Rudolph A. Lorentz

Texas A&M University at Qatar | TAMU Qatar · Science

Cambridge Core - Recreational Mathematics - Connections in Discrete Mathematics - edited by Steve Butler

Bivariate Gončarov polynomials are a basis of the solutions of the bivariate Gončarov Interpolation Problem in numerical analysis. A sequence of bivariate Gončarov polynomials is determined by a set of nodes and is an affine sequence if is an affine transformation of the lattice grid , i.e., for some 2×2 matrix and constants . In this paper we prov...

Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivar...

We introduce the sequence of generalized Gon\v{c}arov polynomials, which is a basis for the solutions to the Gon\v{c}arov interpolation problem with respect to a delta operator. Explicitly, a generalized Gon\v{c}arov basis is a sequence $(t_n(x))_{n \ge 0}$ of polynomials defined by the biorthogonality relation $\varepsilon_{z_i}(\mathfrak d^{i}(t_...

Univariate Gončarov polynomials arose from the Gončarov interpolation problem in numerical analysis. They provide a natural basis of polynomials for working with u-parking functions, which are integer sequences whose order statistics are bounded by a given sequence u. In this paper, we study multivariate Gončarov polynomials, which form a basis of...

Predictive methods of image compression traditionally visit the pixels to be compressed in raster scan order, making a prediction for that pixel and storing the difference between the pixel and its prediction. We introduce a new predictive lossless compression method in which the order in which the pixels are visited is determined using a predictor...

We present a method for lossy compression of three dimensional gray scale images that is based on a 3D linear spline approximation to the image. We have extended an approach that has previously been successfully applied in two dimensions. In our method, we first select significant points in the data, and use them to create a 3D tetrahedralization....

In this paper we present a new method for lossy compression of volumetric data that is based on data dependent triangulation. We have extended an approach that has previously been successfully applied in the case of two dimensional images. In our method we first select significant points in the data, and using them, a three dimensional Delaunay tri...

The aim of this paper is to provide the users of the data format HDF5 with a preprocessor package for lossless compression for all of its predefined numerical data types. Combining this package with the built in compression filters generally compresses the data better than any of them individually, while not being essentially slower than the standa...

A common statement made when discussing the efficiency of compression programs like JPEG is that the transformations used, the discrete cosine or wavelet transform, decorrelate the data. The standard measure used for the information content of the data is the probabilistic entropy. The data can, in this case, be considered as the sampled values of...

We consider interpolation of multivariate functions by algebraic polynomials in ℝS, s ≥ 2. Since our methods and results do not depend on dimension s ≥ 2, we restrict ourselves to bivariate interpolation, s=2. Using methods of Birkhoff interpolation from.

In this paper, a new gridless method for numerically solving hyperbolic partial differential equations is presented. This method uses collocation based on Hermite interpolation at scattered sites at each time step. The basis can be chosen at each time step, which makes the approach adaptive and allows flexibility. The goal of this paper is to demon...

This is a survey of that theory of multivariate Lagrange and Hermite interpolation by algebraic polynomials, which has been developed in the past 20 years. Its purpose is not to be encyclopedic, but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples. It takes “classical...

Introduction In this note we discuss some results concerning multilevel finite element schemes of hierarchical basis (HB) type in connection with discretizing and preconditioning elliptic problems in Sobolev spaces. Roughly speaking, HB-methods require the introduction of a hierarchically defined algebraic basis Psi j of locally supported functions...

Several approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of t...

We characterize the closure of the union of the subspaces of a multiresolution analysis which does not necessarily enjoy the usual density property. One consequence of our development is that in many instances the density hypothesis is redundant. Another consequence is the fact that every multiresolution analysis can be complemented by another so t...

Without Abstract

. Motivated by the use of multilevel Riesz basis preconditioners for elliptic problems in Sobolev spaces, we investigate compactly supported prewavelets with respect to homogeneous Sobolev norms for multiresolution analyses based on box splines in IR d ; d 2. We show that, in contrast to the cases L 2 (IR d ) and H s (IR 1 ), prewavelets of compact...

For any ε > 0, we construct an orthonormal Schauder basis of C(K) consisting of trigonometric polynomials Tn n = 1, 2, . . . , such that deg(Tn) ≤ (1/2)(1 + ε)n. This is best possible with regard to the degree. The construction uses wavelet techniques.

We develop new high-order positive, monotone and convex interpolations, which are to be used in the multigrid context. This means that the value of the interpolant is calculated only at the midpoints lying between the locations of the given values. As a consequence, these interpolants can be calculated very efficiently. They are then tested in a ti...

We give explicit formulas for various scaling functions and wavelets which arise from the dyadic multiresolution analyses generated by box splines. Questions concerning interpolation are also addressed

Univariate interpolation.- Basic properties of Birkhoff interpolation.- Singular interpolation schemes.- Shifts and coalescences.- Decomposition theorems.- Reduction.- Examples.- Uniform Hermite interpolation of tensor-product type.- Uniform Hermite interpolation of type total degree.- Vandermonde determinants.- A theorem of Severi.- Kergin interpo...

We try to solve the bivariate interpolation problem (1.3) for polynomials (1.1), whereS is a lower set of lattice points, and for theq-th interpolation knot,A q is the set of orders of derivatives that appear in (1.3). The number of coefficients |S| is equal to the number of equations |A q |. If this is possible for all knots in general position,...

A whole new class of Chebycheff spaces is introduced. The construction starts with a Chebycheff space on the interval [0, 2π] and yields a periodic Chebycheff space. By a modification of the construction, derivatives can also be made continuous. Practical and theoretical applications are discussed.

Without Abstract

We consider the Hermite interpolation problem of interpolating all derivatives up to a given order at each point of a point set in the plane by bivariate polynomials of a given total degree. Such interpolations are called uniform if the order to which derivatives are interpolated is the same at each point. We show that all such schemes interpolatin...

This paper is devoted to bivariate interpolation. The problem is to find a polynomialP(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matricesE, their regularity, their almost regularity, and finally the regularity of the pairE,...

An $m \times n$ matrix $E$ with $n$ ones and $(m - 1)n$ zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, $E$ is singular with probability that converges to one if $m, n \rightarrow \infty$. Previously, this was known only if $m \geqslant (1 + \delta)n/\log...

An m × n m \times n matrix E E with n n ones and ( m − 1 ) n (m - 1)n zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, E E is singular with probability that converges to one if m m , n → ∞ n \to \infty . Previously, this was known only if m ⩾ ( 1 + δ ) n /...

Let Do be the functional given by Dof = f′(0) on C1(−1, 1). Let Πn be the set of polynomials of degree not exceeding n and let Mn be the polynomial interpolation to f at a given set of points x1, x2,…, xn. We approximate Dof by DoMnf. This is called a numerical differentiation formula. We study the pointwise convergence of DoMn to Do for two choice...

We characterize the closure of the union of a nested sequence {V j } j∈ℤ of closed subspaces of L 2 (ℝ d ) under various assumptions on the subspaces. A typical case is that V 0 is invariant under shifts from ℤ d and that V j consists of the A j -dilates of V 0 for some “expanding” d×d-matrix A. In the case that the closure is not dense, we use the...

Thesis (Ph. D.)--University of Minnesota. Bibliography: leaves 65-66.