The rate of decrease of constants in Jackson type inequalities in dependence of the order of the continuity modulus

Jackson type inequalities for moduli of continuity of arbitrary order are established with the use of linear approximation methods. The constants are smaller than known previously. The results hold in different spaces of periodic and nonperiodic functions. Bibliography: 19 titles.     

On the limit polynomial for a solution of an elliptic equation of the fourth order with constant coefficients

We show that a solution of the Dirichlet problem for an elliptic equation of the fourth order with constant coefficients, whose right-hand side is periodic in all variables except one and exponentially decreases, converges at infinity to a certain polynomial of the first degree in the nonperiodic variable. Coefficients of this polynomial are determined. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 437–444, March, 1998.     

Estimates for functionals with a known moment sequence in terms of deviations of Steklov type means

Some estimates for functionals indicated in the title are established. As implications, Jackson type inequalities with constants smaller than the previously known ones are obtained. The results hold in various spaces of both periodic and nonperiodic functions. Bibliography: 9 titles.     

An algorithm for determining the approximation orders of multivariate periodic spline spaces

We give an algorithm which computes the approximation order of spaces of periodic piece-wise polynomial functions, given the degree, the smoothness and tesselation. The algorithm consists of two steps. The first gives an upper bound and the second a lower bound on the approximation order. In all known cases the two bounds coincide.     

Semilocal Smoothing Splines

The paper is aimed at periodic and nonperiodic semilocal smoothing splines, or S-splines of class C p, formed by polynomials of degree n. The first p?+?1 coefficients of each polynomial are determined by the values of the preceding polynomial and its first p derivatives at the glue-points, while the remaining n???p coefficients of the higher derivatives of the polynomial are found by the method of least squares. These conditions are supplemented with the initial conditions (nonperiodic case) or the periodicity condition on the spline-function on the segment where it is defined. A linear system of equations is obtained for the coefficients of the polynomials constituting the spline. Its matrix has a block structure. Existence and uniqueness theorems are proved and it is shown that that the convergence of the splines to the original function depends on the eigenvalues of the stability matrix. Examples of stable S-splines are given.     

On the order of local approximation of functions by trigonometric polynomials that are partial sums of averaging operators

We study the order of polynomial approximations of periodic functions on intervals which are internal with respect to the main interval of periodicity and on which these functions are sufficiently smooth. The estimates obtained contain parameters which characterize the smoothness and alternation of signs of nuclear functions and parameters that determine classes of approximated functions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp.706–714, May, 1997.     

A multiresolution method for fitting scattered data on the sphere

In this work, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric wavelets. As application of this method, we give an algorithm which allows to compress scattered data on spherelike surfaces. In order to illustrate our results, some numerical examples are presented.     

Intermediate error estimates

Let R[f] be the remainder of some approximation method, having estimates of the form f;R[f]f; ρ_i ; f⁽ⁱ⁾ for i = 0,…, r. In many cases, ρ₀ and ρ_r are known, but not the intermediate error constants ρ₁,…,ρ_r−1. For periodic functions, Ligun (1973) has obtained an estimate for these intermediate error constants by ρ₀ and ρ_r. In this paper, we show that this holds in the nonperiodic case, too. For instance, the estimates obtained can be applied to the error of polynomial or spline approximation and interpolation, or to numerical integration and differentiation.     

TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted l-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series.The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic contin- uous functions spaces.Tractability is the minimal number of function samples required to solve the problem in polynomial in ε~(-1)and d.and the strong tractability is the pres- ence of only a polynomial dependence in ε.This problem has been recently studied for quasi-Monte Carlo quadrature rules.quadrature rules with non-negative coefficients. and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables.The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14]on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative.The arguments are not constructive.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

<Emphasis Type="Italic">S</Emphasis>-units in hyperelliptic fields and periodicity of continued fractions

Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.     

Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.     

Successive polynomial spline function approximation

The use of successive polynomial spline approximation is established as a method of improving the accuracy of estimates of derivatives of periodic functions approximated by interpolating odd order splines defined on a uniformly spaced set of data points. For the various configurations possible with this multiple-approximation method, bounds for the leading error terms are explicitly given. In particular, for the quintic spline, the variety of approximation sequences is described in detail.     

Approximations of almost periodic functions by entire ones

In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions which give on (−∞,∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions with an arbitrary spectrum.     

Transverse Spectral Stability of Small Periodic Traveling Waves for the KP Equation

The Kadomtsev–Petviashvili (KP) equation possesses a four‐parameter family of one‐dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two‐dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one‐dimensional traveling wave, or nonperiodic (localized or bounded). We focus on the so‐called KP‐I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP‐II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations that are periodic in the direction of propagation, and have long wavelengths in the transverse direction.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

A note on orthonormal polynomial bases and wavelets

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.     

Fertile HC models with three states on a cayley tree

We consider fertile hard-core (HC) models with three states on a homogeneous Cayley tree. It is known that four types of such models exist. For these models, we describe the translation-invariant and periodic HC Gibbs measures. We also construct a uncountable set of nonperiodic Gibbs measures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 412–424, September, 2008.