Abstract
A sufficient literature is available for the wavelet error of approximation of certain functions in the -norm. There is no work in context of multiresolution approximation of a function in the sense of sup-error. In this paper, for the first time, wavelet estimator for the approximation of a function belonging to class under supremum norm has been obtained. Working in this direction, four new theorems on the wavelet approximation of a function belonging to class using the projection of its wavelet expansions have been estimated. The calculated estimator is best possible in wavelet analysis.
1. Introduction
Orthogonal wavelet is a new development in analysis but very useful in engineering and technology, especially in high-resolution images and signal processing due to their localization properties in time and frequency. Wavelet expansions are superior to classical orthogonal series like Fourier series. Some properties of wavelet expansions have been studied by Chui [1], Daubechies and Lagarias [2], Meyer [3], Walter [4, 5], Islam et al. [6], and so forth. The idea of approximation of various functional spaces under different norms is obtained by Lal and Kumar [7, 8], Abu-Sirhan [9], Coskun [10], and Shiri and Azadi Kenary [11] which gives the inspiration for the present work. But till now no work seems to have been done to obtain the wavelet approximation of a function using the projection of its wavelet expansion and to discuss its convergence. In an attempt to make an advance study in this direction, in this paper, the best possible wavelet sup-error of a function by of its wavelet expansion has been determined. The convergence of wavelet expansions has been also discussed.
2. Definitions and Preliminaries
2.1. Multiresolution Analysis
Let be the set of all integers. A multiresolution analysis of is defined as a sequence of closed subspaces of , with the following properties:(1)(2)(3)(4) is dense in and (5)Suppose a function exists such that the collection is a Riesz basis of
Let , andThen this family of subspaces of gives direct sum decomposition of in the sense that every has unique decomposition: where for all and we describe this by writing where is a Riesz basis of
A function is called a scaling function, if the subspaces of defined by satisfy properties (1) to (5) stated above in this section. It is important to note that the scaling function generates a multiresolution analysis of (Debnath [12]).
Following Walter [4], each has two representations where convergence is in sense of . The function and in fact is the projection of onto . It is also given in terms of the kernel of as The function is given by where
It is remarkable to note that Then
2.2. Quasi-Positive Delta Sequences
If(1)there is such that ,(2) uniformly on compact subset of , as ,(3)for each , then is known as quasi-positive delta sequence of function.
An example of a quasi-positive delta sequence is the Fejer kernel,while a delta sequence that is not quasi-positive is the Dirichlet kernel of Fourier series, because is not absolutely convergent.
2.3. Function of Class
A function if(Titchmarsh [13], p. 406).
Examples(1) (2) If . Let be a positive integer. Let Then and Now,
Remarks(1)If , then is continuous, indeed, uniformly continuous on .(2) class is a linear space over or .(3)If , then is constant function.
2.4. Projection
, the orthogonal projection of onto , is defined by (Sweldens and Piessens [14]).
2.5. Wavelet Approximation
The wavelet approximation of a function under supremum norm is defined by (Zygmund [15], p. 114).
If as then is called the best approximation of of order (Zygmund [15], p. 115).
3. Theorems
In this paper, we prove the following theorems.
Theorem 11. Let be a scaling function and be a basic wavelet satisfying the admissibility condition If a function is represented by its wavelet expansion as then the wavelet approximation of by satisfies
Theorem 12. If a function and is the projection of onto then as uniformly on
Theorem 13. If and then
Theorem 14. If , then as uniformly on .
4. Lemma
For the proof of our theorems, the following lemma is required.
Lemma 15. If , then is a quasi-positive delta sequence of function in .
4.1. Proof of Lemma 15
Following the proof of Walter ([4] p. 113), Let ; then as and as
Thus, as uniformly on and as , [4].
Therefore is a quasi-positive delta sequence of functions.
5. Proofs
5.1. Proof of Theorem 11
is given by Let Select such that Since it is known that therefore Next, is a quasi-positive delta sequence of function.
Now, collecting (50), (31), and (32), we have Thus
Remark 16 (the converse of Theorem 11 is also true). Choose such that and Since , therefore
Hence, Theorem 11 is completely established.
5.2. Proof of Theorem 12
By the proof of Theorem 11Since,therefore,Thus,Hence, Theorem 12 is completely established.
5.3. Proof of Theorem 13
Since,therefore,Hence, Theorem 13 is completely established.
5.4. Proof of Theorem 14
Following the proof of Theorem 13, Since,therefore,Hence, Theorem 14 is completely established.
6. Notes
(1) as in Theorem 11; the wavelet approximations determined in this theorem are best possible in wavelet analysis (Zygmund, [15], p. 115).
(2) Define a function by Therefore is monotonic increasing in and monotonic decreasing in
Thus,Hence A function is defined by ,
For this function,ThenHence
Thus, the result of Theorem 11 is verified for a function defined by
(3) Hölder class: Let denote the Banach spaces of all periodic continuous functions under sup-norm. For each and some positive constant , the function space is defined by(Das et al. [16] p. 83).
Let Then
The function It is continuous and belonging to but . Then .
Thus
Theorem 11 is also valid for a function
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
Shyam Lal is thankful to DST-CIMS for encouragement to this work. Susheel Kumar is grateful to CSIR (India) in the form of Junior Research Fellowship with Ref. no. 19-06/2011 (i)EU-IV dated 03-10-2011 for this research work.