Abstract
Drawing inspiration from the idea of combining multiwavelets on the interval with frame theory organically, we study minimum-energy multiwavelet frame on the interval (MEMWFI). Firstly, left boundary multiscaling functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward, and the equivalent characterizations of MEMWFI are given. Then, two algorithms of constructing MEMWFI are proposed. Finally, the decomposition formula, reconstruction formulas, and numerical examples are given.
1. Introduction
The notion of frame was introduced by Duffin and Schaeffer [1]. It is a generalization of basis and can be viewed as some kind of “overcomplete basis.” A basis in a Hilbert space allows each element to be written as a linear combination of the elements in the basis, and the combination is unique. An overcomplete frame also allows one to represent each element via it, but the representation is not unique. This property plays an important role in mathematics, signal analysis, time-frequency analysis, and so on [2–20]. In nearly thirty years, frame theory has been growing rapidly. Wavelet frames are a class of important frames among all kinds of frames.
Let be a complex, separable Hilbert space. A sequence of elements in is a frame for if there exist constants such thatThe numbers are called frame bounds. A frame is tight if we can choose as frame bounds. A frame is exact if it ceases to be a frame when an arbitrary element is removed. When , then Chui and He [6] proposed the concept of minimum-energy wavelet frames, which can reduce the computational complexity and maintain the numerical stability, and do not need to search dual frames in the decomposition and reconstruction of functions. Therefore, many people paid a lot of attention to the study of minimum-energy wavelet frames. Petukhov [8] studied minimum-energy wavelet frames with symmetry. Huang and Cheng [16] studied minimum-energy wavelet frames with arbitrary integer dilation factor. Gao and Cao [17] researched minimum-energy wavelets frames on the interval and its applications. Huang and Li [18] studied the algorithms of constructing minimum-energy wavelet frames on the interval. Recently, Huang et al. [19] and Liang and Zhao [20] have been concerned with the algorithms of constructing minimum-energy multiwavelet frames. In [19], the authors gave the definitions of multiwavelet frame multiresolution analysis for and minimum-energy multiwavelet frame. But, in many practical applications, one is often interested in the problems which are confined to a finite interval, such as the differential equation solving, image processing, and signal analysis. Therefore, we should pay more attention to minimum-energy multiwavelet frames on the interval; under this case, we can combine the features (symmetry, compact support, and regularity) of multiwavelets on the interval with the superiority (redundancy, good numerical stability, and lower computational complexity) of minimum-energy wavelet frames.
Throughout this paper, let denote the set of integers and real numbers, respectively. denotes the set of positive integers. Let , and .
The organization of this paper is as follows. In Section 2, we give the main results; left boundary multiscaling functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward; the equivalent characterizations of MEMWFI, two algorithms for constructing MEMWFI, and the decomposition and reconstruction formulas of MEMWFI are proposed. Finally, numerical examples are given in Section 3.
2. Main Results
In order to give the definitions of MEMWFI, left boundary multiscaling functions, and right boundary multiscaling functions, we let .
Definition 1. If a closed subspaces sequence in satisfies; is dense in ;;, ;, ;there exists a vector-valued function with such that forms a frame of , where is the biggest integer number to make , then we say is a multiwavelet frame multiresolution analysis on the interval with dilation factor and multiplicity or a multiwavelet frame multiresolution analysis on the interval and denote it as MWFMRA . The function is vector-valued scaling function (multiscaling function).
For a MWFMRA , we let . If we can find vector-valued functions such that is a frame for , then we say is a multiwavelet frame on the interval, where .
2.1. Left and Right Boundary Multiscaling Functions
Let be a vector-valued scaling function and generate a multiwavelet frame multiresolution analysis for with corresponding multiwavelet frame . Suppose satisfies the refinable equation with .
Let .
Theorem 2. Define left boundary multiscaling functions and right boundary multiscaling functions as follows:where are coefficient matrices. Denote If and make and satisfy frame condition, respectively, then there exist coefficient matrices , such that the matrix-valued refinable equations of are expressed as follows: where , and can form a frame of when .
Proof. The proof is similar to Theorem 2.2 in [17].
Theorem 3. Let and be the left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2. Let vector-valued functions , , be expressed as follows: where . Denote If coefficient matrices such that and satisfy frame condition, respectively, then is a frame of for any .
Proof. The proof is similar to Theorem 3.2 in [17] and Theorem 3.2 in [18].
Definition 4. Let be defined in Theorems 2 and 3, respectively. If any , one has and then the family of functions are called a minimum-energy multiwavelet frame on the interval (MEMWFI) associated with dilation factor and multiplicity .
By Parseval identity, (10) is equivalent to
2.2. Construction of MEMWFI
For convenience, we denote the following:(1).(2).(3) is dimension column vector; and are left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2: (4) are dimension column vector; and are multiwavelet frames defined in Theorem 3: where . If is seen as a set of functions, then .
Define where are block matrices as follows:
Similarly, we can define where have the same form as , respectively.
Then, by Theorems 2 and 3, we have The Fourier transform of (17) is where
With these , we will give the characterizations of MEMWFI.
Theorem 5. The following statements are equivalent:(1) is a MEMWFI.(2)For any , (3)For any , where denote the block matrices from th row to th row and th row to th column in , respectively, and denotes the complex conjugate of the transpose of .
Proof. By using Theorems 2 and 3, (11) can be rewritten as For convenience, for every , we denote as dimension column vector , where the th component of is . Similarly, we denote as dimension column vectors . Then, the former formula is equivalent to which is equivalent to or where .
We multiply the same identities in (20) by , to get and this is equivalent to or In other words, the proof of theorem can be reduced to the proof of the equivalence of (21) and (25). It is obvious that (21) ⇒ (28) ⇒ (25). To show (25) ⇒ (21), take any and define The Fourier transform of (25) is Then ; that is, Because is a compact support function, that is,Then, we can get and this is equivalent to .
Related to the proof, is zero vector with appropriate size, and is zero matrix.
Theorem 5 characterizes the necessary and sufficient condition for the existence of the MEMWFI associated with . But it is not a good choice to use this theorem to construct MEMWFI with arbitrary dilation . For convenience, we need to present some algorithms of constructing MEMWFI.
2.3. Algorithms of Constructing MEMWFI
With matrices , we formulate the following block matrix: and then, (21), the characterization of MEMWFI, can be rewritten as
Now we present two algorithms of constructing MEMWFI starting from a multiscaling function and a scalar scaling function, respectively.
Algorithm 1. (1) Take a proper multiscaling function .
(2) Calculate the parameters .
(3) Choose proper coefficient matrices and , and construct boundary multiscaling functions using (5); in addition, we can get .
(4) Use Theorem 2, ascertain coefficient matrices and to get the block matrix .
(5) Take coefficient matrices ; and to construct the block matrixes , and ensure that they satisfy (21) with .
(6) Use the coefficient matrices which we have got in step (5) to construct MEMWFI , by Theorem 3.
(7) Draw the graphs of and by Matlab.
Algorithm 2. (1) Take a proper scalar scaling function (the quadratic sum of recursion coefficients can not be larger than 1).
(2) Use to get refinable vector-valued function ; for example, and so on; we can find the sequence of matrices which satisfies refinable equation with . We should know that the necessary and sufficient conditions for MEMWFI in Theorem 5 reduce to sufficient conditions when the set of components in the vector-valued function is linearly dependent.
(3) Turn to steps (3)–(7) of Algorithm 1.
We should note that the refinable vector-valued function in Algorithm 2 can not be a multiscaling function.
2.4. Decomposition Formula and Reconstruction Formula of MEMWFI
The decomposition formula and reconstruction formula of MEMWFI are obtained by Theorems 2, 3, and 5, which are similar to those of orthogonal wavelets or multiwavelets on the interval. Firstly, for , denote
Theorem 6. The decomposition formula of MEMWFI isand the reconstruction formula of MEMWFI is
Proof. The decomposition formula can be obtained by Theorems 2 and 3, and the reconstruction formula is gotten from Theorems 2, 3, and 5.
3. Numerical Examples
In this section, we present some numerical examples to show the effectiveness of the proposed algorithm with dilator factors .
is th order cardinal B-spline with dilation factor ; the refinement equation of is the following: where .
3.1.
Example 1. With , the symbol of the B-spline is Take , and satisfies In fact, for , we can choose different set of coefficient matrices such that it satisfies the above equation.
Let . Taking , . Then, we can obtain a set of multiscaling functions on the interval by Theorem 2 and coefficient matrices by Theorem 5, The graphs of , left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 1.
The graphs of the first row in Figure 1 are the multiscaling function and its boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with .
We can discover from Figure 1 that each component of multiscaling function and MEMWFI is (anti)symmetrical. The boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric.
Example 2. With , the symbol of 3rd order B-spline is Take , and satisfies Certainly, for , we can choose different set of coefficient matrices such that it satisfies the above equation.
Let . Taking . By Theorems 2 and 5, we can obtain that The graphs of , left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 2.
The graphs of the first row in Figure 2 are multiscaling function and left (right) boundary multiscaling functions, and the rest of graphs belong to MEMWFI.
We can discover from Figure 2 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.
Example 3. The previous two examples are constructed when ; we will give an example based on B-spline with here.
With , the symbol of 4th order B-spline is Take , and satisfies Let . Taking . Then, by Theorems 2 and 5, we can get that The graphs of , left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 3.
The graphs of the first row in Figure 3 are multiscaling function and left (right) boundary multiscaling functions, and the rest rows belong to MEMWFI.
We can discover from Figure 3 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.
3.2.
Example 4. We will construct this numerical example based on 2nd B-spline . With , the symbol of is Take , and satisfies In fact, we have other choices to get the refinable equations associated with .
Let . Taking where denotes zero matrix, denotes unit matrix. Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 4). It is so lengthy that we do not write the expression of here. The graphs of , left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 4.
The graphs of the first row in Figure 4 are multiscaling function and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI.
We can discover from Figure 4 that each component of multiscaling function and MEMWFI is (anti)symmetric in this example. Left (right) boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric. The second right boundary multiscaling function has the same graph with the first.
Example 5. We will construct this numerical example based on 3rd B-spline . With , the symbol of is Take , and satisfies Certainly, we have other choices to get the refinable equations associated with . The components of the multiscaling function are linear dependence. It does not satisfy the assumption in Section 2. The sufficient and necessary conditions for minimum-energy frames in Theorem 5 reduce to sufficient conditions. We can still use Theorem 5 to get MEMWFI.
Now . Take Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 5). It is so lengthy that we do not write the expression of here. The graphs of , left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 5.
The graphs of the first row in Figure 5 are and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 61261043), the Natural Science Foundation of Ningxia (Grant no. NZ13084), and Research Project of Beifang University for Nationalities (Grant no. 2013QZP07).