Biorthogonal Wavelet on a Logarithm Curve 
 ℂ

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve 
 
 C
 
 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.


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 areapreserving 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. e 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][11][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 lengthpreserving projection p in this paper. is paper will be organized as follows: in Section 2, 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.

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 function Y � AK α L β e μ . Taking the Log on both sides of it, we have lnY � lnA + αlnK + βlnL + μ, μ ∼ N(0, σ 2 ). 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) is model is called a one-dimensional logarithmic regression model. If an explained variable y and an explanatory variable x satisfy the equation then this model is called a one-dimensional semilogarithmic regression model and y � α + β ln x is called a logarithmic regression trend curve.

Length-Preserving Projection via Euler Discretization
Scheme. According to the algorithm of length-preserving projection p on a smooth curve in [5,14], the projection p is given in this section.
Consider a log-regression curve C: By calculating, we have So e inverse projection p − 1 is not explicit, but it could be determined by the numerical method.
For example, Figure 1 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. According to the above example, it can be seen that the inverse p − 1 of the length-preserving projection p can be denoted by numerical method, such as Euler discretization method.

Biorthogonal Wavelet Analysis.
In this section, we introduce multiresolution analysis of L 2 (R), 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 L 2 (R) is an increasing sequence of closed subspace V j ∈ L 2 (R), satisfying the following conditions: (4) ere exists a function ϕ(t) ∈ L 2 (R); the set ϕ(t − k), k ∈ Z is a Riesz basis of V 0 ; that is, there exist two constants A and B such that, for an arbitrary sequence c j j∈N ∈ l 2 , there is where V j � clos L 2 (R) 〈ϕ j,k � 2 (j/2) ϕ(2 j t − k): k ∈ Z〉 and ϕ j,k � 2 (j/2) ϕ(2 j t − k).
Definition 3 (see [10,15]). Assume that V j , ϕ j∈Z and V j , ϕ j∈Z are two multiresolution analyses of L 2 (R). If 〈ϕ j,k , ϕ j′,l 〉 � δ j,j′ δ k,l , then V j , ϕ j∈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.
Assume that V j , ϕ j∈Z and V j , ϕ j∈Z are the mutual dual multiresolution analysis of L 2 (R); then ϕ and ϕ satisfy the following two-scale equations: where the sequences h k and h k are called the two-scale sequences of ϕ(t) and ϕ(t), respectively. Taking the Fourier transform on both sides of equation (11), we have where H(ω) � (1/2) k∈Z h k e − ikω and H(ω) � (1/2) k∈Z h k e − ikω are called the two-scale symbols of ϕ(t) and ϕ(t), respectively.
For the mutual dual wavelets ψ and ψ, it is also obtained that where the sequences g k and g k are called the two-scale sequences of ψ(t) and ψ(t), respectively. G(ω) � (1/2) k∈Z g k e − ikω and G(ω) � (1/2) k∈Z g k e − ikω are called the two-scale symbols of ψ(t) and ψ(t), respectively. Lemma 1 (see [10,15]). If ϕ 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:

). Assume that ϕ(t) ∈ L 2 (R) is a realvalue function, and ϕ(ω) is its Fourier transform. en ϕ(t) has generalized linear phase and a is phase of ϕ(ω) if and only if ϕ(t) is symmetrical or antisymmetrical at t � a.
Lemma 3 (see [10,15]). Assume that a k ∈ l 1 is a real-value sequence, and A(ω) its discrete Fourier transform. en a k has generalized linear phase and a is phase of A(ω) if and only if a k is symmetrical or antisymmetrical at k 0 ; that is,

Biorthogonal Wavelet on a Smooth Curve C
In this section, biorthogonal wavelet analysis on a smooth curve C 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.

Multiresolution Analysis on a Smooth Curve C.
Assume that a smooth curve C satisfies parameter equation: 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 dL(ξ) 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 function f, g ∈ L 2 (C), we have the following equation (seen in [5,9,14]): Analogously, if f, g ∈ L 2 (R), it is obtained that Based on equations (15) and (16), the following results can be established and it is easy to prove these results. e Fourier transform can be regarded as the inner product of function f ∈ L 2 (C) and Fourier basis.
e Fourier transform of f ∈ L 2 (C) can be defined as follows: where e − ip(ϖ)·p(η) is induced by e − iω·X , p(ϖ) � ω, and p(η) � X. And the corresponding inverse transform is Lemma 4 (see [7,9,14]). Assume that J is a countable set and f k k∈J ∈ L 2 (R). For each k ∈ J, we define f k k∈J ∈ L 2 (C) as f k � f k ∘ p. en, we have the following: (1) If f k k∈J is an orthogonal basis of L 2 (R), then f k k∈J is also an orthogonal basis of L 2 (C) and B, then f k k∈J is a Riesz basis of L 2 (C) with the same frame bounds For every function f ∈ L 2 (R), the induced function f C ∈ L 2 (C) can be defined as According to equation (16), if the functions f j,k j,k are orthogonal, so are Choose f j,k � ϕ j,k and f j,k � ψ j,k ; we have the functions on L 2 (C): us, the multiresolution analysis of L 2 (C) can be defined by multiresolution analysis of L 2 (R) and the induced function. For j ∈ Z, we define the space ] j as It is immediate that ] j is a closed subspace sequence of L 2 (C) (seen in [5,9,14]).
us, it is a Hilbert space. Moreover, these spaces have the following properties: where ] j � clos L 2 (C) 〈ϕ C j,k : k ∈ Z〉.
Definition 6 (see [5,9,14]). If a sequence of subspaces ] j of L 2 (C) satisfy the above properties, it is called a multiresolution analysis of L 2 (C).
Once the multiresolution analysis is determined, the wavelet subspaces w j may be constructed in usual way. For every j ∈ Z, assume that w j denotes an orthogonal com- It is easy to prove that, for each j ∈ Z, ψ C j,k , k ∈ Z is an orthogonal basis of w j and so ψ C j,k , j ∈ Z, k ∈ Z is an orthogonal basis of ⊕ j∈Z w j � L 2 (C). We can define ψ C j,k as ψ C j,k � ψ j,k°p , then ϕ C is called the scaling function on the regression trend curve C, and ψ C is called the corresponding wavelets on the regression trend curve C.

Scaling Function on a Logarithmic
Curve C. In this section, we discuss the scaling function and wavelet on a logarithmic curve C. Firstly, the translation operator and dilation operator in the space L 2 (C) can be defined as follows (seen in [1,5,9,14]): According to the induced scaling function, translation operator T b , and dilation operator D a , the two-scale equation of the scaling function ϕ C in L 2 (C), which can be induced by ϕ in L 2 (R), can be deduced as follows: where a sequence h C k is called the two-scale sequence of ϕ C (η). is equation is also called the refinable equation of ϕ C (η).
Based on the length-preserving projection p, the twoscale sequence h C k of ϕ C (η) satisfies the following conclusion.
Lemma 5 (see [1,5,9,14]). Assume that p is a lengthpreserving projection from the regression trend curve C to a subset of real axis R. If ϕ C is the scaling function in L 2 (C), which is induced from ϕ ∈ L 2 (R) by p, then the sequences h k and h C k in equations (11) and (25) where H C (p(ϖ)/2) � (1/2) k∈Z h C k e − (ip(ϖ)·k/2) is called the two-scale symbol of ϕ C (η).
By utilizing equation (11) repeatedly, it is obtained that is concluded in the following lemma.

Biorthogonal Wavelet on a Logarithmic Curve C
Definition 7 Assume that υ j , ϕ C j∈Z and υ j , ϕ C j∈Z are two multiresolution analysis of are called the mutual dual multiresolution analysis. ϕ C and ϕ C are called a pair of biorthogonal scaling functions.
are called a pair of biorthogonal wavelet basis. e functions ψ C and ψ C are called mutual dual wavelet or a pair of biorthogonal wavelet functions.
Definition 9. Assume that ϕ C and ϕ C are two scaling functions of L 2 (C), and ψ C and ψ C are two wavelet functions corresponding to ϕ C and ϕ C , respectively. If then ϕ C and ϕ C are called a pair of biorthogonal scaling functions, and ψ C and ψ C are a pair of biorthogonal wavelet functions corresponding to ϕ C and ϕ C , respectively.
Assume that υ j , ϕ C j∈Z and υ j , ϕ C j∈Z are the mutual dual multiresolution analysis of L 2 (C); then ϕ C and ϕ C satisfy the following two-scale equations: where the sequences h C k and h C k are called the two-scale sequences of ϕ C and ϕ C , respectively.
For the mutual dual wavelets ψ C and ψ C , it is also obtained that where the sequences g C k and g C k are called the two-scale sequences of ψ C and ψ C , respectively. G C (p(ϖ)/2) � (1/2) k∈Z g C k e − (ip(ϖ)·k/2) and G C (p(ϖ)/2) � (1/2) k∈Z g C k e − (ip(ϖ)·k/2) are called the two-scale symbols of ψ C and ψ C , respectively.
(2) ⟺ (3) According to Lemmas 5 and 6, where then ϕ C is symmetrical at α, where α is a fixed point. If then ϕ C is antisymmetrical at α.
Definition 11. If the Fourier transform ϕ C of ϕ C satisfies then it is called that ϕ C has linear phase. Moreover, if , where α, β are fixed points and B(ϖ) is a real-value function, then it is called that ϕ C has generalized linear phase, where α is called phase of ϕ C .

Theorem 2.
Assume that ϕ C ∈ L 2 (C) is a real-value function, and ϕ C is its Fourier transform. en ϕ C has generalized linear phase and α is phase of ϕ C if and only if ϕ C is symmetrical or antisymmetrical at α.
Proof. Assume that ϕ C has generalized linear phase and α is phase of ϕ C . en the inverse Fourier transform of ϕ C can be written as follows: Taking the complex conjugation on both sides of the above equation and B(ϖ) being real-value, we have On the other side, if by taking the Fourier transform on its both sides, we have So, if ϕ C ∈ L 2 (C) is a real-value function, then e i2p(β) � ± 1 and (42) at is, ϕ C is symmetrical or antisymmetrical at α. □ Theorem 3. Assume that h C k ∈ l 2 is a two-scale sequence of ϕ C , and H C (p(ϖ)/2) is defined in its discrete Fourier transform. en h C k has generalized linear phase and k 0 is phase of H C (p(ϖ)/2) if and only if h C k is symmetrical or antisymmetrical at k 0 ; that is, Proof. According to Lemma 3, eorem 1, and the lengthpreserving projection p, it is shown obviously.

Example and Figure
Consider the biorthogonal wavelet "bior2.4" in [10,15], and the relevant data can be obtained in MATLAB. e scaling function of "bior2.4" satisfies the following two-scale equation: And the dual scaling function satisfies According to the length-preserving projection p, the scaling function ϕ(t) and the dual ϕ(t) of biorthogonal wavelet "bior2.4" can be induced onto a regression curve C. And the scaling function ϕ C and the dual ϕ C can be generated by p. According to Lemma 5, the corresponding twoscale sequences h C k and h C k can be obtained as follows:

Journal of Mathematics
According to equation (25), In order to show the figures of scaling function ϕ C 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 ϕ C and its dual can be shown in Figures 2  and 3. Similarly, the wavelet function ψ C 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."

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 curve C, we can decompose and reconstruct Bitcoin transaction data. e 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.
In detail, we choose daily close data of bitcoin transaction data (denoted by SP:USD) from April 30, 2017, to December 1, 2020. e 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. e logarithmic trend curve C of raw data SP and time T can be estimated by the ordinary least squares estimation. e equation of logarithmic trend curve can be obtained as follows: SP � − 3.1345 × 10 3 + 1.7572 × 10 3 × ln(T). (47) According to biorthogonal wavelet "bior2.4" lifted onto the logarithmic curve C 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 lowfrequency 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 highfrequency 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. e 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. e 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. ese are multiresolution analysis of the data SP on logarithmic trend curve C. e blue line indicates low-frequency data and highfrequency data d1, d2, d3, and d4. And the red line indicates its logarithmic trend curve.
By reconstruction algorithm and length-preserving projection, we reconstruct the low-frequency data a4, high-frequency data d1, 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." e blue line indicates original data, reconstructed data, and error data. e red line indicates its logarithmic growth trend curve. 8 Journal of Mathematics

Conclusion and Discussion
According to the length-preserving projection and Euler discretization method, biorthogonal wavelet filters can be lifted onto logarithmic curves C. 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. e 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. ese may have a new inspiration for dealing with some data on manifolds or manifolds learning in signal processing.

Data Availability
Bitcoin transaction data were used to support this study. e data can be derived from the bitcoin transaction website (https://cn.investing.com/crypto/bitcoin/historical-data).