A novel method of constructing compactly supported orthogonal scaling functions from splines

A novel construction of compactly supported orthogonal scaling functions and wavelets with spline functions is presented in this paper. Let Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document} be the center B-spline of order n, except for the case of order one, we know Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document} is not orthogonal. But by the formula of orthonormalization procedure, we can construct an orthogonal scaling function corresponding to Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document}. However, unlike Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document} itself, this scaling function no longer has compact support. To induce the orthogonality while keeping the compact support of Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document}, we put forward a simple, yet efficient construction method that uses the formula of orthonormalization procedure and the weighted average method to construct the two-scale symbol of some compactly supported orthogonal scaling functions.


Introduction
It is well known that B-splines have many useful properties, and they are widely used in practical problems. But except for the case of order one, B-splines of other orders are not orthogonal []. Thus, in order to get the property of orthogonality, many researchers are interested in the study of constructing orthogonal wavelets with B-splines [-]. For instance, Franklin wavelet and Battle-Lemarié wavelets [, ], but these wavelets are not compactly supported. In [] Goodman gave a construction, for any n ≥ , of a space S of spline functions of degree n - with simple knots in   Z which is generated by a triple of refinable, orthogonal functions with compact support. Subsequently, Cho and Lai simplified Goodman's constructive steps for compactly supported orthonormal scaling functions and provided an inductive method for constructing compactly supported orthonormal wavelets []. In [] Nguyen and He presented a method to construct orthogonal splinetype scaling functions with B-splines. They multiplied a class of polynomial function factors to the two-scale symbol of the B-splines so that they become the two-scale symbol of a spline-type orthogonal compactly supported function. Different from above, firstly, we use orthonormalization procedure so that splines become orthogonal scaling functions. Unfortunately, the orthogonal scaling functions are not compactly supported. So, in order to make them have the property of compact support, we use the weighted average method to eliminate the denominator of the two-scale symbol, which is the corresponding orthog-onal scaling function. And from examples in Section , we found that this method is simple and flexible.
The goal of this section is to prepare for the next chapter of theorem proving. For this reason, we need the following auxiliary results.
(v) (Basis) There exists a function φ ∈ V  such that {φ(xk) : k ∈ Z} is an orthonormal basis or a Riesz basis for V  .
The function φ defined as in Definition . is called the scaling function of the given MRA. From (iv), we know that φ ∈ V  is also in V  . Since {φ ,k :=  / φ(xk) : k ∈ Z} is a Riesz basis of V  , then there exists a unique l  -sequence {p k } satisfying the 'two-scale relation' This sequence {p k } is called the 'two-scale sequence' of φ. With this l  -sequence, we define Then the Fourier transform formulation of identity (.) can be written as We call P(ω) the two-scale symbol of the scaling function φ. Noticing that {φ(xk) : k ∈ Z} is an orthonormal basis, we have the following equivalent statements of orthogonality, see also in [, -].
Theorem . Suppose that P(ω) =   k p k e -iωk is a polynomial satisfying the following conditions: Then P(ω) is the two-scale symbol of an orthogonal scaling function.
Riesz lemma Let a  , . . . , a N be real numbers and a N =  such that Then there exists a polynomial with real coefficients and exact degree N satisfying

Constructing compactly supported orthogonal scaling functions
In this section we will give a new method to construct compactly supported orthogonal scaling functions and wavelets by a center cardinal B-spline. The mth order center cardinal B-spline M m is defined as follows, see also [, ].
Furthermore, it is easy to verify that Note that the scaling function M m (x) is semi-orthogonal for m ≥ . Next, we give a method to obtain the orthogonal scaling function through the B-spline M m . We define a function ϕ m (x) through its Fourier transform By (.) and (.), we can obtain P m (ω), which is the two-scale symbol of the scaling function ϕ m (x). It is well known that the scaling function ϕ m (x) determined by (.) is orthogonal but not compactly supported. So next we concentrate our effort on the study of constructing compactly supported orthogonal scaling functions. Note that the presence of the denominator in (.) can bring about scaling functions which are not compactly supported. Therefore, we multiply a function factor to the twoscale symbol P m (ω) and obtain the following Theorem . and some corollaries.
is the two-scale symbol of an orthogonal scaling function, and let where λ i (ω) is a π -periodic function and satisfies the following conditions: Then H(ω) is the two-scale symbol of an orthogonal scaling function.
Proof It is easy to observe that H(ω) satisfies statements (.) and (.) in Theorem ., now we only need to prove that H(ω) also satisfies |H(ω)|  + |H(ω + π)|  = . Since h i (ω) is the two-scale symbol of an orthogonal scaling function, we have Thus which by Theorem . implies that H(ω) is the two-scale symbol of an orthogonal scaling function. and Then H m (ω) is a two-scale symbol of some compactly supported orthogonal scaling function.
Proof By (.) and (.), we have Since P m (ω) is a two-scale symbol of some orthogonal scaling function, we obtain  = P m (ω)  + P m (ω + π) Multiplying  - m- k= M m (k) sin  (kω) on both sides in the above equation, we obtain By Theorem . and the Riesz lemma, we know that H m (ω) is a two-scale symbol of some compactly supported orthogonal scaling function. where λ i (ω) satisfies the following conditions:

Then h(ω) is the two-scale symbol of a compactly supported orthogonal scaling function.
The proof is analogous to that of Corollary ..

Corollary . Define
where the real numbers a, b, c, d, e satisfy Moreover, let where P  (ω) and P  (ω) are defined as in (.). Then P(ω) is the two-scale symbol of a compactly supported orthogonal scaling function.
To facilitate our proof of Corollary ., we need the following result. Proof By (.) we obtain

Lemma . Define
Now, consider the system It is easy to check that the pair number (a, b, c, d, e) satisfying is the solution of (.), also the solution of (.).
This Proof of Corollary . By calculation, we have Denote and note that A(ω) is an even and π -periodic function. We obtain from Lemma . Therefore, |P(ω)|  > . Noticing that P  (ω) and P  (ω) are the two-scale symbols of orthogonal scaling functions and λ  (ω) + λ  (ω) = , we have Now applying Theorem . and the Riesz lemma, we know that P(ω) is a two-scale symbol of some compactly supported orthogonal scaling function.

Examples
In this section, we give three examples to show our construction scheme introduced in the above section.
Example . For m = , from (.) and (.), we have respectively. Moreover, we have Now, we obtain from (.) that Therefore, by the Riesz lemma, we have In Figure , we show the graphs of φ  (x) and ψ  (x), respectively.
Example . Consider m  = , m  = , a  =   and a  =   , then by Corollary ., we have

Conclusion
A simple and flexible method for constructing compactly supported orthogonal scaling functions is presented in this paper. Using this method, we can construct orthonormal compactly supported scaling functions from B-splines. Note that the change of λ i (i = , . . . , N ) can cause the change of the scaling functions corresponding to two-scale symbol H(ω) in (.). Therefore we can provide the user with different scaling functions with the same compact support. Similarly, then we can obtain different compactly supported scaling functions by changing the parameters in (.).