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Juan Su, Xinhan Dong, "On Dimension Extension of a Class of Iterative Equations", Abstract and Applied Analysis, vol. 2013, Article ID 180184, 7 pages, 2013. https://doi.org/10.1155/2013/180184
On Dimension Extension of a Class of Iterative Equations
This investigation aims at studying some special properties (convergence, polynomial preservation order, and orthogonal symmetry) of a class of r-dimension iterative equations, whose state variables are described by the following nonlinear iterative equation: . The obtained results in this paper are complementary to some published results. As an application, we construct orthogonal symmetric multiwavelet with additional vanishing moments. Two examples are also arranged to demonstrate the correctness and effectiveness of the main results.
Giving any compact supported vector-valued function , we define -dimension iterative equation as follows: where is -order real matrix, , , and . Let denote the masks of the iterative equation, then the Fourier transform of , if it exists, can be defined by and the discrete-time Fourier transform (DTFT) of can be defined by where . Therefore, iterative equation (1) takes the following frequency domain form:
We claim that iterative equation (1) converges to a fixed function, and if exists, we denote it by the -dimension vector function , whose frequency domain form is defined as an infinite product as follows:
Obviously, (5) is equivalent to satisfying -dimension refinement equation
Iterative equation (1) is nonlinearity; in the real world, nonlinear problems are not exceptional, but regular phenomena. Nonlinearity is the nature of matter and its development [1, 2]. Recently, iterative equation (1) has attracted increasing interest due to the potential applications in the field of wavelet analysis. In fact, the limit of iterative equation (1) satisfies refinement equation (6) which is fundamental to the theory of the scaling functions, and then we can construct special properties masks of iterative equation (1) to obtain scaling vector function with special properties. For example, in order to construct the wavelet frames with high order vanishing moments, in , Han and Mo investigate the method to factorize a matrix mask of (1). In order to construct multiwavelets with vanishing moments of arbitrarily high order, in , with the help of dimension extension and iterative scheme for revising masks of (1), Chui and Lian investigate the compactly supported orthogonal scaling function with additional polynomial preservation order (p.p.o.).
As we all know, the orthogonal symmetric scaling vector function with high p.p.o. is very important to construct symmetric multiwavelets with high vanishing moments by multiresolution analysis, and yet its construction is very difficult, especially in high dimension. The main objective of this paper is to develop an iterative equation to generate orthogonal symmetric scaling vector function as the limit of (1) and to analysis its convergence. Furthermore, we will introduce an iterative scheme by extending the dimension of iterative equation to obtain orthogonal symmetric scaling function with p.p.o increasing. As an application, we will construct compactly supported orthogonal symmetric multiwavelets to achieve any order vanishing moments.
Just as shown in [5, 6], iterative equation (1) converges to a fixed point or that the refinement equation (6) exists solution if and only if matrix satisfies Condition E (for a matrix , one says that satisfies Condition E if the spectral radius of is equal to 1, where 1 is the unique eigenvalue of on the unit circle, and it is a simple eigenvalue). The limit of (1) is called -scaling vector function if is -stable, meaning that is a Riesz basis of , where, for , which is also called multiresolution analysis (MRA) of , provided that .
According to [6, 7], iterative equation (1) converges to an -dimension scaling vector function if and only if the matrix and the matrix satisfy Condition E, where, , is the matrix given by , and “” denotes the Kronecker product, that is, for two matrices and , .
Definition 1 (see ). Let be -dimension scaling function, and if polynomial , , where , then we get that has polynomial preservation order (p.p.o.).
Polynomial preservation order is a desired feature to a scaling function in application, for example, to construct multiwavelet with high-order vanishing moments. In , Plonka studied the polynomial preservation order properties of refinable function vectors in detail. Lian, in , established certain necessary and sufficient conditions for a multiscaling function with p.p.o. in terms of the eigenvalues and their corresponding eigenvectors of masks of (1). Iterative equation (1) generates -dimension scaling function with p.p.o. , if and only if the matrix masks of (1) satisfy order sum rules; that is, there exists real vector with , for any such that
A function vector is said to be orthogonal if it satisfies , ; . If satisfying refinement equation is orthogonal scaling function vector, then the masks of (1) must satisfy condition
A scaling function vector has symmetry property if all of its components are either symmetric or antisymmetric. The symmetry of is decided by the masks of (1). From , let be symmetric scaling vector function generated by (1) with two-scale symbol , if and only if (see ) where , with , being either 0 or 1, depending on symmetry or antisymmetry of the corresponding components of , respectively.
Let be -dimension orthogonal symmetric scaling function which satisfies (6) with two-scale symbol satisfying where , for a nonnegative integer , . In this case, if satisfies (11) and (1) generates orthogonal symmetric scaling function vectors , then are symmetric about and are antisymmetric about ).
If there exist matrices , such that being orthogonal matrix, one can define as follows: and then we have
Let be r-dimension orthogonal symmetric scaling function satisfying refinement equation (6) with two-scale symbol satisfying (11). If is defined by (14), define by and then is -dimension orthogonal symmetric multi-wavelet function; that is, is the orthogonal basis of . When scaling function has , one obtains that multi-wavelet function has order vanishing moments; that is,
3. Main Results
At first, we give the following convergent lemma.
Lemma 2. Iterative equation (1) with any given compact supported vector-valued function converges to a unique vector function , if and only if the spectral radius of is equal to 1, 1 is the unique eigenvalue of on the unit circle, and 1 is simple.
Proof. Using Fourier transform, from (1), (4), and (5), we obtain that the iterative equation (1) converges to vector function if and only if the infinite product (5) converges. From , it is equivalent to the spectral radius of which is equal to 1, 1 is the unique eigenvalue of on the unit circle, and 1 is simple. This completes the proof.
When iterative equations (1) with masks generate -dimension orthogonal symmetric scaling function with , we can establish the following theorem to increase scaling function p.p.o. by extending the dimension of iterative equation (1).
Theorem 3. Let be -dimension orthogonal symmetric scaling function with generated by iterative equation (1) with mask , where satisfying (11), and Construct -dimension iterative equations mask as follows: where defined by (14). Then there exists matrix such that iterative equation (1) with mask generates -dimension orthogonal symmetric scaling function which has .
Proof. First, if matrix satisfies the conditions of Lemma 2, then obviously, matrix constructed by (18) satisfies all conditions of Lemma 2. That is to say that the -dimension iterative equation (1) with mask constructed by (18) converges to .
By applying the of and sum rules of (8), there exist , with , for satisfying
When , there is no -dimension row vector which satisfies (19). Now we will show that satisfies order sum rules by choosing matrix .
Let , , , , being some -dimension vector, then it is easy to obtain that satisfies order sum rules; that is for , we have When , (20) is equivalent to When , for with , there does not exist satisfying (21), but we will show that by choosing matrix and , there exists satisfying (21), thus satisfies order sum rules. Let
For with , (19) is equivalent to the following equation (23) having no solutions:
Because are symmetric which satisfy (11) and (14), the following matrix is nonsingular:
We claim that the following system of linear equations (25) has solutions :
For (25), (26), and (27), we have that the row vectors , , and matrix satisfy (21), that is to say that the scale symbol satisfies sum rules order .
In the following, we will show that is orthogonal symmetric two-scale symbol. For , satisfying (11), (14), and the defined matrix , it is easy to obtain
By (21) and (26)–(29), we have showed that is orthogonal symmetric two-scale symbol and satisfies at least order sum rules, that is to say that iterative function system (1) with mask generates -dimension orthogonal symmetric scaling function vector with . This completes the proof of Theorem 3.
By applying Theorem 3 to a pair of orthogonal symmetric scaling and wavelet vector functions, not only do we obtain a new scaling vector function with higher p.p.o., but also some corresponding orthogonal symmetric multi-wavelet vector function can be easily constructed. Precisely, we have the following.
Theorem 4. Let , be -dimension orthogonal symmetric scaling and wavelet function vectors with two-scale symbols , satisfying (11) and (14), respectively, where has , and then we can construct according to Theorem 3, and -dimension corresponding two-scale symbol by where , , being and order matrices, respectively, satisfies condition
Defining , generated by , , then , are orthogonal symmetric, and has at least vanishing moments.
Proof. First, we show that satisfies symmetry conditions. In Theorem 3, we have proved that satisfies symmetric condition For matrix given by (26) and (27), with , being and order matrices, respectively, and satisfying (14), we have that satisfies symmetric condition are orthogonal two-scale symbols, then it is easy to get From (34), when condition (31) is satisfied, and constructed by Theorem 3 and Theorem 4 are orthogonal symmetric two-scale symbols and wavelet functions with at least order vanishing moments. This completes the proof of Theorem 4.
One of the important features of the construction procedure described in Theorem 3 and Theorem 4 is that it can be applied repeatedly without increasing the support (or filter length). In Theorem 3, matrix is decided by two-scale symbol . In Theorem 4, matrix is constructed by matrix with condition (31). How can we obtain matrix satisfying (31)? Considering matrix given by (26) and (27), we have the following theorem.
Theorem 5. Let matrix be given by (26) and (27), and let matrix satisfy (31) with being and order matrices, respectively, and then can be obtained by where are unit orthogonal complement vectors of , and are unit orthogonal complement vectors of .
Proof. Condition (31) is equivalent to For the and , we have Matrix has eigenvalue and 1. For (26) and (27), we have . Characteristic unit vectors correspond to eigenvalue and which are orthogonal complement vectors of and characteristic unit vectors correspond to eigenvalue 1 of matrix , and then we can get From (37) and (38), we obtain In the same way, we have This completes the proof of Theorem 5.
Applying Theorems 3, 4, and 5, it is easy to extend the -dimension orthogonal symmetric multi-wavelet to -dimension orthogonal symmetric multi-wavelet with vanishing moments increasing. In Theorem 3, if matrix is constructed by (27) with or , we can extend the -dimension orthogonal symmetric multi-wavelet to -dimension orthogonal symmetric multi-wavelet with vanishing moments increasing without increasing the support of wavelet functions, and then we will give two examples to show it.
Example 6. Let
be two-scale symbols and satisfy
and then (1) generates orthogonal symmetric 2D scaling function with , and . From Theorem 3, let , , and define , , and then we obtain , as follows:
Two-scale symbols , generate 3D scaling and wavelet function vectors with , with vanishing moment with order 2, and , , symmetric, , , antisymmetric. satisfies sum rules with order 2 with , .
Example 7. Let with , be 2D orthogonal symmetric scaling and wavelet function, respectively, generated by two-scale symbols  where which satisfy By applying Theorems 3–5, we obtain a new pair of orthogonal scaling function with and multi-wavelet that can be obtained from two-scale symbols , given by where In addition, satisfies sum rules with order 4 with the four vectors given by
This study is supported by the Natural Sciences Foundation of China (11171100) and the Scientific Research Found of Hunan Provincial Education Department (no. 11W012).
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Copyright © 2013 Juan Su and Xinhan Dong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.