A new proof of some polynomial inequalities related to pseudo-splines

Pseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], which also appeared in [A. Cohen, J.P. Conze, Régularité des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351–365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=N−1. In this note, we will give a new insight into this proposition.     

Pseudo-splines,wavelets and framelets

The first type of pseudo-splines were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46; I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163–181] to construct tight framelets with desired approximation orders via the unitary extension principle of [A. Ron, Z. Shen, Affine systems in $L_2({\mathbb{R}}^d)$: The analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997) 408–447]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provide a rich family of refinable functions. B-splines are one of the special classes of pseudo-splines; orthogonal refinable functions (whose shifts form an orthonormal system given in [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996]) are another class of pseudo-splines; and so are the interpolatory refinable functions (which are the Lagrange interpolatory functions at $\mathbb{Z}$ and were first discussed in [S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–204]). The other pseudo-splines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type. This gives a wide range of choices of refinable functions that meets various demands for balancing the approximation power, the length of the support, and the regularity in applications. This paper will give a regularity analysis of pseudo-splines of the both types and provide various constructions of wavelets and framelets. It is easy to see that the regularity of the first type of pseudo-splines is between B-spline and orthogonal refinable function of the same order. However, there is no precise regularity estimate for pseudo-splines in general. In this paper, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity of the corresponding wavelets and framelets. The asymptotical regularity analysis, as the order of the pseudo-splines goes to infinity, is also provided. Furthermore, we show that in all tight frame systems constructed from pseudo-splines by methods provided both in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46] and this paper, there is one tight framelet from the generating set of the tight frame system whose dilations and shifts already form a Riesz basis for $L_2(\mathbb{R})$.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Affine Frames of Multivariate Box Splines and their Affine Duals

We give a simple and explicit construction of primal and dual wavelet filters based on refinable multivariate splines (with respect to dilation matrices M) such that the corresponding wavelet functions generate dual affine frames of arbitrarily high regularity. Furthermore, the number of wavelets does not depend on the regularity. We apply the method also to generalized B-splines.     

A Construction of Biorthogonal Functions to B-Splines with Multiple Knots

We present a construction of a refinable compactly supported vector of functions which is biorthogonal to the vector of B-splines of a given degree with multiple knots at the integers with prescribed multiplicity. The construction is based on Hermite interpolatory subdivision schemes, and on the relation between B-splines and divided differences. The biorthogonal vector of functions is shown to be refinable, with a mask related to that of the Hermite scheme. For simplicity of presentation the special (scalar) case, corresponding to B-splines with simple knots, is treated separately.     

On stable refinable function vectors with arbitrary support

Refinable function vectors with arbitrary support are considered. In particular, necessary conditions for stability are given and a characterization of the symbol associated with a stable refinable function vector in terms of the transfer operator is provided: this is a generalization of Gundy’s theorem to the vector case. The proof adapts the tools provided in [S. Saliani, On stability and orthogonality of refinable functions, Appl. Comput. Harmon. Anal. 21 (2006) 254–261]. Though complications arise from noncommuting matrix products, the fundamental ideas are the same.     

三元双正交小波滤波器的刻画

研究由三元双正交插值尺度函数构造对应的双正交小波滤波器的矩阵扩充问题.当给定的一对三元双正交尺度函数中有一个为插值函数时,利用提升思想与矩阵多相分解方法,给出一类三元双正交小波滤波器的显示构造公式和一个计算实例.讨论了三元双正交小波包的的性质.     

Generalized Pseudo-Butterworth Refinable Functions

In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L ₂(?) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.     

Some remarks on “On the windowed Fourier transform and wavelet transform of almost periodic functions,” by J.R. Partington and B. Ünalmı

J.R. Partington and B. Ünalmı consider in their 2001 paper [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the windowed Fourier transform and wavelet transform as tools for analyzing almost periodic signals. They establish Parseval-type identities and consider discretized versions of these transforms in order to construct generalized frame decompositions. We have found a gap in the construction of generalized frames in the windowed Fourier transform case; we comment on this gap and give an alternative proof. As for the wavelet transform case, in [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the generalized frame decomposition is done only for the simplest wavelet, the Haar wavelet; we show how to construct generalized frame decompositions for a wide family of wavelets.     

Small Support Spline Riesz Wavelets in Low Dimensions

In Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006), a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in Han and Shen (J. Fourier Anal. Appl. 11:615–637, 2005). Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006) were used in a surface fitting algorithm of Johnson et al. (J. Approx. Theory 159:197–223, 2009), and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in Khodakovsky et al. (Proceedings of SIGGRAPH, 2000).     

A class of multiwavelets and projected frames from two-direction wavelets

This article aims at studying two-direction refinable functions and two-direction wavelets in the setting ?^s, s > 1. We give a sufficient condition for a two-direction refinable function belonging to L²(?^s). Then, two theorems are given for constructing biorthogonal (orthogonal) two-direction refinable functions in L²(?^s) and their biorthogonal (orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal (orthogonal) two-direction wavelets, symmetric biorthogonal (orthogonal) multiwaveles in L²(?^s can be obtained easily. Applying the projection method to biorthogonal (orthogonal) two-direction wavelets in L²(?^s, we can get dual (tight) two-direction wavelet frames in L²(?^m, where. m ≥ s From the projected dual (tight) two-direction wavelet frames in L²(?^m, symmetric dual (tight) frames in L²(?^m can be obtained easily. In the end, an example is given to illustrate theoretical results.     

A construction of interpolating wavelets on invariant sets

We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.

     

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.     

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

Linear independence of pseudo-splines

In this paper, we show that the shifts of a pseudo-spline are linearly independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (refinable) function and its shifts has been studied in several areas of approximation and wavelet theory. Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and sufficient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.

     

Compactly supported wavelet bases for Sobolev spaces

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and in satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ_jk:=2^j/2ψ(2^j·−k) ( ) form a Riesz basis for . If, in addition, φ lies in the Sobolev space , then the derivatives 2^j/2ψ^(m)(2^j·−k) ( ) also form a Riesz basis for . Consequently, is a stable wavelet basis for the Sobolev space . The pair of φ and are not required to be biorthogonal or semi-orthogonal. In particular, φ and can be a pair of B-splines. The added flexibility on φ and allows us to construct wavelets with relatively small supports.     

On Multivariate Compactly Supported Bi-Frames

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.     

Wavelets and framelets from dual pseudo splines

Dual pseudo splines constitute a new class of refinable functions with B-splines as special examples.In this paper,we shall construct Riesz wavelet associated with dual pseudo splines.Furthermore,we use dual pseudo splines to construct tight frame systems with desired approximation order by applying the unitary extension principle.