Compactly Supported Tight and Sibling Frames Based on Generalized Bernstein Polynomials

We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.


Introduction
Because it is highly desirable to construct wavelets within a class of analytically representable functions, compactly supported sibling frames with interorthogonality attract a considerable amount of attention, recently.
In 1997, Ron and Shen completed the structure of the affine system, which can be factored during a multiresolution analysis construction.This leads to a characterization of all tight frames that can be constructed by the methods in [1].In 2000, compactly supported tight frames that correspond to refinable functions were studied and a constructive proof was given by Chui and He [2].In [3], Han gave his investigation of symmetric tight framelet filter banks with a minimum number of generators and systematically studied them with three high-pass filters which are derived from the oblique extension principle.In 2002, compactly supported tight and sibling frames, with symmetry (or antisymmetry), minimum support, shift-invariance, and interorthogonality, were constructed in [4].In 2003, Daubechies et al. discussed wavelet frames constructed via multiresolution analysis, with emphasis on tight wavelet frames.More importantly, they established general principles and specific algorithms for constructing framelets and tight framelets in [5].In 2005, Averbuch et al. [6] obtained tight and sibling frames originated from discrete splines, in which, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets.Next, in [7], symmetric wavelets dyadic sibling and dual frames, where each of the frames consists of three generators obtained using spectral factorization, were given.In 2007, a new type of pseudo-splines was introduced to construct symmetric or antisymmetric tight framelets with desired approximation orders by Dong and Shen [8].And they provided various constructions of wavelets and framelets.In 2013, Shen and Xu [9] give -Spline framelets derived from the unitary extension principle, which led to the result that the wavelet system is generated by finitely many consecutive derivatives.More tight frames have been studied in [10][11][12][13][14][15][16][17][18][19][20], so far.
This paper is concerned with the construction of compactly supported tight and sibling frames based on generalized Bernstein polynomials [21], defined as  ()   () = ( Mathematical Problems in Engineering where  ≥ 0. We complete the convergence of cascade algorithms associated with the new masks.Furthermore, the symmetry, regularity, and approximation orders of corresponding refinable functions are analyzed.At last, we implement interorthogonality of sibling frames.The remainder of this paper is organized as follows.In Section 2, some notations about refinement marks are collected and some technical lemmata are given to use in other sections.We will elaborate on convergence of cascade algorithms based on the masks, which guarantees the existence of refinable functions in Section 3. Section 4 analyzes the symmetry and gives a symmetry proof.In Section 5, regularity and approximation orders are focused on study; at the same time, we obtain the lower bound of the regularity exponents of refinable functions by estimating the decay rates of their Fourier transform.At last, we construct tight and sibling frames and obtain interorthogonality of sibling frames in Section 6.

Preliminaries
For the convenience of the readers, we review some definitions and properties about refinement marks in this section.
New marks based on generalized Bernstein polynomials (1), with order (, , ), for given nonnegative integers , , and  ≥ 0, are defined as follows: ) . ( For notational simplicity, we will introduce the following two definitions: By  2 (R), we denote all the functions () satisfying and  2 (Z) the set of all sequences  defined on Z such that In the following, we will give a compactly supported realvalued refinable function  : R → R with finite mask and real mask coefficients; that is,  satisfies a two-scale relation: for some real numbers   .Assume that the corresponding two-scale Laurent polynomials satisfy for some  ≥ 1, with a Laurent polynomial  0 that satisfies  0 (−1) ̸ = 0.The Fourier transform of  is And,  satisfies With the above, the refinement equation ( 6) can be written in terms of its Fourier transform as where (/2) = (),  =  −/2 .We call  the refinement mask for convenience, too.By the iteration of (11), the corresponding refinable function  can be written in terms of its Fourier transform as In the following, we will adopt some of the notations from [2,4,22].The transition operator  â for 2-periodic functions â and  can be defined as .
The notation (â) is defined by For convenience, assume that  is piecewise Lip , for some  > 0.
A function  belongs to the Hölder class   (T) with  > 0, if  is a 2-periodic continuous function such that  is  times continuously differentiable and there exists a positive number  satisfying for all ,  ∈ T, where  is the largest integer such that  ≤ .We use for approximation of  ∈  2 (R).And a function  satisfies the Strang-Fix condition of order  if Under certain conditions on  (e.g., if it is compactly supported and φ(0) = 1), the Strang-Fix condition is equal to the requirement that τ0 has a zero of order  at each of the points in {0, } \ 0. In [5], if  satisfies the Strang-Fix condition of order  and the corresponding mark  satisfies that 1−|(⋅)| 2 = (|⋅|  1 ) at the origin, then the approximation order is min{,  1 }.
We will provide some lemmas which are necessary for the following theorem.The following lemmas are about the relations of the quantities   (â, ∞) associated with masks and a condition of the convergence of cascade algorithms.
For regularity, our primary goal is to obtain the lower bound of its exponents  ,, 0 of refinable functions  ,, by Mathematical Problems in Engineering estimating the decay rates  ,, 0 of their Fourier transform.The relation is expressed by for any small enough  > 0; see [23].Consequently,  ,, ∈   ,, 0 . Next, we will give an estimate of the decay rates  ,, 0 of the Fourier transform of refinable functions  ,, with the mask  ,, ().By [23,24], for any stable, compactly supported refinable functions  in  2 (R) with φ(0) = 1, the refinement mask  must satisfy (0) = 1 and () = 0. Thus,  can be factorized as where  is the maximal multiplicity of the zeros of  at  and L() is a trigonometric polynomial with L(0) = 1.Therefore, one obtains The following lemmas are useful for obtaining the important tight and sibling frames.Lemma 5 (see [4,Theorem 2]).For any compactly supported refinable function  that satisfies ( 8)-( 16), there exist compactly supported sibling frames { 1 ,  2 }, {ψ 1 , ψ2 } with the property that all of the four functions have  vanishing moments, where  is the order of the root  = −1 of the two-scale Laurent polynomial P. Furthermore, if  is symmetric, then all of the four functions can be chosen to be symmetric for even  and antisymmetric for odd .Lemma 6 (see [4,Theorem 3]).For any compactly supported refinable function  that satisfies ( 8)-( 16), there exists a pair of sibling frames ( 1 ,  2 ) and (ψ 1 , ψ2 ) such that all of the four functions have compact support and the maximum number  of vanishing moments and that ( 1 ,  2 ) is interorthogonal.Lemma 7 (see [4,Theorem 8]).Let {}, {ψ} be a pair of compactly supported sibling frames associated with a VMR function .If  is Laurent polynomial, then the function   ∈  1 with two-scale symbol   , where   = (/√(−1))()(−1/), defines a tight frame of  2 which is associated with the same VMR function .

Convergence of Cascade Algorithms Based on the Masks
In this section, demonstration of the convergence of cascade algorithms in the space  2,∞ (R) is given.To complete it, a useful condition of proving the convergence of cascade algorithms is described as follows.

Symmetry
Symmetric coefficients of the mark are of great significance in image processing.The following lemma is helpful for the demonstration of symmetry.

Regularity and Approximation Orders
This section is devoted to analysis of the regularity and approximation orders of refinable functions  ,, with the mask  ,, () defined by (2) in the following theorem., where  ,, 0 ≥  ,, 0 − 1 − , for any small enough  > 0.

Tight and Sibling Frames
In this section, tight and sibling frames are constructed in the following theorem.At the same time, the interorthogonality of sibling frames is implemented.

Conclusions
In this paper, we study new marks (2)  (sin 2 (/2) + )) with two positive integers , , satisfying  <  − 5, to provide derived properties.The convergence of cascade algorithms in Theorem 9 is obtained, which guarantees the existence of refinable functions.In Theorem 11, we analyze the symmetry of the refinable functions, which is of importance.The regularity and approximation order of the new refinable functions are given; at the same time, the lower bound of the regularity exponents of refinable functions is showed by estimating the decay rates of their Fourier transform.Finally, we construct tight and sibling frames and demonstrate interorthogonality of sibling frames in Section 6.And, numerical examples are given to illustrate the construction of the proposed approach.