Compactly supported tight and sibling frames with maximum vanishing moments

The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline N_m. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry (or antisymmetry), minimum support, “shift-invariance,” and inter-orthogonality. For N_m, it turns out that symmetry can be achieved for even m and antisymmetry for odd m, that minimum support and shift-invariance can be attained by considering the frame generators with two-scale symbols 2^−m(1−z)^m and 2^−mz(1−z)^m, and that inter-orthogonality is always achievable, but sometimes at the sacrifice of symmetry. The results in this paper are valid for all compactly supported refinable functions that are reasonably smooth, such as piecewise Lipα for some α>0, as long as the corresponding two-scale Laurent polynomial symbols vanish at z=−1. Furthermore, the methods developed here can be extended to the more general setting, such as arbitrary integer scaling factors, multi-wavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function).     

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.     

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.     

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.     

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 cardinal wavelets with compact support and their duals

Some people try to construct an orthonormal wavelet such that the corresponding scaling function φ(t) has the cardinal property,i.e. ϕ(n)= σn0, since such wavelets have many good applications. Unfortunately it is impossible to do so, except for a trivial case^[1]. In this work, a family of non-orthogonal cardinal wavelets with compact support is constructed and their duals are investigated. This work is supported by the project of new stars of Beijing     

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.     

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.     

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 $L_2(\mathbb{R})$.     

A Class of Orthogonal Refinable Functions and Wavelets

We give a construction, for any n 2, of a space S of spline functions of degree n – 1 with simple knots in (1/4)Z which is generated by a triple of refinable, orthogonal functions with compact support. Indeed, the result holds more generally by replacing the B-spline of degree n – 1 with simple knots at the integers by any continuous refinable function whose mask is a Hurwitz polynomial of degree n. A simple construction is also given for the corresponding wavelets.     

Orthonormal Wavelets and Tight Frames with Arbitrary Real Dilations

The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L²:=L²( ). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L² function for an arbitrary dilation a>1 that possess “good” time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a>1 is irrational such that a^j remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a>1. Generalization to dual frames is also discussed in this paper.     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

The Sobolev Regularity of Refinable Functions

Refinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms on the other. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L₂-regularity of the refinable function φ in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function φ, the relevant invariant space is proved to be finite dimensional and is completely characterized in terms of the dependence relations among the shifts of φ together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported and that the shifts of φ are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may be abysmal. Our characterization is further extended to the FSI (i.e., vector) case, to the unisotropic dilation matrix case, and to even snore general setups. We also establish corresponding results for refinable distributions.     

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.     

Boundary value problems for second order nonlinear differential equations on infinite intervals

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L²(0,∞) and L²(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established.     

Tight frames and geometric properties of wavelet sets

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝ^d is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L²(ℝ^d) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions. Dedicated to Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C40. John J. Benedetto: Both authors gratefully acknowledge support from ONR Grant N000140210398. The first named author also gratefully acknowledges support from NSF DMS Grant 0139759.     

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

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withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.