A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights

The asymptotic behavior of thenth root of the leading coefficient of orthogonal polynomials on (–,) and the distribution of their zeros is studied for nonsymmetric weights that behave like exp(–2B|x|) whenx>0 and exp(–2Ax ) whenx<0,>AB. These results generalize previous investigations of Rakhmanov and Mhaskar and Saff who handle the symmetric caseA=B.Communicated by Paul Nevai.     

Rational approximation to Lipschitz and Zygmund classes

In this paper, characterizations for lim _n(R _n (f)/(n ^–1)=0 inH and for lim _n(n ^r+ R _n (f)=0 inW ^r Lip ,r1, are given, while, forZ, a generalization to a related result of Newman is established.Communicated by Ronald A. DeVore.     

Interpolatory properties of best rationalL 1-approximations

In this paper the distribution of the zeros of the error function for bestL ₁-approximation by rational functions fromR _n,m is considered. It is shown that the maximal distance between such zeros isO(1/(n–m)), ifn > m.Communicated by Edward B. Saff.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Absolute projection constant of the linear functions in a Lebesgue space

Cheney and Franchetti have computed the projection constant (V; L ¹[a, b]) for the two-dimensional spaceV of lines inL ¹[a, b] relative to the overspaceL ¹[a, b]. The purpose of this note is to show that (V; L ¹[a, b]) is in fact the absolute projection constant ofV.Communicated by Carl de Boor.     

Chebyshev polynomials for disjoint compact sets

We are concerned with the problem of finding among all polynomials of degreen with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for alln. A closely related approximation problem is obtained by considering all polynomials that have degree no larger thann and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for alln.Communicated by Doron S. Lubinsky     

Monotone interpolation of scattered data in R s

We present a method to interpolate scattered monotone data in R ^s using a variational approach. We present both theoretical and practical properties and give a dual algorithm allowing us to compute the resulting function whens=2. The method is specially suited for scattered data but comparison with existing methods for data on grids shows that it is a valid approach even in that case.Communicated by Wolfgang Dahmen.     

On subspaces of exp(N)

Let exp(X) denote the exponential space of a topological spaceX introduced by Vietoris [14]. In this paper, we study the subspaces of the space exp(N), whereN={1, 2, 3, ...} is the discrete space of natural numbers. We also show that if the metrizability number of exp(X) is countable, thenX (and exp(X)) must be compact and metrizable.     

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.     

The numerical range of elementary operators

For n-tuplesA=(A ₁,...,A _n ) andB=(B ₁,...,B _n ) of operators on a Hilbert spaceH, letR _A,B denote the operator onL(H) defined by . In this paper we prove that

Higher order recurrences for analytical functions of Tchebysheff type

Relation of hyperbolons of volume one to generalized Clifford algebras is described in [1b] and there some applications are listed. In this note which is an extension of [8] we use the one parameter subgroups of the group of hyperbolons of volume one in order to define and investigate generalization of Tchebysheff polynomial system. Parallely functions of roots of polynomials of any degree are studied as possible generalization of symmetric functions considered by Eduard Lucas. It is found how functions of roots of polynomial of any degree are related to this generalization of Tchebysheff polynomials. The relation is explicit. In a primary sense the considered generalization is in passing fromZ ₂ toZ _n group decomposition of the exponential. We end up with an application of the discovered generalization to quite large class of dynamical systems with iteration.     

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

On a general class of matrix nearness problems

The problem is considered of finding the nearest rank-deficient matrix to a given rectangular matrix. For a wide class of matrix norms, and arbitrary sparsity imposed on the matrix of perturbations, it is shown that when a solution exists, it can be calculated from the solution of a simpler problem involving fewer variables and only vector norms. Some numerical results are given for a special case involving the Frobenius norm.Communicated by Charles Micchelli.     

Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients

Sufficient conditions are presented for the existence of an absolutely continuous part for the measure of orthogonality for a system of polynomials with unbounded recurrence coefficients. Results are obtained by analyzing the spectral measure of a related self-adjoint operator.Communicated by Paul Nevai.     

Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative

In this paper, we consider the following second-order three-point boundary value problem

Adaptive greedy approximations

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.     

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.     

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.