Estimates of Approximation Error by Legendre Wavelet

This paper first introduces Legendre wavelet bases and derives 
their rich properties. Then these properties are applied to estimation of 
approximation error upper bounded in spaces  and  by norms  and  , respectively. 
These estimate results are valuable to solve integral-differential equations by 
Legendre wavelet method.


Introduction
In recent years, an application of Legendre wavelet to solve integral-differential equations and partial differential equations is deeply considered [1]- [9].Generally, representations of function and operator by Legendre wavelet are exact up to arbitrary but finite precision, then the approximation error should be estimated.Although estimating the approximation error is a tough technique, if the wavelet satisfies certain conditions [5]- [11], then the upper bounded of the wavelet transform coefficients can be estimated.In this article, we use the rich properties of Legendre wavelet bases such as compactly supported, polynomials, orthogonality to estimate the approximation error upper bounded.
In this paper, Section 2 introduces Legendre wavelet bases and its properties.Section 3 estimates the approximation error upper bounded by norms 2 ⋅ and 1 ⋅ for spaces [ ] ( ) , respectively.This paper ends with brief conclusion.

Legendre Wavelet and Its Properties
In this section, we first briefly introduce Legendre wavelet bases and our notations.Secondly, the rich properties and some important results of Legendre wavelet that will be used later are elaborated.

{ }
, : is a polynomial of degree strictly less than and vanishes elsewhere .
We now start to review Legendre polynomials and Legendre wavelet bases [1].Let ( ) k L x denote Legendre polynomial of degree k, which is defined as follows: Then, at the level of resolution 0 n = , let ( ) denote Legendre wavelet bases defined as The whole set { } , 0 1, 0 2 1 which forms an orthonormal basis for Now, let 3, 1 p n = = , then obtain six Legendre wavelet base functions which are given by ( ) and Figure 1 illustrates these base function as

Some Properties of Legendre Wavelet
It is clear that Legendre wavelet bases are compactly supported, polynomial, bounded and orthogonal on each subinterval nl I .These properties are very useful to estimate the approximation error upper bounded.Lemma 1. Legendre wavelet bases satisfy the results .
Lemma 2. For any where k is the order of Legendre wavelet.
Proof.According to the definition of Legendre wavelet bases, Legendre wavelet defined on subinterval nl I are obtained through Legendre polynomials by dilation.With the result of Legendre polynomials ( ) A relation of between Legendre wavelet and their derivative on each subinterval nl I is derived as Proof.Using the result of between Legendre polynomials and their derivative, i.e., ( ) ( ) ( ) ( ) we can obtain the above result.
Using this result, we can obtain , 2 However, when 0 k = , the integration is calculated as Now, the orthogonal property of Legendre wavelet bases is given by Lemma 4. Legendre wavelet bases defined on the interval [ ) which completes the proof.
Thus, any function ( ) , is Legendre wavelet coefficients and ( ) .,. denotes inner product.Accordingly, norm equality is given by ( ) If approximation of the function is analyzed in the space V p n , then the approximation formula is described by where S and ( ) are 2 1 n p × matrices and defined as, respectively which makes the function approximated by arbitrary precision, when numerical computation is adopted by Legendre wavelet method.

Upper Bounded Estimates of Approximation Error by Legendre Wavelet
In this section, the preliminaries of the function spaces with respect to exponential α-Hölder continuity and [ ] ( ) are first introduced, respectively.Secondly, the upper bounded estimates of approximation error in the spaces by Legendre wavelet bases are derived by norms 2 ⋅ and 1 ⋅ , respectively.
( ) space denotes that all the functions f which are bounded and continuously differentiable up to N-order for any α ( ) where 0 0,1, , k N =  .

Approximation Error Estimate by Norm 2 ⋅
The upper bounded of Legendre wavelet transform coefficients is estimated as: , then the upper bounded estimate of Legendre wavelet transform coefficients satisfies where kfl T is a constant with respect to k, f and l.Proof.Taking advantage of the results of ( 6) and ( 7), we have { } ∫ which completes this proof.

Remark:
The upper bounded of Legendre wavelet transform coefficients vanish with exponent in terms of multiplies of the scale index or exponential α-Hölder continuity.
, suppose that wavelet has n vanishing moments, then the upper bounded estimate of wavelets ( ) transform coefficients k such that then the upper bounded estimate of Legendrewavelet transform coefficients satisfies ( where kfl T is a constant with respect to k, f and l.Proof.The proof of this theorem utilizes the 1 k − vanishing moments of Legendre wavelet , k nl φ and Taylor expansion of the function f and then is similar to that of the theorem 1.Now, taking advantage of the results of ( 9), (13) and theorem 1, we can derive the upper bounded estimation by the norm 2 ⋅ .Theorem 3. Let [ ] ( ) , then the upper bounded estimate of approximation error by using Legendre wavelet bases satisfies ( ) where T is a constant with respect to kfl T and ε is an arbitrarily small positive constant.
Proof.From the equality ( ) , there exists positive integers 0 N , 1 N , K, 1 K and arbitrarily small positive constant ε satisfying which completes the proof.Similarly, we can obtain the estimate of approximation error in space [ ] ( ) , the upper bounded estimate of approximation error using Legendre wavelet is described as where T is a constant with respect to kfl T and ε is an arbitrarily small positive constant.
These estimates of the approximation error upper bounded provide computational precision for numerical computation.

Approximation Error Estimate by Norm 1 ⋅
In this subsection, we derive the estimations of approximation error by norm 1 .Theorem , then the estimation of the approximation error upper bounded by the norm where T is a constant with respect to kfl T and ε is an arbitrarily small positive constant.
Proof.Taking advantage of the definition of 1 ⋅ and using (13), it is clear that the approximation error upper bounded is estimated by

Conclusion
As all, this paper considers the compactly supported, polynomial, orthogonal and bounded properties of Legendre wavelet bases.Using these properties, the upper bounded estimates of the approximation error are presented


by dilation and translation, i.e.,

Definition 1 .
Exponential α-Hölder continuity for any α ( ) technique by the norm 1 ⋅ is similar to the theorem 4 and theorem 5.
for the function belonging to exponential α-Hölder continuity and space