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.     

On fundamental solutions for multivariate singular interpolation

The objective of this paper is to give a constructive approach to the solution of the fundamental functions for cardinal interpolation from a shift-invariant space generated by the (multi-)integer translates of some compactly supported function whose polynomial symbol has a non-empty zero set. This problem was first introduced by Chui, Diamond, and Raphael, where explicit solutions were given for various zero sets. Later, de Boor, Höllig, and Riemenschneider gave an existence proof for zero sets which are more general. In this paper, we give an integral representation of the fundamental solutions that can be made explicit in some cases and we will also give a growth condition of such fundamental solutions. The four-directional box splines will be used as an illustrative example.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

Using the refinement equation for the construction of pre-wavelets

A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarié, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method can produce only the Lemarié Ondelettes. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.Research partially supported by DARPA and NSF Grant INT-87-12424     

Fast integral wavelet transform on a dense set of the time-scale domain

Summary. The objective of this paper is to introduce a fast algorithm for computing the integral wavelet transform (IWT) on a dense set of points in the time-scale domain. By applying the duality principle and using a compactly supported spline-wavelet as the analyzing wavelet, this fast integral wavelet transform (FIWT) is realized by applying only FIR (moving average) operations, and can be implemented in parallel. Since this computational procedure is based on a local optimal-order spline interpolation scheme and the FIR filters are exact, the IWT values so obtained are guaranteed to have zero moments up to the order of the cardinal spline functions. The semi-orthogonal (s.o.) spline-wavelets used here cannot be replaced by any other biorthogonal wavelet (spline or otherwise) which is not s.o., since the duality principle must be applied to some subspace of the multiresolution analysis under consideration. In contrast with the existing procedures based on direct numerical integration or an FFT-based multi-voice per octave scheme, the computational complexity of our FIWT algorithm does not increase with the increasing number of values of the scale parameter. Received March 3, 1994     

Multipoint Taylor formulæ

Summary. The main result of this paper is an abstract version of the Kowalewski–Ciarlet–Wagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map. Received March 29, 1996 / Revised version received November 22, 1996     

A Family of Approximation Formulas and some Applications

The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated.

Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis.

Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered.

Some applications to fast calculation of Fourier coefficients and to solution of integrodifferential equations are given. The corresponding numerical results have been obtained by means of MATHEMATICA 3.0 system.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonnegative splines, in particular of degree five

Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given. Received November 16, 1995 / Revised version received February 24, 1997     

Pole Assignment in Hybrid Differential-Difference Systems

We present a general approach to the pole assignment problem for linear stationary hybrid differential-difference systems as a coefficient control problem for their characteristic equations. Various scales (classes) of linear feedback controllers are considered. Special attention is paid to the solvability of the pole assignment problem for such systems in the scale of general differential-difference controllers and in a general scale that, along with differential-difference controllers, contains integral controllers whose kernels are compactly supported functions. A general scheme for constructing such controllers is proposed based on the algebraic properties of the shift operator, the Paley–Wiener theorem on compactly supported functions, and the methods of interpolation theory in the class of entire functions of exponential type. Examples and counterexamples illustrating the results are given.     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

Commutation for Irregular Subdivision

We present a generalization of the commutation formula to irregular subdivision schemes and wavelets. We show how, in the noninterpolating case, the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal compactly supported irregular knot B-spline wavelets starting from Lagrangian interpolation. September 4, 1998. Date revised: July 27, 1999. Date accepted: November 16, 2000.     

Eigenfunctions of Two-Scale Difference Equations with Dilation Parameter and Infinite Products

This paper deals with two-scale difference equations having an arbitrary dilation parameter and a formal power series as symbol. We investigate the equation concerning the existence of nonzero compactly supported distributional solutions. In order to include also continuous solutions it is advantageous to consider the two-scale difference equation as eigenvalue problem where the solutions are either compactly supported or integrals of compactly supported distributions. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure depending on the dilation parameter. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.     

Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of the distribution of the points. These interpolants depend linearly on the data and include a construction of Berrut as a special case.     

Thinning algorithms for scattered data interpolation

Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in ℝ ^d .     

A new formal approach to the rational interpolation problem

Summary An elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods. We allow confluent interpolation points, poles, and infinity as one of the interpolation points. Not only one specific solution is given but a nice parametrization of all solutions. We also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices.     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

One-sided tangential interpolation for operator-valued Hardy functions on polydisks

All solutions of one-sided tangential interpolation problems with Hilbert norm constraints for operator-valued Hardy functions on the polydisk are described. The minimal norm solution is explicitly expressed in terms of the interpolation data.The research of this author is partially supported by NSF grant DMS 9800704, and by the Faculty Research Assignment grant from the College of William and Mary.     

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.