Fix Conditions Research Articles

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 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.

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.

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.

Full rank filters and polynomial reproduction

We study polynomial (re)production properties of full rank stationary subdivision schemes and develop characterizations that are almost identical to the scalar case. Moreover, we give a version of the Strang--Fix conditions for the integer translates of a matrix valued function whose columns are linearly independent.

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.

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.

Ventilatory and gas exchange responses under spontaneous and fixed breathing modes during arm exercise.

To evaluate the difference of ventilatory and gas exchange response differences between arm and leg exercise, six healthy young men underwent ramp exercise testing at a rate of 15 W.min-1 on a cycle ergometer separately under either spontaneous (SPNT) or fixed (FIX) breathing modes, respectively. Controlled breathing was defined as a breathing frequency (fb; 30 breaths.min-1) which was neither equal to, nor a multiple of, cranking frequency (50 rev.min-1) to prevent coupling of locomotion and respiratory movement, i.e., so-called locomotor-respiratory coupling (LRC). Breath-by-breath oxygen uptake (VO2), ventilation (VE), CO2 output (VCO2), tidal volume (VT), fb and end-tidal PCO2 (PETCO2) were determined using a computerized metabolic cart. Arm exercise engendered a higher level of VO2 at each work rate than leg exercise under both FIX and SPNT conditions. However, FIX did not notably affect the VO2 response during either arm or leg exercise at each work rate compared to SPNT. During SPNT a significantly higher fb and lower PETCO2 during arm exercise was found compared with leg exercise up to a fb of 30 breaths.min-1 while VE and VT were nearly the same. During fixed breathing when fb was fixed at a higher rate than during SPNT, a significantly lower PETCO2 was observed during both exercise modes. These results suggest that: 1) FIX breathing does not affect the VO2 response during either arm or leg exercise even when non-synchronization between limb locomotion movement and breathing rate was adopted; 2) at a fb of 30 breaths.min-1 FIX breathing induced a hyperventilation resulting in a lower PETCO2 which was not associated with the metabolic rate during either arm or leg exercise, showing that VE during only leg exercise under the FIX condition was significantly higher than under the SPNT condition.

Nonseparable, Compactly Supported Interpolating Refinable Functions with Arbitrary Smoothness

In this paper we construct families of compactly supported nonseparable interpolating refinable functions with arbitrary smoothness (or regularity). The symbols for the newly constructed scaling functions are given by a simple formula related to the Bernstein polynomials. The emphasis of the paper is to show that under an easy-to-verify geometric condition these families satisfy Cohenrs condition, and they have arbitrarily high regularity. Furthermore, the constructed scaling functions satisfy, under the same geometrical condition, the Strang–Fix conditions of arbitrarily high order, which implies that corresponding interpolating schemes have arbitrarily high accuracy.

Quasi-Orthogonality and Quasi-Projections

Our main concern in this paper is the design of simplified filtering procedures for the quasi-optimal approximation of functions in subspaces of L2generated from the translates of a function ϕ(x). Examples of signal representations that fall into this framework are Schoenberg's polynomial splinesof degree n, and the various multiresolution spaces associated with the wavelet transform. After a brief review of the relation between the order of approximation of the representation and the concept of quasi-interpolation (Strang–Fix conditions), we investigate the implication of these conditions on the various basis functions and their duals (vanishing moment and quasi-interpolation properties). We then introduce the notion of quasi-duality and show how to construct quasi-orthogonal and quasi-dual basis functions that are much shorter than their exact counterparts. We also consider the corresponding quasi-orthogonal projection operator at sampling step h and derive asymptotic error formulas and bounds that are essentially the same as those associated with the exact least-squares solution. Finally, we use the idea of a perfect reproduction of polynomials of degree n to construct short kernel quasi-deconvolution filters that provide a well-behaved approximation of an oblique projection operator.

Approximation by translates of refinable functions

The functions\(f_1(x), \dots, f_r(x)\) are refinable if they are combinations of the rescaled and translated functions\(f_i(2x-k)\) . This is very common in scientific computing on a regular mesh. The space \(V_0\) of approximating functions with meshwidth\(h=1\) is a subspace of \(V_1\) with meshwidth\(h=1/2\) . These refinable spaces have refinable basis functions. The accuracy of the computations depends on \(p\), the order of approximation, which is determined by the degree of polynomials\(1, x, \dots, x^{p-1}\) that lie in \(V_0\). Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions\(f_i(x)\) are known only through the coefficients\(c_k\) in the refinement equation – scalars in the traditional case,\(r \times r\) matrices for multiwavelets. The scalar "sum rules" that determine\(p\) are well known. We find the conditions on the matrices\(c_k\) that yield approximation of order\(p\) from \(V_0\). These are equivalent to the Strang–Fix conditions on the Fourier transforms\(\hat f_i(\omega)\) , but for refinable functions they can be explicitly verified from the \(c_k\).

Cardinal interpolation by multivariate splines

The purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials. In particular, the notion of a commutator is introduced and its connection with the Marsden identity is observed. The order of a commutator is shown to be equivalent to the Strang and Fix conditions that arise in the study of the local approxima- tion orders using quasi-interpolants. We also prove that scaled cardinal interpolants give these local approximation orders. Introduction. Cardinal interpolation by bivariate box splines was first studied by de Boor, Hollig, and Riemenschneider (3). The purpose of our paper is to investigate the cardinal interpolation problem from a different point of view. In particular, the notion of a commutator is introduced. It will be shown that this notion generalizes the Marsden identity for univariate splines to the multivariate setting. The order of a commutator will be shown to be equivalent to the Strang and Fix conditions used in the study of the order of controlled approximation by Dahmen and Micchelli (8) or local approximation by de Boor and Jia (4). An application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper. 1. Preliminaries. This section consists of preliminary material for multivariate cardinal interpolation. Our approach is motivated by the work in (5) and (13) where cardinal interpolation in l2 was connected with certain convolution operators Lr Let Z denote the set of integers and Z+, the nonnegative ones. For any given complex sequence = ( by