an easier way to draw the Cantor functionf (Figure 4). For more details on the Cantor function, we refer the reader to [33]. We note that the method of Chalice can be used to draw the graphics of the functions ψνxν both on the compact interval 0,1 and on the Cantor set.Figure 4
The Cantor function.


## 5.2. Local fractional Calculus

We recall that if the real functionfx is such that
(47)fx−fx0<εν,withx−x0<δ being ε,δ>0, then fx is called fractional continuous at x=x0. As in the classical case, we denote this property by
limx⟶x0fx=fx0.In the same spirit, the functionfx is called local fractional continuous on the interval a,b and denoted by f∈Cνa,b, if it satisfies the condition (46) for x∈a,b. Now, if f∈Cνa,b, the local fractional derivative of the real function fx of order ν at x=x0 is given [34] by
(48)fνx0=dνdxνfxx=x0=Γ1+νlimx⟶x0fx−fx0x−x0ν,0<ν≤1.The functionf is called ν-differentiable (or differentiable of order ν) at x=x0 if the limit in (47) exists and is finite. It is worth noticing that the local fractional derivative (47) does not satisfy the generalized Leibniz rule [21]. Nevertheless, if f and g are both ν-differentiable in the interval a,b in the sense of (47), we have that
(49)fgν=fνg+fgν.In fact, (47) implies that
dνdxνfxgx=Γ1+νlimh⟶0fx+hgx+h−fxgxhα=Γ1+νlimh⟶0fx+h−fxhαgx+fx+hgx+h−gxhα=fνxgx+fxgνx.Thus, theν-derivative of the product of two functions is the fractional equivalent of the product rule for integer derivatives.We note that the definition (47) works for almost all the rules of fractional calculus (except, of course, the generalized Leibniz rule). For instance, we have [35] that
dνdxνsinνxν=cosνxν,dνdxνcosνxν=−sinνxν.Moreover,(50)dνdxνxνk=Γ1+kνΓ1+k−1νxk−1ν,and thus, fork=1, it follows that
(51)dνxν=Γ1+νdxν⇒dxν=Γ1+νdxν.In recent years, the differential operator (50) was used to define integral transformations on Cantor sets (see, e.g., [36, 37]). The results of Section 4 allow us to obtain a series expansion of the local fractional derivative (47) as stated in the following theorem.Theorem 14.
Letf∈Lw20,1 such that the wavelet coefficients (10) exist and are finite. Then, the local fractional derivative of f in the neighborhood of x=0 is given by
dνdxνfx=∑p=1∞ApβΓ1+p−1νxp−1ν,where
Apβ≔2π∑n=1∞∑m,q=0∞βn,mkγn1mqpk1δq+2−1q1−δq.Proof.
The proof follows directly by (45) and (49). In fact, we have
dνdxνfx=2π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δq1Γνp+1dνdxνxνp=502π∑p=1∞∑n=1∞∑m,q=0∞βn,mkγn1mqtk1δq+2−1q1−δq1Γ1+p−1νxp−1ν.Example 2.
Let us consider the functiong:x∈ℝ⟶3x2e2xsin−2πx∈ℝ. This example shows the efficiency of the results proved in this paper. First, we note that g∈Lw20,1. In fact,
∫013x2e2xsin−2πx1−x2dx≈5.0275.
Moreover,g″ is bounded on 0,1. Accordingly, we can expand the function g in Chebyshev wavelets. Now, (11) implies that g can be approximated in terms of Chebyshev wavelets as follows:
(52)3x2e2xsin−2πx≃∑n=12k∗−1∑m=0M∗−1βn,mkψn,mkx,
where increasing the values of bothk∗ and M∗ provides a better approximation in (51). Obviously, wavelet coefficients βn,mk are computed by (10). The approximation error of g by Chebyshev wavelets is clearly shown in Figure 5.Figure 5
Comparison between the functiongx=3x2e2xsin−2πx and its approximation (dashed line) by Chebyshev wavelets with k∗=6∧M∗=21 (top) and k∗=9∧M∗=31 (bottom).On the one hand, Theorem14 implies that
(53)dνdxν3x2e2xsin−2πx≃∑p=1∞ApTβΓ1+p−1νxp−1ν,where the coefficientsApTβ is the truncated version of Apβ, i.e.,
ApTβ=2π∑n=12k∗−1∑m=0M∗−1∑q=0Q∗−1βn,mkγn1mqpk1δq+2−1q1−δq.On the other hand, (48) gives
(54)dνdxνgx=−3dνdxνx2e2xsin2πx=−32πνx2e2xΓ3Γ3−ν12πxν+1πνsin2πx+sin2πx+π2ν,where we used the fact [35] that
dνdxνeax=aνeax,dνdxνsinax=aνsinax+π2ν,a∈ℝ,dνdxνxn=Γn+1Γn+1−νxn−ν,n≥0.Comparison between (52) and (53) shows the efficiency of this method even with rough wavelet approximations (Figure 6).Figure 6
Comparison between the local fractional derivative ofgx=3x2e2xsin−2πx for ν=1/2 and its approximation (dashed line) by Chebyshev wavelets with k∗=9∧M∗=31.



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*Source: 5542054-2022-06-30.xml*