The limits of refinable functions

A function is refinable ( ) if it is in the closed span of . This set is not closed in , and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.

     

Refinable bivariate quartic -splines for multi-level data representation and surface display

In this paper, a second-order Hermite basis of the space of -quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of -quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.

     

Approximation properties of multivariate wavelets

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in provides approximation order .

     

Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on Vilenkin group $\mathfrak{G}$ . We consider a class of refinable step functions for which the mask m ₀(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$ and its modulus |m ₀(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$ . We show the sharpness of this result, too.     

Polynomial interpolation, ideals and approximation order of multivariate refinable functions

The paper identifies the multivariate analog of factorization properties of univariate masks for compactly supported refinable functions, that is, the ``zero at '-property, as containment of the mask polynomial in an appropriate quotient ideal. In addition, some of these quotient ideals are given explicitly.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

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.

     

Refinable subspaces of a refinable space

Local refinable finitely generated shift-invariant spaces play a significant role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begin with a refinable function which is supported on . We are interested in spaces generated by a function built from the shifts of .

     

Directions for computing truncated multivariate Taylor series

Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of variables by fixing a direction in . After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or ``interpolated'. The sequence of univariate calculations is more efficient than multivariate methods, although previous work indicates a space cost for this savings and significant cost for the reconstruction. We completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction. By appropriate choice of directions, the reconstruction reduces to a sequence of Lagrange polynomial interpolation problems in for which a divided difference algorithm computes the coefficients of a Newton form. Another algorithm collects like terms from the Newton form and returns the desired multivariate coefficients.

     

Interpolatory Subdivision Schemes Induced by Box Splines

This paper is devoted to a study of interpolatory refinable functions. If a refinable function φ on ^sis continuous and fundamental, i.e., φ(0)=1 and φ(α)=0 for α ^s\{0}, then its corresponding mask bsatisfies b(0)=1 and b(2α)=0 for all α ^s\{0}. Such a refinement mask is called an interpolatory mask. We establish the existence and uniqueness of interpolatory masks which are induced by masks of box splines whose shifts are linearly independent.     

Frames in spaces with finite rate of innovation

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space is generated by a family of well-localized molecules of similar size located on a relatively separated set using coefficients, and hence is locally finitely generated. Moreover that space includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator , we show that if the generator is a frame for the space and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator for the space with , and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.      

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.     

The number of certain integral polynomials and nonrecursive sets of integers, Part 1

Given 2$">, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the ``cube' with real coordinates from into . This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable.

     

Universal Formulae for SU Casson Invariants of Knots

An Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the Atiyah-Bott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all . Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.

     

Computation of the Newton step for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix

We compute the Newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix as the reciprocal of the trace of an appropriate matrix. We show that, after the Yule-Walker equations are solved, this trace can be computed in additional arithmetic operations, which is in contrast to existing methods, which rely on a recursion, requiring additional arithmetic operations.

     

On homeomorphic Bernoulli measures on the Cantor space

Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability' and show that, if and are refinable and binomially equivalent, then and are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers and such that and are refinable, so the corresponding measures are topologically equivalent.

     

Multivariate matrix refinable functions with arbitrary matrix dilation

Characterizations of the stability and orthonormality of a multivariate matrix refinable function with arbitrary matrix dilation are provided in terms of the eigenvalue and -eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of is equivalent to the order of the vanishing moment conditions of the matrix refinement mask . The restricted transition operator associated with the matrix refinement mask is represented by a finite matrix , with and being the Kronecker product of matrices and . The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

     

On the Preservation of Stability under Convolutions

The preservation of stability under the convolution is shown to be related with the zero set of the Fourier transform of inducing stable function. For example, let φ be in the class Λ₀ of all stable functions ψ such that $\widehat\psi \left( 0 \right) \ne 0{\text{ and }}\widehat\psi$ as well as $E_\psi : = \sum {\left| {\widehat\psi \left( {w + 2{\pi }k} \right)} \right|} ^2$ is continuous. Then Λ₀ is preserved under the convolution by φ if and only if the zero set $Z\left( {\widehat\varphi } \right)$ is contained in 2πZ\{0}. The condition can be transformed into the zero set of the inducing mask trigonometric polynomial in the class Λ^# of compactly supported refinable functions in Λ₀. For example, our result shows that such φ must have its mask of the form $$m_\varphi \left( w \right) = \left( {\frac{{1 + {\text{e}}^{{\text{ - i2}}w} }}{2}} \right)^N \left( {\frac{{1 + {\text{e}}^{{\text{ - i}}w} + {\text{e}}^{{\text{ - i2}}w} }}{3}} \right)^M Q\left( w \right),$$ where integers N≥1 and M≥0, and Q(w) has no real zeros.     

The canonical solution operator to restricted to Bergman spaces

We first show that the canonical solution operator to restricted to -forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in the canonical solution operator to restricted to -forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in 1,$"> the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.

     

The complexity of recursion theoretic games

We show that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured. Several consequences are pointed out; for instance, the set of all -sentences that are uniformly valid in the lattice of recursively enumerable sets is undecidable. Furthermore we show that these games are equivalent to natural subclasses of effectively presented Borel games.