Showing papers in "Constructive Approximation in 1993"

On the Construction of Multivariate (pre) Wavelets

Abstract: A new approach for the construction of wavelets and prewavelets on R d from multiresolution is presented. The method uses only properties of shift- invariant spaces and orthogonal projectors from L2(R d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multi- resolution. We present a new approach for the construction of wavelets and prewavelets on R a from multiresolution. Our method, which is based on our earlier work (BDR), (BDR1), uses only properties of shift-invariant spaces and orthogonal projectors from L2(R a) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows us to derive in a simple way previous constructions of wavelets, as well as new constructions, and to settle completely certain basic questions about multiresolution. A univariate function ~ E L2(R ) is called an orthogonal wavelet if its normalized, translated dilates ~kj.k:= 2k/2r j, k rZ, form an orthonormal basis for L~(R). In other words, this system is complete and satisfies the orthogonality conditions

On trigonometric wavelets

Abstract: Wavelets in terms of sine and cosine functions are constructed for decomposing 2π-periodic square-integrable functions into different octaves and for yielding local information within each octave. Results on a simple mapping into the approximate sample space, order of approximation of this mapping, and pyramid algorithms for decomposition and reconstruction are also discussed.

Wavelets and self-affine tilings

Abstract: Given a self-affine periodic tiling ofR n we construct an associatedr-regular multiresolution analysis and wavelet basis with the same lattice of translations and scaling matrix as the tiling.

Compactly supported bidimensional wavelet bases with hexagonal symmetry

Abstract: We study a class of subband coding schemes allowing perfect reconstruction for a bidimensional signal sampled on the hexagonal grid From these schemes we construct biorthogonal wavelet bases ofL 2(R 2) which are compactly supported and such that the sets of generating functionsψ 1,ψ 2,ψ 3 for the synthesis and $$\tilde \psi _1 , \tilde \psi _2 , \tilde \psi _3 ,$$ for the analysis, as well as the scaling functions φ and $$\tilde \varphi $$ , are globally invariant by a rotation of 2π/3 We focus on the particular case of linear splines and we discuss how to obtain a higher regularity We finally present the possibilities of sharp angular frequency resolution provided by these new bases

Irregular sampling of wavelet and short-time Fourier transforms

Abstract: We obtain irregular sampling theorems for the wavelet transform and the short-time Fourier transform. These sampling theorems yield irregular weighted frames for wavelets and Gabor functions with explicit estimates for the frame bounds.

Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis

Abstract: In this paper we construct a compactly supported biorthogonal wavelet basis adapted to some simple differential operators. Moreover, we estimate the condition numbers of the corresponding stiffness matrices.

Orlicz spaces, spline systems, and brownian motion

Abstract: There are three results proved in this paper. The first one characterizes the Holder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two-dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Holder functions with exponent 1/2 but in the norm of the Orlicz spaceLM* withM(t)=expt(t2)−1.

Banded matrices with banded inverses, II: Locally finite decomposition of spline spaces

Abstract: Given two function spacesV 0,V 1 with compactly supported basis functionsC i, Fi, i∈Z, respectively, such thatC i can be written as a finite linear combination of theF i's, we study the problem of decomposingV 1 into a direct sum ofV 0 and some subspaceW ofV 1 in such a way thatW is spanned by compactly supported functions and that eachF i can be written as a finite linear combination of the basis functions inV 0 andW. The problem of finding such locally finite decompositions is shown to be equivalent to solving certain matrix equations involving two-slanted matrices. These relations may be reinterpreted in terms of banded matrices possessing banded inverses. Our approach to solving the matrix equations is based on factorization techniques which work under certain conditions on minors. In particular, we apply these results to univariate splines with arbitrary knot sequences.

TheC ℓ Bailey transform and Bailey lemma

Abstract: TheC l nonterminating Cl summation theorem is derived by appropriately specializing Gustafson's6ψ6 summation theorem for bilateral basic hypergeometric series very well-poised on symplecticC l groups. From this, the terminating6ϕ5 and, hence, terminating4ϕ3 summation theorem is obtained. A suitably modified4ϕ3 is then used to derive theC l generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of theC l Bailey pair. TheC l generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheC l Bailey lemma is then used to obtain a connection coefficient result for generalC l littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryA l, or equivalentlyU(l+1), case. The classical case, corresponding toA 1 or equivalentlyU(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. TheC l nonterminating6ϕ5 summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.

A note on orthonormal polynomial bases and wavelets

Abstract: A simple and explicit construction of an orthnormal trigonometric polynomial basis in the spaceC of continuous periodic functions is presented. It consists simply of periodizing a well-known wavelet on the real line which is orthonormal and has compactly supported Fourier transform. Trigonometric polynomials resulting from this approach have optimal order of growth of their degrees if their indices are powers of 2. Also, Fourier sums with respect to this polynomial basis are projectors onto subspaces of trigonometric polynomials of high degree, which implies almost best approximation properties.

A Bernstein-type inequality associated with wavelet decomposition

Abstract: Wavelet decomposition and its related nonlinear approximation problem are investigated on the basis of shift-invariant spaces of functions. In particular, a Bernstein-type inequality associated with wavelet decomposition is established in such a general setting. Several examples of piecewise polynomial spaces are given to illustrate the general theory.

Planar curve interpolation by piecewise conics of arbitrary type

Abstract: Five points in general position inR2 always lie on a unique conic, and three points plus two tangents also have a unique interpolating conic, the type of which depends on the data. These well-known facts from projective geometry are generalized: an odd number 2n+1≥5 of points inR2, if they can be interpolated at all by a smooth curve with nonvanishing curvature, will have a uniqueGC2 interpolant consisting of pieces of conics of varying type. This interpolation process reproduces conics of arbitrary type and preserves strict convexity. Under weak additional assumptions its approximation order is ϑ(h5), whereh is the maximal distance of adjacent data pointsf(t i ) sampled from a smooth and regular planar curvef with nonvanishing curvature. Two algorithms for the construction of the interpolant are suggested, and some examples are presented.

Extremal Measures for a System of Orthogonal Polynomials

Abstract: We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.

Construction of compactp-wavelets

Abstract: In [D1] the method for construction of compact wavelets, based on a decimation by 2, was introduced. In [CW] and [A] decimation by an integer different from 2 is discussed but construction for a generalp is not completely treated. Here we review some of the elements of that construction and give an approach that generalizes to thep>2 case in a concrete fashion which requires some new ideas.

Hypergeometric Analogues of the Arithmetic-Geometric Mean Iteration

Abstract: The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iteration a+ 1 = (a + bn)/2 and b+ 1 = axfa~,b, with a0:= 1 and b 0 := x The common limit is 2F1(�89 �89 1; 1 - x2) - 1 and the convergence is quad- ratic This is a rare object with very few close relatives There are however three other hypergeometric functions for which we expect similar iterations to exist, namely: 2F1(�89 - sl, �89 + s; 1; ) with s -- ~,1 g,x g Our intention is to exhibit explicitly these iterations and some of their generalizations These iterations exist because of underlying quadratic or cubic transformations of certain hypergeometric functions, and thus the problem may be approached via searching for invariances of the corresponding second-order differential equations It may also be approached by searching for various quadratic and cubic modular equations for the modular forms that arise on inverting the ratios of the solutions of these differential equations In either case, the problem is intrinsically computational Indeed, the discovery of the identities and their proofs can be effected almost entirely computationally with the aid of a symbolic manipulation package, and we intend to emphasize this computational approach

Quasi-interpolation by thin-plate splines on a square

Abstract: Quasi-interpolation is one method of generating approximations from a space of translates of dilates of a single function ψ. This method has been applied widely to approximation by radial basis functions. However, such analysis has most often been performed in the setting of an infinite uniform grid of centers. In this paper we develop general error bounds for approximation by quasiinterpolation on ann-cube. The quasi-interpolant analyzed involves a finite number, growing ash −n , of translates of dilates of the function ψ, and a bounded number of edge functions. The centers of the translates of dilates of ψ form a uniformly spaced grid within the cube. These error bounds are then applied to approximation by thin-plate splines on a square. The result is an O(ω(f, [-1,1]2,h)) error bound for approximation by thin-plate splines supplemented with eight arctan functions.

Rational approximation in the real Hardy spaceH 2 and Stieltjes integrals: A uniquencess theorem

Abstract: The paper deals with rational approximation over the real Hardy spaceH2, R(V), whereV is the complement of the closed unit disk. The results concern Stieltjes functions $$f(z) = \int {\frac{{d\mu (t)}}{{z - t}},} $$ where μ is a positive measure. It is shown that there is a unique critical point and hence both a unique local and a unique global best rational approximation in each degree, provided the support of μ lies within some absolute bounds which are explicitly estimated.

What distributions of points are possible for convergent sequences of interpolatory integration rules

Abstract: Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} f(x_{jn} )} $$ is aninterpolatory integration rule of numerical integration, that is, $$I_n [f]: = \int\limits_{ - 1}^1 {P(x)dx,} degree(P)< n.$$ Suppose, furthermore, that, for each continuousf:[−1, 1]→R, $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - 1}^1 {f(x)dx.} $$

Tchebycheff polynomials on a triangular region

Abstract: A bivariable polynomial of total degreen that has minimal uniform norm on a triangular region is given explictly.

A minimax principle for the optimal error of Monte Carlo methods

Abstract: Some results are proved showing that the average case error with least favorable distribution and the optimal error of appropriate Monte Carlo methods coincide in several cases. The results are based on a special Minimax Theorem, derived from Ky Fan's Lemma. Finally, an application is given to the case of Banach spaces having “few” extreme points.

On Müntz rational approximation

Abstract: One of the main results of the present paper shows that, for any sequence of real numbers {λn} with infinitely many distinct elements, the monotone rational combinations of {Xλn} always form a dense set in the uniform norm in the subspace of monotone functions fromC [0, 1].

Distribution of points for convergent interpolatory integration rules on (

Abstract: Letw be a “nice” positive weight function on (−∞, ∞), such asw(x)=exp(−⋎x⋎α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I n} t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x jn} ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI n has precision other thann-1.

General interpolation schemes for the generation of irregular surfaces

Abstract: We introduce an interpolation scheme to generate a class of irregular surfaces. The analysis is first carried out for a triangleT. We define the function ϕ on a subsetX, dense inT. In terms of the construction parameters of ϕ, we establish sufficient conditions for its uniform continuity so that it would be possible to extend it to a continuous function on the whole ofT. We do the same analysis in the case of a rectangleR.

Optimality and uniqueness conditions in complex rational Chebyshev approximation with examples

Abstract: It is well known that best complex rational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case.

Harmonic approximation on arcs

Abstract: If a real interval is mapped by a continuous, proper injection into a noncompact Riemannian manifold of dimension at least two, then every continuous function on the image arc can be approximated arbitrarily closely by functions harmonic on the entire manifold.

The density of rational functions in Markov systems: A counterexample to a conjecture of D. J. Newman

Abstract: We construct a family of infinite Markov systems on [−1, 1] with the property that the rational functions from these systems are not dense inC[−1, 1]. This gives counterexamples to a long-standing conjecture of D. J. Newman.

Optimal recovery of the Sobolev-Wiener class of smooth functions by double sampling

Abstract: An optimal interpolation problem is considered on the Sobolev-Wiener class of smooth functions defined on the real line by double samples. We calculate the exact value of the minimal intrinsic error, identify an optimal set of sampling points and constructing an optimal linear estimator (algorithm).

Distribution of interpolation points of bestL 2-approximants (nth partial sums of Fourier series)

Abstract: For a continuous 2π-periodic real-valued functionf, we investigate the asymptotic behavior of the zeros of the errorf(θ)−sn(θ), wheresn(θ) is thenth Fourier section. We prove that there is a subsequence {nk} for which such zeros (interpolation points) are uniformly distributed on [−π, π]. This extends previous results of Saff and Shekhtman. Moreover, results dealing with the maximal distance between consecutive zeros off−snk are obtained. The technique of proof involves coefficient estimates for lacunary trigonometric polynomials in terms of itsLq-norm on a subinterval.