Since 1988 the wavelet theory has been applied in several fields of science and industry, for instance, in signal processing, image processing, and numerical analysis  - . However, the multiwavelets theory and application are less developed. For instance, some authors use Legendre wavelets or Chebyshev wavelets to solve the differential equations    . But these wavelets do not have the property of vanishing moments, which is characteristic for wavelets. Actually, they are multiscaling functions of the Legendre multiwavelets   or Chebyshev multiwavelets in the following. Besides, there are not fast decomposition and reconstruction algorithm for approximating a function. Recently the biorthogonal Jacobi multiwavelet basis for the weighted space is defined, and applied to solve a class of prototypical initial and boundary value problems of fractional differential equations of general order . The complete approximating algorithm is not presented in this paper.
The purpose of this paper is to define Chebyshev biorthogonal multiscaling functions and multiwavelet functions based on Chebyshev polynomials. Although Chebyshev polynomials can be considered a special case of Jacobi polynomials with , they need to be treated independently since usually the condition is imposed on Jacobi polynomials . We will also provide the approximation method for signals (functions) by using the multiscaling functions and wavelets, which is essential to a complete decomposition algorithm. Using the same framework one may construct other biorthogonal multiwavelets based on some orthogonal polynomials like Laguerre polynomials and Hermite polynomials, etc.
The remainder of this paper is organized as follows: We construct Chebyshev biorthogonal multiwavelets in Section 2, and derive the convergence rate for the projection of a signal (function) on the subspace of and the computation formula in Section 3. Finally we give the numerical examples and conclusion remarks in Section 4.
2. Chebyshev Biorthogonal Multiwavelets
We will define Chebyshev multiscaling functions and multiwavelets, and obtain two approximations to the functions in the weighted space , which can be converted each other by the fast decomposition and reconstruction algorithms.
2.1. Chebyshev Multiscaling Functions
Classical Chebyshev polynomials can be defined by the formula
They are orthogonal with respect to the weight function on the interval :
is a polynomial of order n, and is the one among all polynomials of order n with leading coefficient which has the minimal error to zero . It has all its zeros in the interval :
We define Chebyshev multiscaling functions by
then are orthonormal with respect to the weight function :
If we denote for
then the polynomials can be expressed by
Let be an integer, then the first r multiscaling functions form an orthonormal base of the function space
which is composed of all linear combination of the functions .
For an integer , and for , we denote the dilates and translates of
which is supported on the interval . We define the function space
then form an orthonormal base of this space with respect to the inner product which is defined by
Furthermore, we define a function space
where denotes the orthogonal direct sum, and an inner product on as following:
where and , then the functions form an orthonormal base of , and any can be expressed as
2.2. Chebyshev Multiwavelet Functions
Since , we denote the orthonormal complement of in , , then an orthonormal base of can be constructed by the Gram-Schmidt process  . These functions are called Chebyshev multiwavelet functions, they have r vanishing moments:
Let , , , , then we have
Following the line of (10)-(14), we denote the dilates and translates of , the support of is , and then we define the function spaces and . It follows that the functions form an orthonormal base of with respect to the inner product .
Equation (17) can be easily generalized to the equations for integers :
Conversely, we have the dilation equations
The above two Equations (18)-(19) mean that for any , and any function as in (15) can also be expressed as
We have decomposition algorithm by (18)
and the reconstruction algorithm by (19)
From , we inductively obtain
For a function , its orthogonal projection on can be expanded in the orthonormal bases :
By virtue of (23), it can also be expanded in the multiwavelet bases :
As in , a biorthogonal dual bases (where ) to the Chebyshev multiwavelet bases of with respect to the inner product can be defined. Therefore the expansion (25) can be reformulated as
3. The Function Approximation Error in
Let be the weighted function space defined with inner product and norm
For a function , a positive integer r, and , we define the orthogonal projection of f (with respect to inner product as defined in (14)) onto by the formula
The projection converges (in the mean) to f as . If the function f is serval times differentiable, we can bound the error, as established by the following lemma.
Theorem 3.1 Suppose that the function is r times differentiable, . Then approximates f with error bounded as follows:
Proof. We divide the interval into subintervals , the restriction of to one such subinterval is the polynomial of degree less than r that approximates f with minimal mean error. We then use the maximum error estimate for the polynomial which interpolates f at Chebyshev nodes of order r on . We define for , and obtain
and by taking square roots we have the bound (29). Here denotes the polynomial of degree r, which agrees with f at the Chebyshev nodes of order r on , and we have used the well-known maximum error bound for Chebyshev interpolation (see  ).
The number of coefficients in the expressions (24) or (25) is , thus the above theorem predicts a convergence rate r for the approximation if is sufficiently smooth. Besides, this convergence rate can be achieved by using the polynomial from above proof. The Chebyshev nodes of order r for on [0, 1] are (see (3)-(4))
Then the nodes on the interval are . Plugging them into
leads to the linear system of equations
Let be the coefficient matrix, which is independent of k. Therefore, the cost for calculating all coefficients of (28) is only as . Then the coefficients of multiwavelet expansion (25) are obtained from the decomposition algorithm (21).
Table 1. Numerical error results for Example 1.
Table 2. Numerical error results for Example 2.
4. Numerical Examples and Conclusion
We present some numerical examples and give the conclusions in this last section.
Example 4.1 We take , which is a smooth function. For illustration we fix or , and compute for . The numerical results are summarized in Table 1. The convergence rate on this table is defined by the formula
We see in the table clearly that Cvge. rates are close to 3.00 for and 5.00 for . This result is in accordance with the error estimates of Theorem 3.1 since when the signal f(x) is smooth, Theorem 3.1 predicts a convergence rate of or .
Example 4.2 We take , fix or , and compute for . The numerical results are summarized in Table 2, which is in accordance with the error estimates of Theorem 3.1 since the signal f(x) is only two times differentiable, the convergence rate is less than r if .
5. Concluding Remarks
In this paper we defined Chebyshev biorthogonal multiwavelet basis for the weighted space , and showed how to use this basis to approximate functions. The algorithm is efficient, accurate and stable. Thus we set a foundation for its further applications in numerical methods for partial differential equations. As well-known, when the multiwavelets are applied instead of multiscaling functions (or wavelets as named in several papers), the resulting linear system of algebraic equations will have a bounded condition number (   ). Other applications may include signal processing and computational geometry.
This work is subsidized by Tan Kah Kee College scientific research incubation project 2018L06.
The author wishes to express the sincere thanks to the anonymous referees for valuable suggestions that helped improve the quality of this paper.
 Alpert, B., Beylkin, G., Gines, D. and Vozovoi, L. (2002) Adaptive Solution of Partial Differential Equations in Multiwavelet Basis. Journal of Computational Physics, 182, 149-190.
 Daubechis, I. (1992) Ten Lectures on Wavelets. Book Code: CB61, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia.
 Heydari, M.H., Hooshmandsl, M.R., and Maalek Ghaini, F.M. (2014) A New Approach of the Chebyshev Wavelets Method for Partial Differential Equations with Boundary Conditions of the Telegraph Type. Applied Mathematical Modeling, 38, 1597-1606.
 Li, Y.L. (2010) Solving a Nonlinear Fractional Differential Equation Using Chebyshev Wavelets. Communications in Nonlinear Science and Numerical Simulation, 15, 2284-2292.
 Mohammadi, F., Hosseini, M.M. and Mohyud-Din, S.T. (2011) Legendre Wavelet Galerkin Method for Solving Ordinary Differential Equations with Non-Analytic Solution. International Journal of Systems Science, 42, 579-585.