# Biorthogonal Wavelet on a Logarithm Curveℂ

Authors: Xiaohui Zhou; Gang Wang
Journal: Journal of Mathematics
 (2021)

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## Abstract

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curveC is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C. Finally, a numerical application is given for dealing with financial data.


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## Body

## 1. Introduction

It is known to all that wavelet analysis is widely applied in various fields, such as signal analysis and processing, the image compression, object detection [1, 2], facial biometrics [2], face recognition [2, 3], stock market fluctuations [4], and so on. A unique feature of wavelet analysis is that it can process non-stationary data, localize in time domain, and perform multiscale analysis of signals, according to the required time scale. In recent years, some scholars have tried to use wavelet analysis to deal with some problems in economics and finance. For example, Francis In and Sangbae Kim (2012) have discussed an instruction to wavelet theory in finance and some examples of its application in economics and finances in their monograph [4]. Some new research interpretation are presented in dealing with financial data, such as “volatility on regression growth trend,” “error on regression growth trend,” “wavelet approximate data on the regression trend line,” and so on [5]. At the same time, manifold learning is an important method for parameter reduction and dimensionality reduction in machine learning [6, 7]. And it is also an effective method to deal with high-dimensional data or some data on a manifold. So wavelet analysis on some manifolds is a new significant method to deal with some data on manifolds. Some scholars have done much work in “wavelet on the manifolds,” such as the continuous wavelet transform on conic sections [8], wavelet analysis on some surfaces of revolution via area-preserving projection [9], wavelet transform on regression trend curve [5], and so on. A feasible method to discuss the wavelet on the manifold is by using bijective projection, such as the vertical projection, the radial projection, stereographic projection, area-preserving projection, length-preserving projection, and so on. The orthogonal wavelets with compact support (except Haar wavelets) are not symmetric; that is, the corresponding filters cannot have linear phase. In order to overcome the shortcoming of orthogonal wavelet, the concept of biorthogonal wavelet is introduced and discussed widely [10–12]. And biorthogonal wavelet filters are both compact support and linear-phase. “Orthogonal wavelet filters on a manifold” is in many literatures. So, the biorthogonal wavelet on a logarithm curve C is discussed by constructing length-preserving projection p in this paper.This paper will be organized as follows: in Section2, the preliminary is given, such as concepts of biorthogonal wavelet, length-preserving projection, and so on; in Section 3, the biorthogonal wavelet function on a logarithm curve C is discussed by length-preserving projection and multiresolution analysis; finally, we give an example and a numerical application in financial data.


## 2. Preliminary

In this section, we introduce length-preserving projection via Euler discretization scheme and biorthogonal wavelet analysis. Many mathematical models may be used in electronic signal, economic data, and financial data, such as linear model, time series model, logarithmic model, semi-logarithmic model, and so on. We can estimate parameters of the models by statistical and numerical methods. Furthermore, the regression trends of these data can be obtained by parameters. Common trends include linear regression curve, logarithmic regression curve, and so on. Orthogonal wavelet filters on a linear regression curve have been discussed and applied in financial data [5]. We introduce the logarithmic regression curve firstly.Logarithmic regression models are often used in some economic models, such as Cobb–Douglas production functionY=AKαLβeμ. Taking the Log on both sides of it, we have lnY=lnA+αlnK+βlnL+μ, μ∼N0,σ2. At this time, the model is transformed into a linear model, which is also a special case of logarithmic regression model.Definition 1.
(see [13]). If an explained variable y and an explanatory variable x satisfy the following equation:(1)lny=α+βlnx+μ,μ∼N0,σ2.This model is called a one-dimensional logarithmic regression model. If an explained variable y and an explanatory variable x satisfy the equation(2)y=α+βlnx+μ,μ∼N0,σ2,then this model is called a one-dimensional semilogarithmic regression model and y¯=α¯+β¯lnx is called a logarithmic regression trend curve.


### 2.1. Length-Preserving Projection via Euler Discretization Scheme

According to the algorithm of length-preserving projectionp on a smooth curve in [5, 14], the projection p is given in this section.Consider a log-regression curveC:(3)y=alnx,a∈R,x∈0,∞.By calculating, we have(4)Lx1=∫0x11+a2x2dx=x12+a2−alna+x12+a2x1.So(5)Lx=x2+a2−alna+x2+a2x=l,p:x,y⟶l,0=Lx,0.The inverse projectionp−1 is not explicit, but it could be determined by the numerical method.(6)x=L−1ly=alnL−1l.that is,p−1:l,0⟶L−1l,alnL−1l.Consider the Euler discretization method: for a finite intervalT0,T1, T0=x0<x1<⋯<xn=T1, then(7)1+a2x2dx=dl⟺1+a2xi2Δxi=Δli,where Δxi=xi+1−xi,Δli=li+1−li.The length-preserving projectionp can be denoted as follows:(8)p:xi,yi⟶li=li−1+1+a2xi−12xi−xi−1,where x0,alnx0⟶l0=0. And its inverse p−1 is(9)p−1:Xi⟶xi,yi=xi−1+xi−12xi−12+a2li−li−1,alnxi−1+xi−12xi−12+a2li−li−1,where l0⟶x0,alnx0=T0,alnT0.For example, Figure1 shows that an interval [0.5, 10] is projected onto a logarithmic curve by the Euler discretized inverse p−1 and the length of [0.5, 10] is equal to the length of the logarithmic curve.Figure 1
Projecting interval [0.5, 10] onto a logarithmic curve byp−1.According to the above example, it can be seen that the inversep−1 of the length-preserving projection p can be denoted by numerical method, such as Euler discretization method.


### 2.2. Biorthogonal Wavelet Analysis

In this section, we introduce multiresolution analysis ofL2R, the scaling function, wavelet function, and their dual function in wavelet theory.Definition 2.
(see [10, 15]). Assume that a multiresolution analysis generated by ϕ on L2R is an increasing sequence of closed subspace Vj∈L2R, satisfying the following conditions:(1)
Vj⊂Vj+1, ∀j∈Z.(2)
∩j∈ZVj=0, ∪j∈ZVj¯=L2R.(3)
ft∈Vj⇒f2t∈Vj+1.(4)
There exists a functionϕt∈L2R; the set ϕt−k,k∈Z is a Riesz basis of V0; that is, there exist two constants A and B such that, for an arbitrary sequence cjj∈N∈l2, there is(10)A∑k∈Zck2≤∑j∈Zckϕt−kL2R2≤B∑k∈Zck2,where Vj=closL2Rϕj,k=2j/2ϕ2jt−k:k∈Z and ϕj,k=2j/2ϕ2jt−k.Definition 3.
(see [10, 15]). Assume that Vj,ϕj∈Z and V˜j,ϕ˜j∈Z are two multiresolution analyses of L2R. If ϕj,k,ϕ˜j′,l=δj,j′δk,l, then Vj,ϕj∈Z and V˜j,ϕ˜j∈Z are called the mutual dual multiresolution analysis, and ϕ and ϕ˜ are called a pair of biorthogonal scaling functions.Definition 4.
(see [10, 15]). Assume that ψ,ψ˜∈L2R, ψj,k=2j/2ψ2jt−k,ψ˜j,k=2j/2ψ˜2jt−k. If ψj,kt|j,k∈Z, ψ˜j,kt|j,k∈Zare all the Riesz basis of L2R, and ψj,k,ψ˜j′,l=δj,j′δk,l; then ψj,kt|j,k∈Z, ψ˜j,kt|j,k∈Z are called a pair of biorthogonal wavelet basis. The functions ψ and ψ˜ are called mutual dual wavelet or a pair of biorthogonal wavelet functions.
Assume thatVj,ϕj∈Z and V˜j,ϕ˜j∈Z are the mutual dual multiresolution analysis of L2R; then ϕ and ϕ˜ satisfy the following two-scale equations:(11)ϕt=∑k∈Zhkϕ2t−k,ϕ˜t=∑k∈Zh˜kϕ˜2t−k,where the sequences hk and h˜k are called the two-scale sequences of ϕt and ϕ˜t, respectively.
Taking the Fourier transform on both sides of equation (11), we have(12)ϕ^ω=Hω2ϕ^ω2,ϕ˜^ω=H˜ω2ϕ˜^ω2,where Hω=1/2∑k∈Zhke−ikω and H˜ω=1/2∑k∈Zh˜ke−ikω are called the two-scale symbols of ϕt and ϕ˜t, respectively.
For the mutual dual waveletsψ and ψ˜, it is also obtained that(13)ψt=∑k∈Zgkϕ2t−k,ψ˜t=∑k∈Zg˜kϕ˜2t−kandψ^ω=Gω2ϕ^ω2,ψ˜^ω=G˜ω2ϕ˜^ω2,where the sequences gk and g˜k are called the two-scale sequences of ψt and ψ˜t, respectively. Gω=1/2∑k∈Zgke−ikω and G˜ω=1/2∑k∈Zg˜ke−ikω are called the two-scale symbols of ψt and ψ˜t, respectively.Lemma 1 (see [10, 15]).
Ifϕ and ϕ˜ are called a pair of biorthogonal scaling functions, and ψ and ψ˜ are a pair of biorthogonal wavelet functions corresponding to ϕ and ϕ˜, respectively, then the following statements are equivalent:(1)
ϕt,ϕ˜t−k=δ0,k,ϕt,ψ˜t−k=0,ϕ˜t,ψt−k=0,ψt,ψ˜t−k=δ0,k.(2)
∑k∈Zhkh˜¯k+2n=2δ0,n,∑k∈Zhkg˜¯k+2n=0,∑k∈Zh˜kg¯k+2n=0,∑k∈Zgkg˜¯k+2n=2δ0,n.(3)
HωH˜ω¯+Hω+πH˜ω+π¯=1,HωG˜ω¯+Hω+πG˜ω+π¯=0,H˜ωGω¯+H˜ω+πGω+π¯=0,GωG˜ω¯+Gω+πG˜ω+π¯=1.
For an arbitrary functionϕt, if ϕa+t=ϕa−t, ∀t∈R, then ϕt is symmetrical at t=a where a is a constant. If ϕa+t=−ϕa−t, ∀t∈R, then ϕt is antisymmetrical at t=a. If the Fourier transform ϕ^ω of ϕt satisfies ϕ^ω=±ϕ^ωe−iaω, then it is called that ϕt has linear phase. Moreover, if ϕ^ω=Bωe−iaω+b, where a, b are constants and Bω is a real-value function, then it is called that ϕt has generalized linear phase, where a is called phase of ϕ^ω.Lemma 2 (see [10, 15]).
Assume thatϕt∈L2R is a real-value function, and ϕ^ω is its Fourier transform. Then ϕt has generalized linear phase and a is phase of ϕ^ω if and only if ϕt is symmetrical or antisymmetrical at t=a.Lemma 3 (see [10, 15]).
Assume thatak∈l1 is a real-value sequence, and Aω its discrete Fourier transform. Then ak has generalized linear phase and a is phase of Aω if and only if ak is symmetrical or antisymmetrical at k0; that is, ak=a2k0−k.


## 2.1. Length-Preserving Projection via Euler Discretization Scheme

According to the algorithm of length-preserving projectionp on a smooth curve in [5, 14], the projection p is given in this section.Consider a log-regression curveC:(3)y=alnx,a∈R,x∈0,∞.By calculating, we have(4)Lx1=∫0x11+a2x2dx=x12+a2−alna+x12+a2x1.So(5)Lx=x2+a2−alna+x2+a2x=l,p:x,y⟶l,0=Lx,0.The inverse projectionp−1 is not explicit, but it could be determined by the numerical method.(6)x=L−1ly=alnL−1l.that is,p−1:l,0⟶L−1l,alnL−1l.Consider the Euler discretization method: for a finite intervalT0,T1, T0=x0<x1<⋯<xn=T1, then(7)1+a2x2dx=dl⟺1+a2xi2Δxi=Δli,where Δxi=xi+1−xi,Δli=li+1−li.The length-preserving projectionp can be denoted as follows:(8)p:xi,yi⟶li=li−1+1+a2xi−12xi−xi−1,where x0,alnx0⟶l0=0. And its inverse p−1 is(9)p−1:Xi⟶xi,yi=xi−1+xi−12xi−12+a2li−li−1,alnxi−1+xi−12xi−12+a2li−li−1,where l0⟶x0,alnx0=T0,alnT0.For example, Figure1 shows that an interval [0.5, 10] is projected onto a logarithmic curve by the Euler discretized inverse p−1 and the length of [0.5, 10] is equal to the length of the logarithmic curve.Figure 1
Projecting interval [0.5, 10] onto a logarithmic curve byp−1.According to the above example, it can be seen that the inversep−1 of the length-preserving projection p can be denoted by numerical method, such as Euler discretization method.


## 2.2. Biorthogonal Wavelet Analysis

In this section, we introduce multiresolution analysis ofL2R, the scaling function, wavelet function, and their dual function in wavelet theory.Definition 2.
(see [10, 15]). Assume that a multiresolution analysis generated by ϕ on L2R is an increasing sequence of closed subspace Vj∈L2R, satisfying the following conditions:(1)
Vj⊂Vj+1, ∀j∈Z.(2)
∩j∈ZVj=0, ∪j∈ZVj¯=L2R.(3)
ft∈Vj⇒f2t∈Vj+1.(4)
There exists a functionϕt∈L2R; the set ϕt−k,k∈Z is a Riesz basis of V0; that is, there exist two constants A and B such that, for an arbitrary sequence cjj∈N∈l2, there is(10)A∑k∈Zck2≤∑j∈Zckϕt−kL2R2≤B∑k∈Zck2,where Vj=closL2Rϕj,k=2j/2ϕ2jt−k:k∈Z and ϕj,k=2j/2ϕ2jt−k.Definition 3.
(see [10, 15]). Assume that Vj,ϕj∈Z and V˜j,ϕ˜j∈Z are two multiresolution analyses of L2R. If ϕj,k,ϕ˜j′,l=δj,j′δk,l, then Vj,ϕj∈Z and V˜j,ϕ˜j∈Z are called the mutual dual multiresolution analysis, and ϕ and ϕ˜ are called a pair of biorthogonal scaling functions.Definition 4.
(see [10, 15]). Assume that ψ,ψ˜∈L2R, ψj,k=2j/2ψ2jt−k,ψ˜j,k=2j/2ψ˜2jt−k. If ψj,kt|j,k∈Z, ψ˜j,kt|j,k∈Zare all the Riesz basis of L2R, and ψj,k,ψ˜j′,l=δj,j′δk,l; then ψj,kt|j,k∈Z, ψ˜j,kt|j,k∈Z are called a pair of biorthogonal wavelet basis. The functions ψ and ψ˜ are called mutual dual wavelet or a pair of biorthogonal wavelet functions.
Assume thatVj,ϕj∈Z and V˜j,ϕ˜j∈Z are the mutual dual multiresolution analysis of L2R; then ϕ and ϕ˜ satisfy the following two-scale equations:(11)ϕt=∑k∈Zhkϕ2t−k,ϕ˜t=∑k∈Zh˜kϕ˜2t−k,where the sequences hk and h˜k are called the two-scale sequences of ϕt and ϕ˜t, respectively.
Taking the Fourier transform on both sides of equation (11), we have(12)ϕ^ω=Hω2ϕ^ω2,ϕ˜^ω=H˜ω2ϕ˜^ω2,where Hω=1/2∑k∈Zhke−ikω and H˜ω=1/2∑k∈Zh˜ke−ikω are called the two-scale symbols of ϕt and ϕ˜t, respectively.
For the mutual dual waveletsψ and ψ˜, it is also obtained that(13)ψt=∑k∈Zgkϕ2t−k,ψ˜t=∑k∈Zg˜kϕ˜2t−kandψ^ω=Gω2ϕ^ω2,ψ˜^ω=G˜ω2ϕ˜^ω2,where the sequences gk and g˜k are called the two-scale sequences of ψt and ψ˜t, respectively. Gω=1/2∑k∈Zgke−ikω and G˜ω=1/2∑k∈Zg˜ke−ikω are called the two-scale symbols of ψt and ψ˜t, respectively.Lemma 1 (see [10, 15]).
Ifϕ and ϕ˜ are called a pair of biorthogonal scaling functions, and ψ and ψ˜ are a pair of biorthogonal wavelet functions corresponding to ϕ and ϕ˜, respectively, then the following statements are equivalent:(1)
ϕt,ϕ˜t−k=δ0,k,ϕt,ψ˜t−k=0,ϕ˜t,ψt−k=0,ψt,ψ˜t−k=δ0,k.(2)
∑k∈Zhkh˜¯k+2n=2δ0,n,∑k∈Zhkg˜¯k+2n=0,∑k∈Zh˜kg¯k+2n=0,∑k∈Zgkg˜¯k+2n=2δ0,n.(3)
HωH˜ω¯+Hω+πH˜ω+π¯=1,HωG˜ω¯+Hω+πG˜ω+π¯=0,H˜ωGω¯+H˜ω+πGω+π¯=0,GωG˜ω¯+Gω+πG˜ω+π¯=1.
For an arbitrary functionϕt, if ϕa+t=ϕa−t, ∀t∈R, then ϕt is symmetrical at t=a where a is a constant. If ϕa+t=−ϕa−t, ∀t∈R, then ϕt is antisymmetrical at t=a. If the Fourier transform ϕ^ω of ϕt satisfies ϕ^ω=±ϕ^ωe−iaω, then it is called that ϕt has linear phase. Moreover, if ϕ^ω=Bωe−iaω+b, where a, b are constants and Bω is a real-value function, then it is called that ϕt has generalized linear phase, where a is called phase of ϕ^ω.Lemma 2 (see [10, 15]).
Assume thatϕt∈L2R is a real-value function, and ϕ^ω is its Fourier transform. Then ϕt has generalized linear phase and a is phase of ϕ^ω if and only if ϕt is symmetrical or antisymmetrical at t=a.Lemma 3 (see [10, 15]).
Assume thatak∈l1 is a real-value sequence, and Aω its discrete Fourier transform. Then ak has generalized linear phase and a is phase of Aω if and only if ak is symmetrical or antisymmetrical at k0; that is, ak=a2k0−k.


## 3. Biorthogonal Wavelet on a Smooth CurveC

In this section, biorthogonal wavelet analysis on a smooth curveC is defined and discussed systematically. It mainly includes multiresolution analysis on a smooth curve C, biorthogonal wavelet functions on a smooth curve C, and so on.


### 3.1. Multiresolution Analysis on a Smooth CurveC

Assume that a smooth curveC satisfies parameter equation:(14)ξ=ξx,y=xt,yt,t∈R,where x and y are functions of parameter t. Assume that a projection p:x,y⟶X,0 is the length-preserving projection in Section 2.1 such that the length element dLξ of C is equal to the length element dX of R. And it is bijective, whose inverse is p−1:X,0⟶x,y.If the functionf˜,g˜∈L2C, we have the following equation (seen in [5, 9, 14]):(15)f˜,g˜L2C=f˜∘p−1,g˜∘p−1L2R.Analogously, iff,g∈L2R, it is obtained that(16)f,gL2R=f∘p,g∘pL2C.Based on equations (15) and (16), the following results can be established and it is easy to prove these results. The Fourier transform can be regarded as the inner product of function f˜∈L2C and Fourier basis.Definition 5.
(see [5, 9, 14]). The Fourier transform of f˜∈L2C can be defined as follows:(17)f˜^ϖ=∫Cf˜ηe−ipϖ⋅pηdLη=∫pCf˜p−1Xe−iω⋅XdX,where e−ipϖ⋅pη is induced by e−iω·X, pϖ=ω, and pη=X. And the corresponding inverse transform is(18)f˜η=12π∫Cf˜^ϖeipϖ⋅pηdLη=12π∫pCf˜^p−1ωe−iω⋅XdX.Lemma 4 (see [7, 9, 14]).
Assume thatJ is a countable set and fkk∈J∈L2R. For each k∈J, we define f˜kk∈J∈L2C as f˜k=fk∘p. Then, we have the following:(1)
Iffkk∈J is an orthogonal basis of L2R, then f˜kk∈J is also an orthogonal basis of L2C(2)
Iffkk∈J is a Riesz basis of L2R with Riesz constants A and B, then f˜kk∈J is a Riesz basis of L2C with the same Riesz constants(3)
Iffkk∈J is a frame of L2R with frame bounds A and B, then f˜kk∈J is a Riesz basis of L2C with the same frame boundsFor every functionf∈L2R, the induced function fC∈L2C can be defined as(19)fC=f∘p.According to equation (16), if the functions fj,kj,k are orthogonal, so are(20)fj,kC=fj,k∘p,∀j,k∈Z.Choosefj,k=ϕj,k and fj,k=ψj,k; we have the functions on L2C:(21)ϕj,kC=ϕj,k∘p,ψj,kC=ψj,k∘p.Thus, the multiresolution analysis ofL2C can be defined by multiresolution analysis of L2R and the induced function. For j∈Z, we define the space νj as(22)νj=fjC=fj∘p|fj∈Vj.It is immediate thatνj is a closed subspace sequence of L2C (seen in [5, 9, 14]). Thus, it is a Hilbert space. Moreover, these spaces have the following properties:(1)
νj⊂νj+1, ∀j∈Z.(2)
∩j∈Zνj=0, ∪j∈Zνj¯=L2C.(3)
fC∈νj⇒D2fC∈νj+1, where D2 is defined in Section 3.2.(4)
There exists a functionϕC∈L2C; the set TkϕC,k∈Z is a Riesz basis of V0; that is, there exist two constants AC and BC such that, for an arbitrary sequence cjCj∈N∈l2, there is(23)AC∑k∈ZckC2≤∑j∈ZckCTkϕCL2R2≤BC∑k∈ZckC2,where νj=closL2Cϕj,kC:k∈Z.Definition 6.
(see [5, 9, 14]). If a sequence of subspaces νj of L2C satisfy the above properties, it is called a multiresolution analysis of L2C.
Once the multiresolution analysis is determined, the wavelet subspaceswj may be constructed in usual way. For every j∈Z, assume that wj denotes an orthogonal complement of coarse space νj into νj+1 such that νj+1=νj⊕wj. It is easy to prove that, for each j∈Z, ψj,kC,k∈Z is an orthogonal basis of wj and so ψj,kC,j∈Z,k∈Z is an orthogonal basis of ⊕j∈Zwj¯=L2C. We can define ψj,kC as ψj,kC=ψj,k°p, then ϕC is called the scaling function on the regression trend curve C, and ψC is called the corresponding wavelets on the regression trend curve C.


### 3.2. Scaling Function on a Logarithmic CurveC

In this section, we discuss the scaling function and wavelet on a logarithmic curveC. Firstly, the translation operator and dilation operator in the space L2C can be defined as follows (seen in [1, 5, 9, 14]):(1)
Translation operatorTb: Tbf˜η=f˜∘p−1pη−b,b∈R(2)
Dilation operatorDa: Daf˜η=f˜∘p−1a⋅pη,a>0wheref˜∈L2C. Combining Tb and Da, a unitary operator Ub,a=DaTb can be denoted by(24)Ub,af˜η=f˜°p−1a⋅pη−b.According to the induced scaling function, translation operatorTb, and dilation operator Da, the two-scale equation of the scaling function ϕC in L2C, which can be induced by ϕ in L2R, can be deduced as follows:(25)ϕℂη=∑k∈ZhkℂUk,2ϕℂη=∑k∈Zhkℂϕℂ∘p−12⋅pη−k,where a sequence hkC is called the two-scale sequence of ϕCη. This equation is also called the refinable equation of ϕCη.Based on the length-preserving projectionp, the two-scale sequence hkC of ϕCη satisfies the following conclusion.Lemma 5 (see [1, 5, 9, 14]).
Assume thatp is a length-preserving projection from the regression trend curve C to a subset of real axis R. If ϕCis the scaling function in L2C, which is induced from ϕ∈L2R by p, then the sequences hk and hkC in equations (11) and (25) satisfy hkC=hk.
Taking the Fourier transform on both sides of equation (25), we have(26)ϕ^Cϖ=HCpϖ2ϕC∘p−1^pϖ2,where HCpϖ/2=1/2∑k∈ZhkCe−ipϖ·k/2 is called the two-scale symbol of ϕCη.
By utilizing equation (11) repeatedly, it is obtained that(27)ϕ^Cϖ=∏j=1∞HCpϖ2jϕC°p−1^0.
The convergence of the infinite product∏j=1∞HCpϖ/2j is concluded in the following lemma.Lemma 6 (see [1, 5, 9, 14]).
Assume thatp is a length-preserving projection from the regression trend curve C to a subset of real axis R. The infinite product ∏j=1∞HCpϖ/2j is uniformly convergent to a continuous function ϕ^Cϖ on compact support if and only if HC0=1 and HCpϖ/22+HCpϖ/2+π2=1.


### 3.3. Biorthogonal Wavelet on a Logarithmic CurveC

Definition 7.
Assume thatυj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are two multiresolution analysis of L2C. If ϕj,kC,ϕ˜j,lCL2C=δj,j′δk,l, then υj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are called the mutual dual multiresolution analysis. ϕ˜C and ϕ˜C are called a pair of biorthogonal scaling functions.Definition 8.
Assume thatψC,ψ˜C∈L2C, ψj,kC=ψj,k∘p, ψ˜j,kC=ψ˜j,k∘p. If ψj,kC|j,k∈Z, ψ˜j,kC|j,k∈Z are all the Riesz basis of L2C, and ψj,kC,ψ˜j,lC=δj,j′δk,l, then ψj,kC|j,k∈Z, ψ˜j,kC|j,k∈Z are called a pair of biorthogonal wavelet basis. The functions ψC and ψ˜C are called mutual dual wavelet or a pair of biorthogonal wavelet functions.Definition 9.
Assume thatϕC and ϕ˜C are two scaling functions of L2C, and ψC and ψ˜C are two wavelet functions corresponding to ϕC and ϕ˜C, respectively. If(28)TkϕC,Tlϕ˜CL2C=δk,l,TkϕC,Tlψ˜CL2C=0,Tkϕ˜C,TlψCL2C=0,TkψC,Tlψ˜CL2C=δk,l,then ϕC and ϕ˜C are called a pair of biorthogonal scaling functions, and ψC and ψ˜C are a pair of biorthogonal wavelet functions corresponding to ϕC and ϕ˜C, respectively.
Assume thatυj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are the mutual dual multiresolution analysis of L2C; then ϕ˜C and ϕ˜C satisfy the following two-scale equations:(29)ϕCη=∑k∈ZhkCUk,2ϕCη,ϕ˜Cη=∑k∈Zh˜kCUk,2ϕ˜Cη,or(30)ϕCη=∑k∈ZhkCϕC∘p−12⋅pη−k,ϕ˜Cη=∑k∈Zh˜kℂϕ˜C∘p−12⋅pη−k,where the sequences hkC and h˜kC are called the two-scale sequences of ϕ˜C and ϕ˜C, respectively.
Taking the Fourier transform on both sides of equation (29), we have(31)ϕ^Cϖ=HCpϖ2ϕC∘p−1^pϖ2,ϕ˜^Cϖ=HCpϖ2ϕ˜C∘p−1^pϖ2,where HCpϖ/2=1/2∑k∈ZhkCe−ipϖ⋅k/2 and H˜Cpϖ/2=1/2∑k∈Zh˜kCe−ipϖ⋅k/2 are called the two-scale symbols of ϕ˜C and ϕ˜C, respectively.
For the mutual dual waveletsψC and ψ˜C, it is also obtained that(32)ψCη=∑k∈ZgkCUk,2ϕCη,ψ˜Cη=∑k∈Zg˜kCUk,2ϕCη,orψCη=∑k∈ZgkCϕC∘p−12⋅pη−k,ψ˜Cη=∑k∈Zg˜kCϕ˜C∘p−12⋅pη−k,ψ^Cϖ=GCpϖ2ϕC∘p−1^pϖ2,ψ˜^Cϖ=G˜Cpϖ2ϕ˜C∘p−1^pϖ2,where the sequences gkC and g˜kC are called the two-scale sequences of ψC and ψ˜C, respectively. GCpϖ/2=1/2∑k∈ZgkCe−ipϖ⋅k/2 and G˜Cpϖ/2=1/2∑k∈Zg˜kCe−ipϖ⋅k/2 are called the two-scale symbols of ψC and ψ˜C, respectively.Theorem 1.
IfϕC and ϕ˜C are a pair of biorthogonal scaling functions, and ψC and ψ˜C are a pair of biorthogonal wavelet functions corresponding to ϕC and ϕ˜C, respectively, then the following statements are equivalent:(1)
TkϕC,Tlϕ˜CL2C=δk,l,TkϕC,Tlψ˜CL2C=0,Tkϕ˜C,TlψCL2C=0,TkψC,Tlψ˜CL2C=δk,l.(2)
∑k∈ZhkCh˜¯k+2nC=2δ0,n,∑k∈ZhkCg˜¯k+2nC=0,∑k∈Zh˜kCg¯k+2nC=0,∑k∈ZgkCg˜¯k+2nC=2δ0,n.(3)
HCpϖ/2H˜Cpϖ/2¯+HCpϖ/2+πH˜Cpϖ/2+π¯=1,HCpϖ/2G˜Cpϖ/2¯+HCpϖ/2+πG˜Cpϖ/2+π¯=0,H˜Cpϖ/2GCpϖ/2¯+H˜Cpϖ/2+πGCpϖ/2+π¯=0,GCpϖ/2G˜Cpϖ/2¯+GCpϖ/2+πG˜Cpϖ/2+π¯=1.Proof.
We only need to prove one equation of them. The proof of others is similar.1⇒2(33)δ0,k⟺ϕCη,Tkϕ˜Cη=∫CϕCηTkϕ˜Cη¯dLη=∫CϕCηϕ˜C∘p−1pη−k¯dLη=∫pCϕCp−1Xϕ˜C∘p−1pp−1X−k¯dX=∫pCϕXϕ˜X−k¯dX=12∑l∈Zhlh˜k+2n¯=12∑l∈ZhlCh˜k+2nC¯.whereϕC∘p−1=ϕ, pη=X.2⟺3 According to Lemmas 5 and 6,(34)1=HCpϖ2H˜Cpϖ2¯+HCpϖ2+πH˜Cpϖ2+π¯=12∑k∈ZhkCe−ikpϖ/212∑k∈Zh˜kCe−ikpϖ/2¯+12∑k∈ZhkCe−ikpϖ/2+π12∑k∈Zh˜kCe−ikpϖ/2+π¯=12∑k∈Zhke−ikω/212∑k∈Zh˜ke−ikω/2¯+12∑k∈Zhke−ikω/2+π12∑k∈Zh˜ke−ikω/2+π¯=Hω2H˜ω2¯+Hω2+πH˜ω2+π¯,where pϖ=ω,hkC=hk,h˜kC=h˜k.
According to Lemma1, we have 2⟺3.Definition 10.
For an arbitrary functionϕC, if(35)T−pαϕC=U−pα,−1ϕC,∀α∈C,then ϕC is symmetrical at α, where α is a fixed point. If(36)T−pαϕC=−U−pα,−1ϕC,∀α∈C,then ϕC is antisymmetrical at α.Definition 11.
If the Fourier transformϕ^C of ϕC satisfies(37)ϕ^Cϖ=±ϕ^Cϖe−ipα⋅pϖ,then it is called that ϕC has linear phase. Moreover, if ϕ^Cϖ=Bϖe−ipα⋅pϖ+pβ, where α,β are fixed points and Bϖ is a real-value function, then it is called that ϕC has generalized linear phase, where α is called phase of ϕ^C.Theorem 2.
Assume thatϕC∈L2C is a real-value function, and ϕ^C is its Fourier transform. Then ϕC has generalized linear phase and α is phase of ϕ^C if and only if ϕC is symmetrical or antisymmetrical at α.Proof.
Assume thatϕC has generalized linear phase and α is phase of ϕ^C. Then the inverse Fourier transform of ϕ^C can be written as follows:(38)ϕCη=12π∫CBϖe−ipα⋅pϖ+pβeipϖ⋅pηdLη,eipβU−pα,−1ϕCη=12π∫CBϖe−ipη⋅pϖdLη.
Taking the complex conjugation on both sides of the above equation andBϖ being real-value, we have(39)eipβU−pα,−1ϕℂη¯=12π∫CBϖeipη⋅pϖdLη=12π∫Cϕ^ℂϖeipα⋅pϖ+pβeipη⋅pϖdLη=eipβT−pαϕℂη.
On the other side, if(40)eipβT−pαϕCη=eipβU−pα,−1ϕCη¯,by taking the Fourier transform on its both sides, we have(41)eipβϕ^ℂϖeipα⋅pϖ=e−ipβ∫ℂU−pα,−1ϕℂη¯e−ipη⋅pϖdLη=eipβϕ^ℂϖeipα⋅pϖ¯.
LetBϖ=eipβϕ^Cηeipα⋅pϖ, then Bϖ is real-value, and ϕC has generalized linear phase, where α is phase of ϕ^C.
So, ifϕC∈L2C is a real-value function, then ei2pβ=±1 and(42)T−pαϕC=U−pα,−1ϕCorTpαϕC=−U−pα,−1ϕC.
That is,ϕC is symmetrical or antisymmetrical at α.Theorem 3.
Assume thathkC∈l2 is a two-scale sequence of ϕC, and HCpϖ/2 is defined in its discrete Fourier transform. Then hkC has generalized linear phase and k0 is phase of HCpϖ/2 if and only if hkC is symmetrical or antisymmetrical at k0; that is, hkC=h2k0−kC.Proof.
According to Lemma3, Theorem 1, and the length-preserving projection p, it is shown obviously.


## 3.1. Multiresolution Analysis on a Smooth CurveC

Assume that a smooth curveC satisfies parameter equation:(14)ξ=ξx,y=xt,yt,t∈R,where x and y are functions of parameter t. Assume that a projection p:x,y⟶X,0 is the length-preserving projection in Section 2.1 such that the length element dLξ of C is equal to the length element dX of R. And it is bijective, whose inverse is p−1:X,0⟶x,y.If the functionf˜,g˜∈L2C, we have the following equation (seen in [5, 9, 14]):(15)f˜,g˜L2C=f˜∘p−1,g˜∘p−1L2R.Analogously, iff,g∈L2R, it is obtained that(16)f,gL2R=f∘p,g∘pL2C.Based on equations (15) and (16), the following results can be established and it is easy to prove these results. The Fourier transform can be regarded as the inner product of function f˜∈L2C and Fourier basis.Definition 5.
(see [5, 9, 14]). The Fourier transform of f˜∈L2C can be defined as follows:(17)f˜^ϖ=∫Cf˜ηe−ipϖ⋅pηdLη=∫pCf˜p−1Xe−iω⋅XdX,where e−ipϖ⋅pη is induced by e−iω·X, pϖ=ω, and pη=X. And the corresponding inverse transform is(18)f˜η=12π∫Cf˜^ϖeipϖ⋅pηdLη=12π∫pCf˜^p−1ωe−iω⋅XdX.Lemma 4 (see [7, 9, 14]).
Assume thatJ is a countable set and fkk∈J∈L2R. For each k∈J, we define f˜kk∈J∈L2C as f˜k=fk∘p. Then, we have the following:(1)
Iffkk∈J is an orthogonal basis of L2R, then f˜kk∈J is also an orthogonal basis of L2C(2)
Iffkk∈J is a Riesz basis of L2R with Riesz constants A and B, then f˜kk∈J is a Riesz basis of L2C with the same Riesz constants(3)
Iffkk∈J is a frame of L2R with frame bounds A and B, then f˜kk∈J is a Riesz basis of L2C with the same frame boundsFor every functionf∈L2R, the induced function fC∈L2C can be defined as(19)fC=f∘p.According to equation (16), if the functions fj,kj,k are orthogonal, so are(20)fj,kC=fj,k∘p,∀j,k∈Z.Choosefj,k=ϕj,k and fj,k=ψj,k; we have the functions on L2C:(21)ϕj,kC=ϕj,k∘p,ψj,kC=ψj,k∘p.Thus, the multiresolution analysis ofL2C can be defined by multiresolution analysis of L2R and the induced function. For j∈Z, we define the space νj as(22)νj=fjC=fj∘p|fj∈Vj.It is immediate thatνj is a closed subspace sequence of L2C (seen in [5, 9, 14]). Thus, it is a Hilbert space. Moreover, these spaces have the following properties:(1)
νj⊂νj+1, ∀j∈Z.(2)
∩j∈Zνj=0, ∪j∈Zνj¯=L2C.(3)
fC∈νj⇒D2fC∈νj+1, where D2 is defined in Section 3.2.(4)
There exists a functionϕC∈L2C; the set TkϕC,k∈Z is a Riesz basis of V0; that is, there exist two constants AC and BC such that, for an arbitrary sequence cjCj∈N∈l2, there is(23)AC∑k∈ZckC2≤∑j∈ZckCTkϕCL2R2≤BC∑k∈ZckC2,where νj=closL2Cϕj,kC:k∈Z.Definition 6.
(see [5, 9, 14]). If a sequence of subspaces νj of L2C satisfy the above properties, it is called a multiresolution analysis of L2C.
Once the multiresolution analysis is determined, the wavelet subspaceswj may be constructed in usual way. For every j∈Z, assume that wj denotes an orthogonal complement of coarse space νj into νj+1 such that νj+1=νj⊕wj. It is easy to prove that, for each j∈Z, ψj,kC,k∈Z is an orthogonal basis of wj and so ψj,kC,j∈Z,k∈Z is an orthogonal basis of ⊕j∈Zwj¯=L2C. We can define ψj,kC as ψj,kC=ψj,k°p, then ϕC is called the scaling function on the regression trend curve C, and ψC is called the corresponding wavelets on the regression trend curve C.


## 3.2. Scaling Function on a Logarithmic CurveC

In this section, we discuss the scaling function and wavelet on a logarithmic curveC. Firstly, the translation operator and dilation operator in the space L2C can be defined as follows (seen in [1, 5, 9, 14]):(1)
Translation operatorTb: Tbf˜η=f˜∘p−1pη−b,b∈R(2)
Dilation operatorDa: Daf˜η=f˜∘p−1a⋅pη,a>0wheref˜∈L2C. Combining Tb and Da, a unitary operator Ub,a=DaTb can be denoted by(24)Ub,af˜η=f˜°p−1a⋅pη−b.According to the induced scaling function, translation operatorTb, and dilation operator Da, the two-scale equation of the scaling function ϕC in L2C, which can be induced by ϕ in L2R, can be deduced as follows:(25)ϕℂη=∑k∈ZhkℂUk,2ϕℂη=∑k∈Zhkℂϕℂ∘p−12⋅pη−k,where a sequence hkC is called the two-scale sequence of ϕCη. This equation is also called the refinable equation of ϕCη.Based on the length-preserving projectionp, the two-scale sequence hkC of ϕCη satisfies the following conclusion.Lemma 5 (see [1, 5, 9, 14]).
Assume thatp is a length-preserving projection from the regression trend curve C to a subset of real axis R. If ϕCis the scaling function in L2C, which is induced from ϕ∈L2R by p, then the sequences hk and hkC in equations (11) and (25) satisfy hkC=hk.
Taking the Fourier transform on both sides of equation (25), we have(26)ϕ^Cϖ=HCpϖ2ϕC∘p−1^pϖ2,where HCpϖ/2=1/2∑k∈ZhkCe−ipϖ·k/2 is called the two-scale symbol of ϕCη.
By utilizing equation (11) repeatedly, it is obtained that(27)ϕ^Cϖ=∏j=1∞HCpϖ2jϕC°p−1^0.
The convergence of the infinite product∏j=1∞HCpϖ/2j is concluded in the following lemma.Lemma 6 (see [1, 5, 9, 14]).
Assume thatp is a length-preserving projection from the regression trend curve C to a subset of real axis R. The infinite product ∏j=1∞HCpϖ/2j is uniformly convergent to a continuous function ϕ^Cϖ on compact support if and only if HC0=1 and HCpϖ/22+HCpϖ/2+π2=1.


## 3.3. Biorthogonal Wavelet on a Logarithmic CurveC

Definition 7.
Assume thatυj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are two multiresolution analysis of L2C. If ϕj,kC,ϕ˜j,lCL2C=δj,j′δk,l, then υj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are called the mutual dual multiresolution analysis. ϕ˜C and ϕ˜C are called a pair of biorthogonal scaling functions.Definition 8.
Assume thatψC,ψ˜C∈L2C, ψj,kC=ψj,k∘p, ψ˜j,kC=ψ˜j,k∘p. If ψj,kC|j,k∈Z, ψ˜j,kC|j,k∈Z are all the Riesz basis of L2C, and ψj,kC,ψ˜j,lC=δj,j′δk,l, then ψj,kC|j,k∈Z, ψ˜j,kC|j,k∈Z are called a pair of biorthogonal wavelet basis. The functions ψC and ψ˜C are called mutual dual wavelet or a pair of biorthogonal wavelet functions.Definition 9.
Assume thatϕC and ϕ˜C are two scaling functions of L2C, and ψC and ψ˜C are two wavelet functions corresponding to ϕC and ϕ˜C, respectively. If(28)TkϕC,Tlϕ˜CL2C=δk,l,TkϕC,Tlψ˜CL2C=0,Tkϕ˜C,TlψCL2C=0,TkψC,Tlψ˜CL2C=δk,l,then ϕC and ϕ˜C are called a pair of biorthogonal scaling functions, and ψC and ψ˜C are a pair of biorthogonal wavelet functions corresponding to ϕC and ϕ˜C, respectively.
Assume thatυj,ϕCj∈Z and υ˜j,ϕ˜Cj∈Z are the mutual dual multiresolution analysis of L2C; then ϕ˜C and ϕ˜C satisfy the following two-scale equations:(29)ϕCη=∑k∈ZhkCUk,2ϕCη,ϕ˜Cη=∑k∈Zh˜kCUk,2ϕ˜Cη,or(30)ϕCη=∑k∈ZhkCϕC∘p−12⋅pη−k,ϕ˜Cη=∑k∈Zh˜kℂϕ˜C∘p−12⋅pη−k,where the sequences hkC and h˜kC are called the two-scale sequences of ϕ˜C and ϕ˜C, respectively.
Taking the Fourier transform on both sides of equation (29), we have(31)ϕ^Cϖ=HCpϖ2ϕC∘p−1^pϖ2,ϕ˜^Cϖ=HCpϖ2ϕ˜C∘p−1^pϖ2,where HCpϖ/2=1/2∑k∈ZhkCe−ipϖ⋅k/2 and H˜Cpϖ/2=1/2∑k∈Zh˜kCe−ipϖ⋅k/2 are called the two-scale symbols of ϕ˜C and ϕ˜C, respectively.
For the mutual dual waveletsψC and ψ˜C, it is also obtained that(32)ψCη=∑k∈ZgkCUk,2ϕCη,ψ˜Cη=∑k∈Zg˜kCUk,2ϕCη,orψCη=∑k∈ZgkCϕC∘p−12⋅pη−k,ψ˜Cη=∑k∈Zg˜kCϕ˜C∘p−12⋅pη−k,ψ^Cϖ=GCpϖ2ϕC∘p−1^pϖ2,ψ˜^Cϖ=G˜Cpϖ2ϕ˜C∘p−1^pϖ2,where the sequences gkC and g˜kC are called the two-scale sequences of ψC and ψ˜C, respectively. GCpϖ/2=1/2∑k∈ZgkCe−ipϖ⋅k/2 and G˜Cpϖ/2=1/2∑k∈Zg˜kCe−ipϖ⋅k/2 are called the two-scale symbols of ψC and ψ˜C, respectively.Theorem 1.
IfϕC and ϕ˜C are a pair of biorthogonal scaling functions, and ψC and ψ˜C are a pair of biorthogonal wavelet functions corresponding to ϕC and ϕ˜C, respectively, then the following statements are equivalent:(1)
TkϕC,Tlϕ˜CL2C=δk,l,TkϕC,Tlψ˜CL2C=0,Tkϕ˜C,TlψCL2C=0,TkψC,Tlψ˜CL2C=δk,l.(2)
∑k∈ZhkCh˜¯k+2nC=2δ0,n,∑k∈ZhkCg˜¯k+2nC=0,∑k∈Zh˜kCg¯k+2nC=0,∑k∈ZgkCg˜¯k+2nC=2δ0,n.(3)
HCpϖ/2H˜Cpϖ/2¯+HCpϖ/2+πH˜Cpϖ/2+π¯=1,HCpϖ/2G˜Cpϖ/2¯+HCpϖ/2+πG˜Cpϖ/2+π¯=0,H˜Cpϖ/2GCpϖ/2¯+H˜Cpϖ/2+πGCpϖ/2+π¯=0,GCpϖ/2G˜Cpϖ/2¯+GCpϖ/2+πG˜Cpϖ/2+π¯=1.Proof.
We only need to prove one equation of them. The proof of others is similar.1⇒2(33)δ0,k⟺ϕCη,Tkϕ˜Cη=∫CϕCηTkϕ˜Cη¯dLη=∫CϕCηϕ˜C∘p−1pη−k¯dLη=∫pCϕCp−1Xϕ˜C∘p−1pp−1X−k¯dX=∫pCϕXϕ˜X−k¯dX=12∑l∈Zhlh˜k+2n¯=12∑l∈ZhlCh˜k+2nC¯.whereϕC∘p−1=ϕ, pη=X.2⟺3 According to Lemmas 5 and 6,(34)1=HCpϖ2H˜Cpϖ2¯+HCpϖ2+πH˜Cpϖ2+π¯=12∑k∈ZhkCe−ikpϖ/212∑k∈Zh˜kCe−ikpϖ/2¯+12∑k∈ZhkCe−ikpϖ/2+π12∑k∈Zh˜kCe−ikpϖ/2+π¯=12∑k∈Zhke−ikω/212∑k∈Zh˜ke−ikω/2¯+12∑k∈Zhke−ikω/2+π12∑k∈Zh˜ke−ikω/2+π¯=Hω2H˜ω2¯+Hω2+πH˜ω2+π¯,where pϖ=ω,hkC=hk,h˜kC=h˜k.
According to Lemma1, we have 2⟺3.Definition 10.
For an arbitrary functionϕC, if(35)T−pαϕC=U−pα,−1ϕC,∀α∈C,then ϕC is symmetrical at α, where α is a fixed point. If(36)T−pαϕC=−U−pα,−1ϕC,∀α∈C,then ϕC is antisymmetrical at α.Definition 11.
If the Fourier transformϕ^C of ϕC satisfies(37)ϕ^Cϖ=±ϕ^Cϖe−ipα⋅pϖ,then it is called that ϕC has linear phase. Moreover, if ϕ^Cϖ=Bϖe−ipα⋅pϖ+pβ, where α,β are fixed points and Bϖ is a real-value function, then it is called that ϕC has generalized linear phase, where α is called phase of ϕ^C.Theorem 2.
Assume thatϕC∈L2C is a real-value function, and ϕ^C is its Fourier transform. Then ϕC has generalized linear phase and α is phase of ϕ^C if and only if ϕC is symmetrical or antisymmetrical at α.Proof.
Assume thatϕC has generalized linear phase and α is phase of ϕ^C. Then the inverse Fourier transform of ϕ^C can be written as follows:(38)ϕCη=12π∫CBϖe−ipα⋅pϖ+pβeipϖ⋅pηdLη,eipβU−pα,−1ϕCη=12π∫CBϖe−ipη⋅pϖdLη.
Taking the complex conjugation on both sides of the above equation andBϖ being real-value, we have(39)eipβU−pα,−1ϕℂη¯=12π∫CBϖeipη⋅pϖdLη=12π∫Cϕ^ℂϖeipα⋅pϖ+pβeipη⋅pϖdLη=eipβT−pαϕℂη.
On the other side, if(40)eipβT−pαϕCη=eipβU−pα,−1ϕCη¯,by taking the Fourier transform on its both sides, we have(41)eipβϕ^ℂϖeipα⋅pϖ=e−ipβ∫ℂU−pα,−1ϕℂη¯e−ipη⋅pϖdLη=eipβϕ^ℂϖeipα⋅pϖ¯.
LetBϖ=eipβϕ^Cηeipα⋅pϖ, then Bϖ is real-value, and ϕC has generalized linear phase, where α is phase of ϕ^C.
So, ifϕC∈L2C is a real-value function, then ei2pβ=±1 and(42)T−pαϕC=U−pα,−1ϕCorTpαϕC=−U−pα,−1ϕC.
That is,ϕC is symmetrical or antisymmetrical at α.Theorem 3.
Assume thathkC∈l2 is a two-scale sequence of ϕC, and HCpϖ/2 is defined in its discrete Fourier transform. Then hkC has generalized linear phase and k0 is phase of HCpϖ/2 if and only if hkC is symmetrical or antisymmetrical at k0; that is, hkC=h2k0−kC.Proof.
According to Lemma3, Theorem 1, and the length-preserving projection p, it is shown obviously.


## 4. Example and Figure

Consider the biorthogonal wavelet “bior2.4” in [10, 15], and the relevant data can be obtained in MATLAB. The scaling function of “bior2.4” satisfies the following two-scale equation:(43)ϕx=364ϕ2x+4−332ϕ2x+3−14ϕ2x+2+1932ϕ2x+1+4532ϕ2x+1932ϕ2x−1−14ϕ2x−1−332ϕ2x−3+364ϕ2x−4.And the dual scaling function satisfies(44)ϕ˜t=12ϕ˜2t+1+ϕ˜2t+12ϕ˜2t−1.According to the length-preserving projectionp, the scaling function ϕt and the dual ϕ˜t of biorthogonal wavelet “bior2.4” can be induced onto a regression curve C. And the scaling function ϕC and the dual ϕ˜C can be generated by p. According to Lemma 5, the corresponding two-scale sequences hkC and h˜kC can be obtained as follows:(45)h−4C=364,h−3C=−332,h−2C=−14,h−1C=1932,h0C=4532,h1C=1932,h2C=−14,h3C=−332,h4C=364,h˜−1C=12,h˜0C=1,h˜1C=12.According to equation (25),(46)ϕCη=364ϕC∘p−12⋅pη+4−332ϕC∘p−12⋅pη+3−14ϕC∘p−12⋅pη+2+1932ϕC∘p−12⋅pη+1+4532ϕC∘p−12⋅pη+1932ϕC∘p−12⋅pη−1−14ϕC∘p−12⋅pη−2−332ϕC∘p−12⋅pη−3+364ϕC∘p−12⋅pη−4,ϕ˜Cη=12ϕ˜C∘p−12⋅pη+1+ϕ˜C∘p−12⋅pη+12ϕ˜C∘p−12⋅pη−1.In order to show the figures of scaling functionϕC and the dual ϕ˜C, we choose a logarithmic curve C, such as y=2lnx. According to the length-preserving projection in Section 2.1 and biorthogonal wavelet theorem on C, the scaling function ϕC and its dual can be shown in Figures 2 and 3.Figure 2
Figures of scaling functionϕC. (a) Scaling function phi of bior2.4 and (b) scaling function phiC on a logarithmic curve.
(a)(b)Figure 3
Figures of dual scaling functionϕ˜C. (a) Scaling function phi∼ of bior2.4 and (b) scaling function phi∼C on a logarithmic curve.
(a)(b)Similarly, the wavelet functionψC and the dual ψ˜C can also be induced and generated by the wavelet function ψ and the dual ψ˜ of biorthogonal wavelet “bior2.4.”


## 5. Application in Financial Data

Since the birth of bitcoin on January 3, 2009, it has become a hot issue in economy and finance. In this section, according to multiresolution analysis on a logarithmic curveC, we can decompose and reconstruct Bitcoin transaction data. The candlestick chart of bitcoin transaction data (BTC/USD) can be derived from the bitcoin transaction website (https://cn.investing.com/crypto/bitcoin/historical-data). By observing candlestick chart in Figure 4, the bitcoin transaction data (BTC/USD) shows a significant logarithmic growth trend curve.Figure 4
Candlestick chart of bitcoin daily transaction data.In detail, we choose daily close data of bitcoin transaction data (denoted by SP:USD) from April 30, 2017, to December 1, 2020. The historical data is derived from the bitcoin transaction website (https://cn.investing.com/crypto/bitcoin/historical-data), and the number of samples is 1312. For convenience, the figure of raw data (denoted by SP) and its graph is given by a solid curve in Figure 5. By observing the graph, the daily data shows a significant logarithmic growth trend. In Figure 5, the period is from April 30, 2017, to December 1, 2020, and the dotted logarithmic curve indicates its growth trend.Figure 5
Figure of bitcoin daily transaction close data and its trend curveC.The logarithmic trend curveC of raw data SP and time T can be estimated by the ordinary least squares estimation. The equation of logarithmic trend curve can be obtained as follows:(47)SP=−3.1345×103+1.7572×103×lnT.According to biorthogonal wavelet “bior2.4” lifted onto the logarithmic curveC in equation (47), the raw data SP can be decomposed to scale 4 on a scale-by-scale basis by decomposition and reconstruction algorithm. In Figure 6(b), the low-frequency a4 captures the approximate data of SP on the logarithmic trend curve C in a 3-dimension space, compared to traditional approximate data in Figure 6(a). It provides more information, including logarithmic growth trend curve of Y-axis and wavelet approximate data of Z-axis. So, this can be interpreted as “wavelet approximate data on logarithmic growth trend curve.” High-frequency data d1, d2, d3, and d4 of the raw data SP are shown in Figures 7–10. In Figure 7(a), the first high-frequency data d1 of SP is obtained by the tradition wavelet “bior2.4.” In Figure 7(b), the first high-frequency data d1 captures volatility with a period length of 2 to 4 days on logarithmic trend curve C in a 3-dimension space. The 3-dimension space is composed of time as X-axis, logarithmic trend curve as Y-axis, and data d1 as Z-axis. It provides more information, including regression growth trend of Y-axis and volatility of Z-axis. So, this can be interpreted as “volatility with logarithmic growth trend.” In Figure 7(c), d1 is turned to XOY plane. In Figure 8(a), the second high-frequency data d2 of SP is obtained by the tradition wavelet “bior2.4.” In Figure 8(b), the second high-frequency data d2 captures volatility with a period length of 4 to 8 days on logarithmic trend curve C in 3-dimension space. In Figure 8(c), d2 is turned to XOY plane. In Figure 9, the third high-frequency data d3 captures volatility with a period length of 8 to 16 days on logarithmic trend curve C. The interpretation is similar to Figure 7. Moreover, the fourth high-frequency data d4 captures volatility with a period length of 16 to 32 days on logarithmic trend curve C in Figure 10. These are multiresolution analysis of the data SP on logarithmic trend curve C. The blue line indicates low-frequency data and high-frequency data d1, d2, d3, and d4. And the red line indicates its logarithmic trend curve.Figure 6
Low-frequency a4 of SP. (a) Low-frequency a4 and (b) approximate data a4 of SP on logarithmic curveC.
(a)(b)Figure 7
High-frequency datad1 of SP. (a) High-frequency data d1, (b) high-frequency data d1 on logarithmic trend curve C, and (c) turning high-frequency data d1 on C to XOY.
(a)(b)(c)Figure 8
High-frequency datad2 of SP. (a) High-frequency data d2, (b) high-frequency data d2 on logarithmic trend curve C, and (c) turning high-frequency data d2 on C to XOY.
(a)(b)(c)Figure 9
High-frequency datad3 of SP. (a) High-frequency data d3, (b) high-frequency data d3 on logarithmic trend curve C, and (c) turning high-frequency data d3 on C to XOY.
(a)(b)(c)Figure 10
High-frequency datad4 of SP. (a) High-frequency data d4, (b) high-frequency data d4 on logarithmic trend curve C, and (c) turning high-frequency data d4 on C to XOY.
(a)(b)(c)By reconstruction algorithm and length-preserving projection, we reconstruct the low-frequency data a4, high-frequency datad1, and high-frequency data d2, d3, and d4 to obtain the reconstructed data (shown in Figure 11). Compared to the original data, reconstructed data and error data are shown in Figure 11 by traditional “bior2.4” wavelet. A 3-dimension space is composed by time as X-axis, logarithmic trend curve as Y-axis, and data as Z-axis. In Figure 12, the original data, reconstructed data, and error data are shown in the 3-dimension space, by “bior2.4” wavelet on logarithmic trend curve C. It provides more information, including logarithmic growth trend of Y-axis, reconstructed data of Z-axis, and error data of Z-axis. So, this can be interpreted as “error with logarithmic growth trend curve.” The blue line indicates original data, reconstructed data, and error data. The red line indicates its logarithmic growth trend curve.Figure 11
Reconstructed data of SP. (a) Original data, (b) reconstructed data, and (c) error data.
(a)(b)(c)Figure 12
Reconstructed data of SP with logarithmic growth trend. (a) Original data on logarithmic trend curve, (b) reconstructed data on logarithmic trend curve, and (c) error data on logarithmic trend curve.
(a)(b)(c)


## 6. Conclusion and Discussion

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet filters can be lifted onto logarithmic curvesC. Under the length-preserving projection via Euler discretization scheme, biorthogonal wavelet filters on curves C can be projected to biorthogonal wavelet filters on real axis R. And the construction of biorthogonal wavelet filters on C is equivalent to biorthogonal wavelet filters on R by length-preserving projection. The properties of biorthogonal wavelet filters on C are also discussed in the paper, and some conclusions are similar to biorthogonal wavelet filters on R. Moreover, an example and figures are given. Finally, a numerical application is given for dealing with financial data. According to the wavelet method on a logarithmic trend curve, some new research interpretations are presented for dealing with financial data, such as “volatility with logarithmic growth trend curve,” “error with logarithmic growth trend curve,” and so on. These may have a new inspiration for dealing with some data on manifolds or manifolds learning in signal processing.



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*Source: 7985069-2021-03-04.xml*