Wavelet approximation of a function using Chebyshev wavelets

In this paper, we estimate the best wavelet approximations of a function f having bounded second derivatives and bounded higher-order derivatives using Chebyshev wavelets of third and fourth kinds.


Introduction
In recent years, wavelets have found their way into many different fields of science and engineering; particularly, wavelets are very successfully used in signal analysis for waveform representation and segmentation, time-frequency analysis, and fast algorithms for easy implementation. Wavelets allow an accurate representation of variety of functions and operators.
The wavelet approximation technique is a recent tool to detect and analyze abrupt change in seismic signal processing. The wavelet approximation of a function by Haar wavelet has been determined by Devore [2], Debnath [1], Meyer [7], Morlet [11], and Lal and Kumar [4].
Chebyshev polynomials have become increasingly crucial in approximation theory. It is well known that there are four kinds of Chebyshev polynomials, and they all are particular cases of the more widely known class of Jacobi polynomials. The first and second kind Chebyshev polynomials are particular cases of symmetric Jacobi polynomials (i.e., ultraspherical polynomials), whereas third and fourth kinds of Chebyshev polynomials are particular cases of the nonsymmetric Jacobi polynomials (see Mastroianni and Milovanović [6,).
Note that a good amount of work on Chebyshev polynomials of the first kind T n (x) and the second kind U n (x) and their applications has already been done. But a very few research work has appeared on the Chebyshev polynomials of third and fourth kinds. We see that the Chebyshev polynomials of third kind V n (x) and fourth kind W n (x) and their applications are highly important in many areas, including wavelet approximation of certain functions.
It is important to note that V n (x) and W n (x) can be useful in situations in which singularities occur at one end point (+1 or -1) but not at the other.
The Chebyshev wavelet approximation method provides the best approximation of a certain function belonging to an approximate class. This motivates us to consider the Chebyshev wavelets of third and fourth kinds to estimate the error of approximation of a function.
Therefore, in this paper, we obtain the best wavelet approximation of a function f by shifted Chebyshev wavelets. In fact, we prove four theorems. In the first two theorems, we obtain the approximation of a function f having bounded second-order derivative and bounded mth derivative using shifted third kind Chebyshev wavelets. In the other two theorems, we obtain the best wavelet approximation of a function f having second-order derivative and bounded mth derivative using shifted fourth kind Chebyshev wavelets. It is important to note that the estimate of a function having more bounded derivatives is better and sharper than the estimate having less bounded derivatives, so comparison of estimated approximation has a significant importance in wavelet analysis.
The outline of the paper is as follows. In Sect. 2, we describe the Chebyshev polynomials and shifted Chebyshev polynomials of third and fourth kinds. In this section, we also define the functional approximation, projection, and wavelet approximation. Four our main theorems are given in Sect. 3. In Sect. 4, we present their proofs. Two corollaries are deduced in Sec. 5. In the last Sect. 6, we conclude our results.

Chebyshev polynomials of third and fourth kinds
The Chebyshev polynomial of third kind is a polynomial of degree n given by and the Chebyshev polynomial of fourth kind is a polynomial of degree n given by where x = cos θ .

Examples of Chebyshev polynomials of third and fourth kinds
Using (1), we get and using (2), we get W 0 (x) = 1, W 1 (x) = 2x + 1, W 2 (x) = 4x 2 + 2x -1, Remark 1 The polynomials V n (x) and W n (x) are, in fact, rescalings of two particular Jacobi polynomials P α,β n (x) with α = -1 2 and β = 1 2 and vice versa. Explicitly, These polynomials also may be efficiently generated by using the recurrence relation W n (x) = (-1) n V n (-x) (see [3,8,10] for application in numerical integration). Since and it immediately follows that with with It is clear from (5) and (6) that both V n (x) and W n (x) are polynomials of degree n in x, in which all powers of x are present, and in which the leading coefficients (of x) are equal to 2 n . The polynomials V n (x) and W n (x) are orthogonal on (-1, 1), that is, where

Shifted Chebyshev polynomials of third and fourth kinds
The shifted polynomials V * n and W * n of third and fourth kinds, respectively, are defined as The orthogonal relations of V * n (t) and W * n (t) on [0, 1] are given by where [5] and [9]).
According to (10) and (11) and the relation W n (x) = (-1) n V n (-x), we can conclude that so that the orthogonal polynomials with respect to w * 2 can be obtained from those orthogonal with respect to w * 1 by the previous simple substitution x := 1x and the factor (-1) n (in order to get all positive leading coefficients). Therefore it suffices to consider only one of these weights, say, w * 1 . The polynomials V * n (x) satisfy the following three-term recurrence relation: and so on.

Shifted Chebyshev wavelets of third and fourth kind
When the dilation parameter a and the translation parameter b vary continuously, then we have the following family of continuous wavelets: Each of the third and fourth kind of Chebyshev wavelets ψ n,m (t) := ψ(k, n, m, t) has four arguments with k, n ∈ N, m is the order of the polynomial V * m (t) or W * m (t), and t is the normalized time. The Chebyshev wavelets of third and fourth kinds are defined explicitly on the interval [0, 1] by

Functional approximation
A function f ∈ L 2 (R) defined over [0, 1] is expanded in terms of Chebyshev wavelet series as where with weights w * i , i = 1, 2, defined in (13). If an infinite series in (15) is truncated, then it can be written as

Multiresolution analysis
A sequence of closed subspaces V j of L 2 (R), j ∈ Z, is called a multiresolution in L 2 (R) if it satisfies the following conditions: ).

Wavelet approximation
The wavelet approximation under the supremum norm is defined by The degree of wavelet approximation E n (f ) of f by P n f under the norm · r is given by Remark 2 If E n (f ) → 0 as n → ∞, then E n (f ) is called the best approximation of f [13].

Main theorems
In this paper, we prove the following theorems. , M > 1.

Proof of the main theorems
where From Chebyshev wavelet we havê Let n = 2 k-2 + 1. Then (19) becomes (other terms vanish due to the orthogonality of ψ n,m ).