For level of decomposition and translation, we define subinterval. For, define as a subspace of piecewise polynomial functions satisfying

We now start to review Legendre polynomials and Legendre wavelet bases [1] . Let denote Legendre polynomial of degree k, which is defined as follows:

Then, at the level of resolution, let denote Legendre wavelet bases defined as

The whole set forms an orthonormal basis for.Generally, the subspace is spanned by functions which are obtained from by dilation and translation, i.e.,

which forms an orthonormal basis for and

.

Now, let, then obtain six Legendre wavelet base functions which are given by

,

,

,

,

and Figure 1 illustrates these base function as

It is clear that Legendre wavelet bases are compactly supported, polynomial, bounded and orthogonal on each subinterval. These properties are very useful to estimate the approximation error upper bounded.

Lemma 1. Legendre wavelet bases satisfy the results

Lemma 2. For any, Legendre wavelet are bounded by the form

where k is the order of Legendre wavelet.

Proof. According to the definition of Legendre wavelet bases, Legendre wavelet defined on subinterval are obtained through Legendre polynomials by dilation. With the result of Legendre polynomials, the bound of Legendre wavelet is easily proved.

Lemma 3. A relation of between Legendre wavelet and their derivative on each subinterval 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

However, when, 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 are orthogonal.

Proof. According to the compactly supported of Legendre wavelet bases, we know that any two such base functions and with the same scale index n and different, are orthogonal. If any two bases functions are only different in, then for any, we have

which completes the proof.

Thus, any function belonging to can be expanded as

where 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 matrices and defined as, respectively

,

which makes the function approximated by arbitrary precision, when numerical computation is adopted by Legendre wavelet method.

The preliminaries of exponential a-Hölder continuity and spaces are defined by

Definition 1. Exponential a-Hölder continuity for any a denotes the function f satisfying

for some positive constant A.

Definition 2. space denotes that all the functions f which are bounded and continuously differentiable up to N-order for any a, i.e.,

where.

The upper bounded of Legendre wavelet transform coefficients is estimated as:

Theorem 1. Let, then the upper bounded estimate of Legendre wavelet transform coefficients satisfies

where 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 a-Hölder continuity.

Theorem 2. Let, 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 is a constant with respect to k, f and l.

Proof. The proof of this theorem utilizes the vanishing moments of Legendre wavelet 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.

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 and is an arbitrarily small positive constant.

Proof. From the equality, there exists positive integers, , K, and arbi-

trarily small positive constant satisfying

which completes the proof.

Similarly, we can obtain the estimate of approximation error in space.

Theorem 4. Let, the upper bounded estimate of approximation error using Legendre wavelet is described as

where T is a constant with respect to and is an arbitrarily small positive constant.

These estimates of the approximation error upper bounded provide computational precision for numerical computation.

In this subsection, we derive the estimations of approximation error by norm.

Theorem 5. Let, then the estimation of the approximation error upper bounded by the norm satisfies

where T is a constant with respect to and is an arbitrarily small positive constant.

Proof. Taking advantage of the definition of norm and using (13), it is clear that the approximation error upper bounded is estimated by

For,the estimate technique by the norm is similar to the theorem 4 and theorem 5.