Space Of Periodic Functions Research Articles

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε−1 and d. Strong tractability means that we have only a polynomial dependence in ε−1. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γj moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑∞j=1γj<∞, whereas tractability holds iff limsupd→∞∑dj=1γj/lnd<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.

On the solvability of generalized Stokes equations in the spaces of periodic functions

The paper is concerned with the solvability theory of the generalized Stokes equations arising in the study of the motion of non-newtonian fluids.

Sharp inequality for deviation of Rogozinski sums and the second continuity modulus in the space of periodic continuous functions

The sharp constant (uniformly in n) is found in a Jackson-type inequality involving the Rogozinski sums of order n and the second modulus of continuity with step π/(n+1). Bibliography: 6 titles.

Discrete-time system modelling in [formula omitted] with orthonormal basis functions

In this paper, model sets for linear time-invariant systems spanned by fixed pole orthonormal bases are investigated. The obtained model sets are shown to be complete in L p( T) (1<p<∞), the Lebesque spaces of functions on the unit circle T , and in C( T) , the space of periodic continuous functions on T . The L p norm error bounds for estimating systems in L p( T) by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the mean growth of the Fourier series are also derived. These results have application in estimation and model reduction.

THE PERIODIC SOLUTIONS FOR TIME DEPENDENT AGE-STRUCTURED POPULATION MODELS

In this paper, the existence of periodic solutions for a time dependent age-structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical behaviors of the model. The existence of a global parameterized branch of periodic solutions of the model is obtained by-using the contracting mapping theorem in a periodic and continuous function space. The global stability of the trivial equilibrium is studied and a very practical stability criteria for the model is obtained. The dynamics of the linear time-periodic model is similar to that of the linear model.

A Class of Periodic Function Spaces and Interpolation on Sparse Grids

We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces.

The asymptotic efficiency of randomized nets for quadrature

An L 2 -type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x,y), which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0,m,s)-nets in base b. For moderately smooth K(x,y) the discrepancy is O(N -1 [log(N)] (s-1)/2 ), and for K(x,y) with greater smoothness the discrepancy is O(N-3/2[log(N)] (s-1)/2 ), where N = b m is the number of points in the net. Numerical experiments indicate that the (t,m,s)-nets of Faure, Niederreiter and Sobol' do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.

Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation

The generalized forced Boussinesq equation, utt − uxx + [f(u)]xx + uxxxx = h0, and its periodic traveling wave solutions are considered. Using the transform z = x − ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green′s function method. This integral equation generates compact operators in a Banach space of real‐valued continuous periodic functions with a given period 2T. The Schauder′s fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.

Fourier Multipliers for Besicovitch Spaces

In this paper a generalization of some results from Fourier analysis on periodic function spaces to the almost periodic case is given. We consider almost periodic distributions which constitute a subclass of tempered distributions. Under suitable conditions on the spectrum \Lambda \subset \mathbb R^s , a distribution T \in S'(\mathbb R^s) is almost periodic if it can be represented as \sum_{\lambda \in \Lambda} a_{\lambda} e^{i \lambda x} , where the sequence (a_{\lambda})_{\lambda \in \Lambda} is tempered. The main result states that any Fourier multipliers for L^q(\mathbb R^s) of the Michlin-Hörmander type is also a Fourier multiplier for the Besicovich spaces B^q_{ap} (\mathbb R^s, \Lambda) , if it is restricted to the spectrum \Lambda . Finally, we prove that the Sobolev-Besicovich spaces H^{N,q}_{sp} (\mathbb R^s, \Lambda) coincide if N \in \mathbb N .

Symmetry and scaling properties of the von Kármán equations

We study symmetries in the von Karman equations with simply supported boundary conditions on rectangular domains. By embedding this fourth order plate problem into a space of periodic functions, we obtain hidden symmetries and scaling properties in its solution manifold. These properties are exploited for efficient numerical approximation of the solution branches at the bifurcation points, and are demonstrated with numerical examples.

Periodic interpolation and wavelets on sparse grids

Nested spaces of multivariate periodic functions forming a non-stationary multiresolution analysis are investigated. The scaling functions of these spaces are fundamental polynomials of Lagrange interpolation on a sparse grid. The approach based on Boolean sums leads to sample and wavelet spaces of significantly lower dimension and good approximation order. The algorithms for complete decomposition and reconstruction are of simple structure and low complexity.

Morin singularities and global geometry in a class of ordinary differential operators

(∗) u′(t) + f(t, u(t)) = g(t), where the unknown u is a real function on S1 and the nonlinearity f : S1×R → R can assume a number of forms. Our approach is to study the global geometry of the operator F : B1 → B0, u → u′ + f(t, u) where the domain is either C1(S1) (the Banach space of periodic functions with continuous derivatives) or the Hilbert space H1(S1) of periodic functions with square integrable derivative. Ideally, we search for global changes of variables in both domain and image taking the operator F to a simple normal form. This goal has been achieved in previous occasions, starting with the seminal work of A. A. Ambrosetti and G. Prodi ([AP]) and its geometric interpretation by M. S. Berger and P. T. Church ([BC]), who showed that the operator associated to a certain nonlinear Dirichlet problem gives rise to a global fold between infinite

Regular Level Sets of Averages of Nemytskiı Operators Are Contractible

LetH1(S1) be the space of periodic real functions with derivative inL2andf:R→R be a smooth function with no double roots. Then there is a diffeomorphism ofH1(S1) taking the setZ={v∈H1(S1)∣∫S1f(v(t))dt=0} to a hyperplane. In this paper we state and prove a general version of this example. We consider a Banach spaceVof functions from some manifoldMtoRn′and a functionf:M×Rn′→Rn: under suitable hypothesis, there is a homeomorphism ofVtakingZ={v∈V∣∫Mf(m, v(m))dm=0} to a closed subspace of codimensionn.

A nonexistenee theorem of small periodic traveling wave solutions to the generalized Boussinesq equation

THE GENERALIZED Boussinesq equation, u tt - u xx + [f(u)] xx + u xxxx = 0, and its periodic traveling wave solutions are considered. Using the transform z = x - wt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein-type integral equation is then established by using the Green's function method. This integral equation generates compact operators in (C 2T , ∥.∥), a Banach space of real-valued continuous periodic functions with a given period 2T. We prove that for small T > 0, there exists an r > 0 such that there is no nontrivial solution to the integral equation in the ball B(0,r) ⊆ C 2T . And hence, the generalized Boussinesq equation has no 2T-periodic traveling wave solutions having amplitude less than r.

Optimal Approximation of Periodic Analytic Functions with Integrable Boundary Values

LetS=[z∈C: |Im(z)|<β] be a strip in the complex plane.Hq, 1⩽q<∞, denotes the space of functions, which are analytic and 2π-periodic inS, real-valued on the real axis, and possessq-integrable boundary values. Letμbe a positive measure on [0, 2π] andLp(μ) be the corresponding Lebesgue space of periodic real-valued functions on the real axis. The even dimensional Kolmogorov, Gel'fand, and linear widths of the unit ball ofHqinLp(μ) are determined exactly, when 1⩽p⩽q<∞ or when=q<p<∞ andβis sufficiently large. It is shown that all threen-widths coincide and a characterization of the widths in terms of Blaschke products is established.

A Fully Discrete Galerkin Method for Integral and Pseudo differential Equations on Closed Curves

We propose a cheep fully discrete version of the trigonometric Galerkin method of optimal accuracy order for integral and pseudodifferential equations on closed curves. A practical implementation of the method leads to a band system of linear algebraic equations. The error analysis is based on a thorough study of related integral operators in Sobolev spaces of periodic functions.

Bloch oscillations and other dynamical phenomena of electrons in semiconductor superlattices.

We solve the time-dependent Schr\"odinger equation based on the complete Hamiltonian for independent electrons in ideal GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As superlattices, described by the conventional one-dimensional flat-band picture, subject to a uniform electric field F. Our approach consists of using high-accuracy numerical methods, thereby enabling us to avoid truncating the Hamiltonian or adopting other uncontrolled approximations. In suitable circumstances the electrons exhibit long-lived, time-periodic Bloch oscillations, in the form of either sinusoidal center-of-mass oscillations or a coherent breathing mode. Depending on the miniband structure of the superlattice, the value of F, and the form of the initial wave function, other dynamical phenomena can coexist with or even totally mask the Bloch oscillations. These are unbounded acceleration of a portion of the electron wave packet antiparallel to the electric field and intrawell oscillations. We provide the conditions under which each of these three basic dynamical elements occurs. We summarize the major results of our systematic investigation of the dependence of the electron dynamics on the entire parameter space of periodic potentials, field strengths, and initial wave functions.

Optimal rates of convergence of some iterative approximation methods for the solution of fredholm equations in spaces of periodic analytic functions

We consider some classes of Fredholm equations with integral operators acting in spaces of periodic analytic functions. For these classes, we establish the exact order of the optimal rates of convergence for some versions of the method of iterative projections and KP methods.

The spectral theory for a pencil of skewsymmetrical differential operators of the third order

AbstractWe consider the linear algebra of a pair of skewsymmetrical forms in the space of periodic functions defined by differential operators. By linear transform in the space of functions we reduce this pair to the simplest possible form. In this process, we prove the theorem of reduction in rather general context. © 1994 John Wiley &amp; Sons, Inc.

Wavelet Transform of Periodic Generalized Functions

The aim of this paper is to define the wavelet transform for spaces of periodic functions, then extend this definition to spaces of generalized functions larger than the space of periodic Schwartz distributions, such as spaces of periodic Beurling ultradistributions and hyperfunctions on the unit circle. It is shown that the wavelet transforms of such generalized functions are nice and smooth functions defined on an infinite cylinder, provided that the analyzing wavelet is also nice and smooth. For example, it is shown that the growth rate of the derivatives of the wavelet transform is almost as good as that of the analyzing wavelet. More precisely, if the mother wavelet g satisfies Sup x ∈ R | x k g ( q) ( x)| ≤ CA k B q k kβ q qα ( k, q = 0, 1, 2, …), then the wavelet transform W g (ƒ) of a periodic Beurling ultradistribution ƒ satisfies sup( r,θ) ∈ Y ϵ | r k ∂ p θ ∂ q r W g (ƒ)( r, θ)| ≤ DA k k α k B p C q p pα q q (α + β); k, p, q ≥ 0, where Y ϵ = {( r, θ): r ≥ ϵ > 0, θ ∈ T}.