Strang Fix Research Articles

Approximation properties of wavelets on p-adic fields

Refinable functions play an important role in the construction and properties of wavelets. Basically, most of the wavelets are generated from refinable functions. In this paper, a study on the approximation properties of refinable functions on [Formula: see text]-adic fields is carried out with necessary theoretical background. Various equivalent forms of approximation order and the connection between the approximation order and the Strang–Fix condition are derived. Finally a characterization for the approximation order of a refinable function is given in terms of order of the sum rules associated with the refinement mask.

Reproducing fractional monomials: Weakening of the Strang–Fix conditions

A method to reproduce causal and symmetric monomials of fractional degree by integer shifts of the corresponding fractional B-splines, introduced by Unser and Blue, is presented. Thus, the traditional relation between the degree of reproduced monomials and the order of approximation holds. Bivariate, obtained by tensor product, fractional B-splines are introduced, and reproducing of bivariate causal and symmetric monomials is shown. Demonstration that the method is based on a weakening of the Strang–Fix conditions is presented.

Reproducing solutions to PDEs by scaling functions

A generalization of the multivariate Strang–Fix conditions to no scale-invariant (only shift-invariant) polynomial spaces multiplied by exponents is introduced. A method to construct nonstationary compactly supported interpolating scaling functions that the scaling functions reproduce polynomials multiplied by exponents is presented. The polynomials (multiplied by exponents) are solutions to systems of linear constant coefficient PDEs, where the symbols of the differential operators that define PDEs can be no scale-invariant and can contain constant terms. Analytically calculated graphs of the scaling functions, including nonstationary, are presented. A concept of the so-called [Formula: see text]-separate MRAs is considered; and it is shown that, in the case of isotropic dilation matrices, the [Formula: see text]-separate scaling functions appear naturally.

Approximation Properties of Systems of Periodic Wavelets on the Cantor Group

We construct systems of periodic dyadic wavelets and study approximation properties of such systems by using the notion of a multiresolution analysis. We prove a Jackson type inequality for the spaces constituting the multiresolution analysis and obtain a periodic analogue of the Strang–Fix condition.

Approximation of Non-decaying Signals from Shift-Invariant Subspaces

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted- $$L_p$$ spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted- $$L_p$$ space, the weighted norm of the approximation error can be made to go down as $$O(h^L)$$ when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the $$O(h^L)$$ behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of $$d/p+\varepsilon $$ , for arbitrary $$\varepsilon >0$$ . This extra amount of derivatives is used to make sure that the direct sampling is stable.

Multivariate sampling-type approximation

Approximation properties of the expansions [Formula: see text], where [Formula: see text] is a linear differential operator and [Formula: see text] is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in [Formula: see text]-norm, [Formula: see text], are given in terms of the Fourier transform of [Formula: see text]. The approximation order depends on the smoothness of [Formula: see text], the order of [Formula: see text], the order of Strang–Fix condition for [Formula: see text] and [Formula: see text]. A wide class of [Formula: see text] including both band-limited and compactly supported functions is considered, but a special condition of compatibility [Formula: see text] with [Formula: see text] is required. Such differential expansions may be useful for engineers.

Band-limited scaling and wavelet expansions

Operators Qjf=∑n∈Z〈f,φ˜jn〉φjn are studied for a class of band-limited functions φ and a wide class of tempered distributions φ˜. Convergence of Qjf to f as j→+∞ in the L2-norm is proved under a very mild assumption on φ, φ˜, and the rate of convergence is equal to the order of Strang–Fix condition for φ. To study convergence in Lp, p>1, we assume that there exists δ∈(0,1/2) such that φˆ¯φ˜ˆ=1 a.e. on [−δ,δ], φˆ=0 a.e. on [l−δ,l+δ] for all l∈Z∖{0}. For appropriate band-limited or compactly supported functions φ˜, the estimate ‖f−Qjf‖p⩽Cωr(f,2−j)Lp, where ωr denotes the r-th modulus of continuity, is obtained for arbitrary r∈N. For tempered distributions φ˜, we proved that Qjf tends to f in the Lp-norm, p⩾2, with an arbitrary large approximation order. In particular, for some class of differential operators L, we consider φ˜ such that Qjf=∑n∈ZLf(2−j⋅)(n)φjn. The corresponding wavelet frame-type expansions are found.

Mortar spectral method in axisymmetric domains

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. We reduce the original problem by a Fourier expansion in the angular variable to a countable family of two-dimensional problems. We decompose the meridian domain, assumed polygonal, in a finite number of rectangles and we discretize by a spectral method. Then we describe the main features of the mortar method and use the algorithm Strang Fix to improve the accuracy of our discretization.

OPERATOR-ADAPTED WAVELETS: CONNECTION WITH THE STRANG–FIX CONDITIONS

In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang–Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients.

Hyperplace arrangements and box splines

The purpose of this paper is to explain some explicit formulas that one can develop in the theory of the box spline and the corresponding algorithms of approximation by functions—in particular, (2.11) and (3.2). The theory of splines is a large subject, and even the part on the box spline is rather well developed. The reader should consult the fundamental book by de Boor, Hollig, and Riemenschneider [7] or the recent notes by Ron [19]. Of this large theory we concentrate on some remarkable theorems of Dahmen and Micchelli (see [3; 4; 5]) and on the theory of quasi-interpolants and the Strang–Fix conditions, for which we refer to de Boor [6]. In essence, here we make explicit certain constructs that are already present in [6]. Thus, from a purely computational point of view, there is probably no real difference with that paper, yet we believe that the explicit formulas (2.11) and (3.2) shed a light on the whole procedure. In fact, the main new formula is (3.2) since (2.11) is essentially in Dahmen–Micchelli (although not so explicit). We also show how some facts about matroids, which are recalled in an appendix written by A. Bjorner, give a proof of one of the basic theorems of the theory on the dimension of a certain space of polynomials describing box splines locally.

Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function $$\psi \in W^{m,p}({\mathbb{R}}^d)$$ such that if $$\widehat{\psi}(0) = 1$$ and ψ satisfies the Strang–Fix conditions to order m − 1, then a scale averaged approximation formula holds for all $$f \in W^{m,p}({\mathbb{R}}^d)$$ : $$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$$ The dilations { a j } are lacunary, for example a j = 2 j , and the coefficients c j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in $${W^{m - 1,p}({\mathbb{R}}^d)}$$ the scale averaging is unnecessary and one has the simpler formula $$f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)$$ . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system $$\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}$$ in $$W^{m,p}({\mathbb{R}}^d)$$ . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

Multivariate refinable functions, differences and ideals — a simple tutorial

This paper summarizes the algebraic quotient ideal approach to polynomial generation by refinable functions and connects it to Strang–Fix conditions and factorization with respect to difference operators. Motivated by the latter one, we also consider vector subdivision schemes with matrix valued coefficients and review some of their properties.

Approximation from shift-invariant spaces by generalized sampling formulas

In this paper we investigate approximation from shift-invariant spaces by using generalized sampling formulas. These sampling formulas involve samples of filtered versions of the function itself. The considered systems include averaging samplers, and classical sampling of the function and its derivatives. Under appropriate hypotheses on the generator φ of the shift-invariant space and on the involved systems, we derive stable generalized sampling formulas in a shift-invariant subspace of L p ( R n ) . From these generalized sampling formulas we construct approximation schemes valid for smooth functions. The approximation order depends both on the order for which the Strang–Fix conditions are satisfied by φ, and on the largest order of the derivatives appearing in the systems if any.

Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix

Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely . These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an RC circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤd→ℤd, and G a complete set of coset representatives of 2π(A −Tℤd/ℤd), we consider polynomial solutions M to the equation ∑ g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order.

Generalized Strang?Fix condition for scattered data quasi-interpolation

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too.

On asymptotic behaviour of modelling time in the importance sampling method

We deal with evaluation of an integral with the positive integrand g(v) : V → R (here is a bounded domain) by standard Monte Carlo method with density p(v), proportional to the Strang–Fix approximation of function g(v), built on the uniform grid with step h in domain V . We consider the mean value of time required for modelling sampling value of random variable with respect to the density p(v) and investigate its asymptotic behaviour as h → 0.

On asymptotic behaviour of modelling time in the importance sampling method

We deal with evaluation of an integral with the positive integrand g(v) : V → R (here is a bounded domain) by standard Monte Carlo method with density p(v), proportional to the Strang–Fix approximation of function g(v), built on the uniform grid with step h in domain V . We consider the mean value of time required for modelling sampling value of random variable with respect to the density p(v) and investigate its asymptotic behaviour as h → 0.

Symmetric prefilters for multiwavelets

When applying discrete multiwavelets, prefiltering is necessary because the initial multiscaling coefficients cannot be trivially derived from the samples of scalar signals. There have been many studies on the design of prefilters, and one main approach is to use a superfunction. The idea is to construct a low-pass function from the multiscaling functions that inherits their approximation power for scalar signals. However, none of the existing prefilters give linear phase combined filters, which is important for many practical applications. The authors analyse the conditions on which the prefilters and the combined filters are symmetric. A method is proposed for the design of good multiwavelet prefilters that allow the superfunction to be symmetric, satisfying the Strang–Fix conditions and the resulting combined filters are linear phase. Design examples using DGHM and Chui–Lian multiwavelets are given.

Convergence of Vector Subdivision Schemes in Sobolev Spaces

In this paper we consider functional equations of the form Φ=∑α∈Zsa(α)Φ(M·−α), where Φ=(φ1,…,φr)T is an r×1 vector of functions on the s-dimensional Euclidean space, a(α), α∈Zs, is a finitely supported sequence of r×r complex matrices, and M is an s×s isotropic integer matrix such that limn→∞M−n=0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φj, j=1,…,r, belonging to the Sobolev space Wkp(Rs)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Qa associated with the sequence a is defined by QaF:=∑α∈Zsa(α)F(M·−α),F∈(Wkp(Rs))r. Let Φ0 be a nontrivial r×1 vector of compactly supported functions in Wkp(Rs). The iteration scheme Φn=QaΦn−1, n=1,2,…, is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0)=m−1∑α∈Zsa(α), m=∣detM∣, and are related to polynomial reproducibility and the classical Strang–Fix conditions.The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution Φ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang–Fix type conditions that determine the feasible set of initial vectors.