Sharpness of estimates for the least uniform derivation from the class of functions with bounded second derivative

We study the uniform approximation of continuous functions on an interval by functions with bounded second derivative. We prove the sharpness of estimates for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

On the stability of spatially uniform Langmuir oscillations of electronic plasmas

We investigate the stability of spatially uniform, time-periodic solutions of the one-dimensional Vlasov–Maxwell system describing the longitudinal oscillations of an electronic plasma in an uniform neutralizing ion background. We show that such a stability problem can be trivially solved since the zero wave number mode of the electric field, i.e. its space average, performs pure Langmuir oscillations independently of the other modes. We however point out that such oscillations do affect on time average the evolution of the velocity distribution function in the frame at rest.     

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.     

Uniform approximation by harmonic functions on compact subsets of ℝ<Superscript>3</Superscript>

We consider uniform approximation by harmonic functions on compact subsets in ℝ³. Under an additional assumption that the approximated function is Dini-continuous, we prove a natural analog of well-known Vitushkin’s uniform approximation lemma for an individual analytic function. Bibliography: 14 titles.     

Statistics of the Edge Green's Function for a One-Dimensional Disordered System with Binary or Uniform Diagonal Disorder

For a one-dimensional diagonally disordered chain, we study the statistics of the edge Green's function (EGF) constructed using the random matrix of the Hamiltonian of this system. We assume that the disorder is either binary or uniform. We show that the EGF distribution function is not analytic in the case of binary disorder and propose a simple algorithm for constructing this function. We calculate the EGF distribution function exactly on some interval in the case of uniform disorder and propose a simple, effective method for determining this function completely and for calculating the mean EGF. We verify all the obtained results using direct computer diagonalization and observe a good agreement.     

Approximation by the class of functions with bounded second derivative

The paper studies the problem of uniform approximation of a continuous function on a closed interval by the class of functions with bounded second derivative. We prove an estimate of the value of best approximation of the function by this class via its second modulus of continuity. The obtained estimate is sharp for the class of continuous functions.     

Bayesian shrinkage prediction for the regression problem

We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function.Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant.Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior.Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.     

On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We study the uniform convergence, on a closed interval, of spectral expansions of Hölder functions in a given complete and minimal system of eigenfunctions corresponding to a spectral problem with spectral parameter in a boundary condition. We consider boundary conditions of the third kind and subject the function to be expanded to a condition of nonlocal type ensuring the uniform convergence. We prove a theorem stating that expansions in the entire system of eigenfunctions of the problem are possible without any additional conditions.     

General interpolation schemes for the generation of irregular surfaces

We introduce an interpolation scheme to generate a class of irregular surfaces. The analysis is first carried out for a triangleT. We define the function ? on a subsetX, dense inT. In terms of the construction parameters of ?, we establish sufficient conditions for its uniform continuity so that it would be possible to extend it to a continuous function on the whole ofT. We do the same analysis in the case of a rectangleR.     

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.     

Polynomials orthogonal in the Sobolev sense,generated by Chebyshev polynomials orthogonal on a mesh

We consider the problem of constructing polynomials, orthogonal in the Sobolev sense on the finite uniform mesh and associated with classical Chebyshev polynomials of discrete variable. We have found an explicit expression of these polynomials by classicalChebyshev polynomials. Also we have obtained an expansion of new polynomials by generalized powers ofNewton type. We obtain expressions for the deviation of a discrete function and its finite differences from respectively partial sums of its Fourier series on the new system of polynomials and their finite differences.     

Best Uniform Approximation of a Family of Functions Continuous on a Compact Set

We investigate the problem of the best uniform approximation of a function continuous on a compact set. We generalize the principal results of this investigation to the problem of the best simultaneous uniform approximation of a family of functions continuous on a compact set.     

A Criterion for the Uniform Convergence of the Lagrange-Jacobi Interpolation Process

We obtain a uniform and sufficient condition for the convergence of the Lagrange interpolation process with Jacobi nodes on a closed interval [a, b] ? (?1, 1). The condition is stated in terms of the second differences of the interpolated function and uses its values only at the interpolation nodes. Some well-known criteria for uniform convergence are obtained as a consequence of our result.     

Approximation of Subharmonic Functions in the Half-Plane by the Logarithm of the Modulus of an Analytic Function

We approximate subharmonic functions defined in the open half-plane in the uniform metric outside the exceptional set by the logarithm of the modulus of an analytic (in the half-plane) function for the cases of a finite order and of an infinite lower order. We also obtain an estimate for the size of the exceptional set. It is shown that, in the case of a finite order, the obtained accuracy of the approximation cannot be essentially improved.     

Stability of spline approximation and the restoration of grid functions

For a functionf(x) ∈ H _ω ^r , defined on a uniform grid approximately, we propose a stable method for approximately restoring the function with the aid of polynomial splines. We derive uniform estimates for the deviations of the spline and its derivatives from the function and its derivatives.     

Statistical modeling for discrete patterns in a sequence of exchangeable trials

This paper proposes a new method for constructing a sequence of infinitely exchangeable uniform random variables on the unit interval. For constructing the sequence, we utilize a Pólya urn partially. The resulting exchangeable sequence depends on the initial numbers of balls of the Pólya urn. We also derive the de Finetti measure for the exchangeable sequence. For an arbitrarily given one-dimensional distribution function, we generate sequences of exchangeable random variables with the one-dimensional marginal distribution by transforming the exchangeable uniform sequences with the inverse function of the distribution function. Among them we mainly investigate sequences of exchangeable discrete random variables. They differ from the well-known exchangeable sequence generated only by the Pólya urn scheme. Some examples are also given as applications of the results to exact distributions of some statistics based on sequences of exchangeable trials. Further, from the above exchangeable uniform sequence we construct partial or Markov exchangeable sequences. We also provide numerical examples of statistical inference based on the exchangeable and Markov exchangeable sequences.     

Sharp estimates of the norms of fractional derivatives of functions of several variables satisfying Hölder conditions

We prove a new sharp Kolmogorov-type inequality that estimates the uniform norm of a mixed derivative of fractional order (in the sense of Marchaud) of a function of several variables via the uniform norm of the function and its norm on Hölder spaces.     

On an estimate for the rearrangement of a function from the Muckenhoupt class <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class A ₁.     

Asymptotics near the boundary of spectral functions of elliptic self-adjoint boundary problems

We obtain the asymptotic behavior of the spectral function of an elliptic self-adjoint boundary problem in a closed domain and give a uniform estimate for the remainder. The estimate for the remainder is the same as the interior estimates of Agmon-Kannai and Hormander.     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.