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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 818907, 7 pages
Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators
1Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
2Departamento de Análisis Matemático, Universidad de Alicante, 03080 Alicante, Spain
3Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, USA
Received 17 December 2012; Accepted 11 April 2013
Academic Editor: Sung Guen Kim
Copyright © 2013 A. San Antolín and R. A. Zalik. 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.
For any dilation matrix with integer entries and , we construct a family of smooth compactly supported tight wavelet frames with three generators in . Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.
Given a dilation matrix with integer entries, such that , we construct smooth compactly supported tight framelets with three generators in associated to such a dilation, and with any desired degree of smoothness. Tight wavelet frames have recently become the focus of increased interest because they can be computed and applied just as easily as orthonormal wavelets, but are easier to construct.
We begin with notation and definitions. The sets of strictly positive integers, integers, and real numbers will be denoted by , , and , respectively. Given a (Lebesgue) measurable set , will denote its Lebesgue measure on and will be its characteristic function. Given a matrix , its transpose will be denoted by and the conjugate of its transpose will be denoted by . The identity matrix will be denoted by .
We say that is a dilation matrix preserving the lattice if all eigenvalues of have modulus greater than and . The set of all dilation matrices preserving the lattice will be denoted by . Note that if , then is an integer greater than . The quotient group is well defined, and by we will denote a full collection of representatives of the cosets of . Recall that there are exactly cosets .
A sequence of elements in a separable Hilbert space is a frame for if there exist constants such that where denotes the inner product on . The constants and are called frame bounds. The definition implies that a frame is a complete sequence of elements of . A frame is tight if we may choose .
Let . A set of functions is called a wavelet frame or framelet with dilation if the system is a frame for . If this system is a tight frame for , then is called a tight framelet.
Let denote the Fourier transform of the function . Thus, if , , where denotes the dot product of vectors and . The Fourier transform is extended to in the usual way.
Han , and independently Ron and Shen , found necessary and sufficient conditions for translates and dilates of a set of functions to be a tight framelet. Ron and Shen also formulated what is known as the Unitary Extension Principle (UEP), which, in addition to its other applications, provides a method for constructing compactly supported framelets. In  (see also ), Ron and Shen show two bivariate constructions of compactly supported tight framelets in with dilation matrix and state that constructions of compactly supported tight framelets with dilation matrix can be made in an analogous way. They also describe an algorithm for constructing compactly supported tight affine frames in with any dilation matrix. That algorithm works particularly well with box splines. Using Ron and Shen’s method, Gröchenig and Ron  show how to construct, for any dilation matrix, compactly supported framelets with any desired degree of smoothness. Furthermore, based on works by Ron and Shen and by Gröchenig and Ron, Han  also constructs compactly supported tight wavelet frames with degree of smoothness and vanishing moments of order as large as desired. Another method for constructing smooth compactly supported tight framelets was described in . Note that while in [4, 6, 8] the number of generators of the tight framelets increases with the degree of smoothness, in  the number of generators may be bounded by a constant depending on the dimension and the determinant of the dilation matrix.
The remainder of this paper is organized as follows. In Section 2 we summarize results we will use for the proof of the main results. In Section 3, for any dilation matrix with integer entries and determinant we describe an algorithm to construct a family of compactly supported tight framelets with three generators and with any desired degree of smoothness and dilation factor . In Section 4 we illustrate the results of this paper by means of examples.
In this section we summarize the results we will use in our construction of tight framelets.
The UEP led to the oblique extension principle (OEP), a method based on the UEP; it was developed by Chui et al. , and independently by Daubechies et al. , who gave the method its name. The OEP may be formulated as follows.
Theorem A. Let . Let be compactly supported and refinable, that is, where is a trigonometric polynomial. Assume moreover that . Let be another trigonometric polynomial such that and . Assume that there are trigonometric polynomials or rational functions , , that satisfy the OEP condition If then is a tight framelet in with dilation factor and frame constant 1.
With an additional decay condition, Theorem A follows from [10, Proposition 1.11], except for the value of the frame constant, which follows from, for example, [3, Theorem 6.5]. However, recent results of Han imply that this decay condition is redundant. Indeed, Theorem A in its present formulation is a consequence of Proposition 4, Corollary 12, and Theorem 17 in  (for a simpler version of Han’s results in dimension, see ).
We also need the following slight generalization of Theorem 3.4 of Lai and Stöckler . The proof is similar and will be omitted. We have also included in the statement a generalization of the algorithm implicit in the proof of Theorem 3.4.
Theorem B. Let and let and be full collections of representatives of the cosets of and , respectively, with . Let be a trigonometric polynomial defined on that satisfies the condition:
be the polyphase matrix, where denotes the row index and denotes the column index.
Let the matrix function be defined by and suppose that there exist trigonometric polynomials such that Let and let the matrix function be defined by Let denote the first block matrix of , and let denote the first row of . Then the trigonometric polynomials and , , satisfy the identity (6) with .
Using the OEP and Theorem B, we will obtain a general method for constructing compactly supported framelets in valid for any dilation matrix.
We will use the following theorem of Gröchenig and Madych .
Theorem C. Let and let be a full collection of representatives of the cosets of with . Then the characteristic function , where the set is defined by is a nonnull compactly supported measurable function, , and it satisfies the refinement equation:
The following statement may be found in [14, Appendix A.2]. The proof is straightforward and will be omitted.
Lemma D. Let be the class of continuous functions in , and let , be the class of functions such that all partial derivatives of of order not greater than are continuous and in . If for some integer and , then is in .
Proposition E. Let and let be a full collection of representatives of the cosets of with , and let be the set defined by (15). Then there exist two positive constants and such that
3. A Family of Tight Framelets
Let be a dilation matrix with integer entries such that . In this section we construct smooth compactly supported tight framelets with three generators in with dilation factor and any desired degree of smoothness.
Two matrices and with integer coefficients are integrally similar if there exists a matrix with integer entries such that and . Let The following complete classification of all matrices in with was found by Lagarias and Wang [16, Lemma 5.2].
Lemma F. Let . If , then is integrally similar to . If , then is integrally similar to one of the matrices , .
We now focus on the dilation matrices . For , let
It is easy to see that for each the set is a full collection of representatives of the cosets of , , and that is a full collection of representatives of the cosets of , . Moreover, we have For simplicity we shall consider as in (20) with , and as in (21). Then the corresponding matrix , , defined by (10), where is any of the matrices , , is We now construct a family of smooth compactly supported refinable functions in with dilation factor .
Proposition 1. Let the matrices , be defined as in (19). Let , and . Let where is the set defined by (15) with , and is one of the matrices . Then , and the function whose Fourier transform is defined by (24) has the following properties: is nonnull, compactly supported, and square integrable on , , and satisfies the refinement equation: Moreover, if , where is defined in Proposition E, then is in continuity class .
Proof. From Theorem C we deduce that is a nonnull compactly supported function.
We now prove that . Let and for let denote the -fold convolution of with itself. Since , by Young’s inequality for convolutions and bearing in mind that , we conclude that We now show that satisfies the refinement equation (25). From (16) we have where .
Since , the definition of implies that .
We now prove the estimates for the degree of smoothness of . By Proposition E, we have Moreover, since is continuous, Hence, if , Lemma D implies that is in continuity class .
Finally, replicating an argument of Wojtaszczyk [14, page 79] it is easy to see that is a compactly supported function on .
We now construct tight framelets in with dilation matrices such that the functions are smooth and compactly supported. For this purpose we use the refinable functions that we obtained in Proposition 1, Theorem B, the Oblique Extension Principle, and an appropriate change of variables.
Let , be the matrices defined in (19), let be defined by (21), assume that , , and that defined as in Proposition 1. Using (22) we readily see that and by an elementary computation we conclude that and that equality holds only if . Since the values of the trigonometric polynomial in Proposition 1 only depend on one variable, from a lemma of Riesz (cf., e.g., [17, Lemma 6.1.3], [14, Lemma 4.6]) we know there is a nonnull trigonometric polynomial on such that The coefficients of may be obtained by spectral factorization ().
The following theorem describes the construction of smooth tight framelets of compact support and arbitrary degree of smoothness with dilation matrix .
Theorem 2. Let , . Let , , let , and let where is a trigonometric polynomial that satisfies (32). If is defined by (24), and is the set of inverse Fourier transforms of the functions defined in the preceding displayed identity, then is a tight framelet with dilation factor and frame constant 1, and the functions have compact support. Moreover, if , where is defined in Proposition E, then the functions are in continuity class .
Proof. Recall that the set defined by (20) is a full collections of representatives of the cosets of , and that the set defined by (21) is a full collection of representatives of the cosets of . Thus, the inequality (31) allows us to apply Theorem B.
From (9) we see that whence (23) implies that Thus, we have
Let be defined as in (32). Setting we therefore conclude that Applying the algorithm described in Theorem B with and , and bearing in mind that , we obtain
Theorem B guarantees that the trigonometric polynomials , , and , satisfy the identity (6) with and . This may also be verified by direct computation. Applying Theorem A we conclude that is a tight framelet in .
Since the functions are trigonometric polynomials and therefore bounded on , the smoothness of the functions follows from (29) and Lemma D.
Finally, note that the functions are compactly supported because is compactly supported and the are trigonometric polynomials.
Using Theorem 2 and Lemma F we now obtain an algorithm for constructing tight framelets in with dilation factor , for any matrix preserving the lattice and with . These framelets have three compactly supported generators of arbitrary degree of smoothness.
Corollary 3. Let with and let be such that there exists a matrix with , such that . Let , , , let be defined by (24), and let be the set of inverse Fourier transforms of the functions defined by (34). If then is a tight framelet with , and the functions have compact support. Moreover, if , where is defined in Proposition E, then the functions are in continuity class .
Proof. Since , the assertion that is a tight framelet in with frame constant 1 readily follows by a change of variable of the form .
Let be an integer such that , where is defined in Proposition E. Since is in continuity class , applying the chain rule we conclude that also is in continuity class .
In this section, we illustrate the results of this paper showing examples of tight framelets where the use of spectral factorization may be avoided.
Example 4. Let , , , , , and be the set defined by (21). We can obtain a tight framelet by elementary computations. Since a choice for is . Thus, if we set and if is defined by (24) with , and is the set of inverse Fourier transforms of the functions defined in the preceding displayed identity, then is a tight framelet in with dilation factor , and the functions are compactly supported.
Example 5. Let , , , , , and be the set defined by (21). The case when is also sufficiently simple that it does not require the use of spectral factorization algorithms to compute . We will just apply the arguments used to prove the lemma of Riesz and simple trigonometric identities.
From (37) we readily see that
We define If , we see that , where . Since , where , we conclude that Setting and bearing in mind that , this yields where
Let Then is a zero of and is a zero of .
From the discussion in  or  we see that if , then Since , we have Thus, a choice for is We now set where is defined by (52) with given by (49). If is defined by (24) with , and is the set of inverse Fourier transforms of the functions defined in the preceding displayed identity, then is a tight framelet in with dilation factor , and the functions have compact support.
A. San Antolín was partially supported by no. PB94-0153. R. A. Zalik is grateful to Instituto Argentino de Matemática for its hospitality during the completion of this work.
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