Sufficient Conditions For Uniform Convergence Research Articles

On transforming hybrid nonlinear control problems with model uncertainty and input disturbance to mixed integer-linear programs

Control of multi-input multi-output (MIMO) hybrid nonlinear dynamic systems which are affine in control inputs is studied in this paper. It is assumed that each control input of the system can be continuous or discrete with a bound constraint. It is also assumed that the controller is discrete-time and updates the control(s) with a constant frequency. Based on these assumptions, a theorem which represents the necessary and sufficient condition for uniform convergence of the sequence of samples of the state vector of system towards the desired point is presented and proved. As a result of this theorem, a (mixed integer-) linear programming for calculation of control(s) is proposed. Two well-known nonlinear hybrid control problems with multiple inputs are solved by using the proposed method and the results are presented.

A Sufficient Condition for Uniform Convergence of Trigonometric Series with p-Bounded Variation Coefficients

In this paper we consider trigonometric series with p-bounded variation coefficients. We presented a sufficient condition for uniform convergance of such series in case p>1. This condition is significantly weaker than these obtained in the results on this subject known in the literature.

Isolation of a periodic component by singular wavelet decomposition

In this paper, we propose to use a discrete wavelet transform with a singular wavelet to isolate the periodic component from the signal. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet is zero). For singular wavelets, the validity condition is not met. As a singular wavelet, you can use the Delta-shaped functions, which are involved in the estimates of Parzen-Rosenblatt, Nadaraya-Watson. Using singular value of a wavelet is determined by the discrete wavelet transform. This transformation was studied earlier for the continuous case. Theoretical estimates of the convergence rate of the sum of wavelet transformations were obtained; various variants were proposed and a theoretical justification was given for the use of the singular wavelet method; sufficient conditions for uniform convergence of the sum of wavelet transformations were formulated. It is shown that the wavelet transform can be used to solve the problem of nonparametric approximation of the function. Singular wavelet decomposition is a new method and there are currently no examples of its application to solving applied problems. This paper analyzes the possibilities of the singular wavelet method. It is assumed that in some cases a slow and fast component can be distinguished from the signal, and this hypothesis is confirmed by the numerical solution of the real problem. A similar analysis is performed using a parametric regression equation, which allows you to select the periodic component of the signal. Comparison of the calculation results confirms that nonparametric approximation based on singular wavelets and the application of parametric regression can lead to similar results.

Sufficient conditions for uniform convergence of random series

Sufficient conditions for uniform convergence of random series

Sufficient conditions for uniform convergence of random series

Sufficient conditions for the uniform convergence of random series are obtained.

On the convergence of Cesàro means of negative order of Vilenkin-Fourier series

AbstractIn 1971 Onnewer and Waterman establish a sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In this paper we consider different classes of functions of generalized bounded oscillation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesàro means of negative order.

Uniform convergence of basic Fourier\u2013Bessel series on a q-linear grid

We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.

APPROXIMATELY SINGULAR WAVELET

The problem of approximation is relevant for most engineering applications. In this connection, the universal methods of approximation are of interest. The method of nonparametric approximation is developing in the paper – the method of singular wavelets. The method includes an effective numerical algorithm based on the summation of a recursive sequence of functions. The universal algorithm of approximation makes it possible to apply it to approximate one-dimensional and multidimensional functions, in decision support systems, in the processing of stochastic information, pattern recognition, and solution of boundary-value problems.The introduction explain the idea of the method of singular wavelets – to combine the theory of wavelets with the Nadaraya-Watson kernel regression estimator. Usually, Nadaraya-Watson kernel regression are considered as an example of non- parametric estimation. However, one parameter, the smoothing parameter, is still present in the traditional kernel regression algorithm. The choice of the optimal value of this parameter is a complex mathematical problem, and numerous studies have been devoted to this question. In the approximation by the method of singular wavelets, summation of Nadaraya-Watson kernel regression estimates with the smoothing parameter takes place, which solves the problem of the optimal choice of this parameter.In the main part of the paper theorems are formulated that determine the properties of the regularized wavelet transform. Sufficient conditions for uniform convergence of the wavelet series are obtained for the first time. To illustrate the effectiveness of the numerical approximation algorithm, we consider an example of the quasi-interpolation of the Runge function by wavelets with a uniform distribution of interpolation nodes.

Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs

In this paper, a class of hybrid difference schemes with variable weights on Bakhvalov–Shishkin mesh is proposed to compute both the solution and the derivative in quasilinear singularly perturbed convection–diffusion boundary value problems. The parameter-uniform second-order convergence of approximating to the solution and the derivative on Bakhvalov–Shishkin mesh and that of nearly second-order on Shishkin mesh are proved clearly by use of an (l∞,l1)-stability property, where the former sufficient conditions for uniform convergence are modestly relaxed on Bakhvalov–Shishkin mesh and are clarified on Shishkin mesh. The numerical examples support the proposed schemes with new sufficient conditions and their error estimates.

On the uniform convergence of the eigenfunction expansions

We study sufficient conditions for uniform convergence of eigenfunction expansions associated with Schrodinger's operator.

Topological properties of measurable structures and sufficient conditions for uniform convergence of frequencies to probabilities

The event-uniform convergence of frequencies to probabilities is one of the main tools of the today theory of learning pattern recognition which enables one to substantiate the generalizability of the learning algorithms. Proposed was a new approach to derivation of the sufficient conditions for such convergence on the basis of only the topological properties of the event ?-algebra induced by the class under consideration.

On summability of N-fold Fourier integrals corresponding to pseudodifferential operators

It is well known, that if N ≥ 3, then spherical partial sums of N-fold Fourier integrals (eigenfunction expansions of Laplace operator) of the characteristic function of the unit ball diverge at the origin. Note, here level surface of Laplace operator and the surface of discontinuity of the considered piecewise smooth function are both spheres. It was first noted by Pinsky and Taylor, that if we consider nonspherical partial sums (eigenfunction expansions of elliptic pseudodifferential operators), then to obtain the same effect, we should change the above second sphere with the dual set to level surface of the pseudodifferential operator. Namely, nonspherical partial sums of a piecewise smooth function, supported inside the dual surface converge everywhere except the origin. In this paper we investigate summability of these expansions by Riesz method and show that the order s > (N − 3)/2 of Riesz means guarantees convergence everywhere, where the function is smooth. Since piecewise smooth functions are in Nikolskii class H 1 1 (R N ), we also establish necessary and sufficient conditions for uniform convergence of expansions of H -functions.

On Convergence of Kernel Learning Estimators

The paper studies convex stochastic optimization problems in a reproducing kernel Hilbert space (RKHS). The objective (risk) functional depends on functions from this RKHS and takes the form of a mathematical expectation (integral) of a nonnegative integrand (loss function) over a probability measure. The problem is generally ill-posed, a difficulty that in statistical learning is addressed through Tihonov regularization, with Monte Carlo approximation of integrals, which also makes it possible to solve the problem by finite dimensional (convex) quadratic optimization. The approximate solutions are referred to as kernel learning estimators and are expressed as a linear combination of kernels evaluated at the sample points. They are functional random variables that depend on the full sample. The paper studies a probabilistic convergence of these approximate solutions under a gradual elimination of the regularization parameter with rising number of observations. Its intended contribution is to derive novel nonasymptotic bounds on the minimization error and exponential bounds on the tail distribution of errors and to establish novel sufficient conditions for uniform convergence of kernel estimators to the true (normal) solution with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size. Applications to least squares, median, and quantile regression estimation, as well as to binary classification, are discussed.

On Stochastic Optimization and Statistical Learning in Reproducing Kernel Hilbert Spaces by Support Vector Machines (SVM)

The paper studies stochastic optimization problems in Reproducing Kernel Hilbert Spaces (RKHS). The objective function of such problems is a mathematical expectation functional depending on decision rules (or strategies), i.e. on functions of observed random parameters. Feasible rules are restricted to belong to a RKHS. This kind of problems arises in on-line decision making and in statistical learning theory. We solve the problem by sample average approximation combined with Tihonov's regularization and establish sufficient conditions for uniform convergence of approximate solutions with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size.

A robust fitted operator finite difference method for a two-parameter singular perturbation problem1

We consider singularly perturbed two-point BVPs with two small parameters ϵ and μ multiplying the derivatives. It is pointed out in P.A. Farrel (Sufficient conditions for uniform convergence of a class of difference schemes for singular perturbation problems, IMA J. Numer. Anal. 7 (1987), pp. 459–472), that ‘in general, the exponentially fitted finite difference methods (EFFDMs) are more effective inside the layers. However, though these methods are uniformly convergent, they do not give fairly good approximations in the whole interval of interest’. In this paper, we study that the non-standard finite difference method (NSFDM) that we develop overcomes this weakness. Like EFFDMs, the NSFDM is also a method of fitted operator type. Secondly, unlike several earlier works (see, for example Gracia et al., Appl. Numer. Math. 56 (2006), pp. 962–980) where the authors use a combination of approaches in various regions, the method presented in this paper consists of just one scheme throughout the domain of interest. This is very important because it increases the possibilities of extending the approach both for higher dimensional and higher order problems. Combination of schemes usually suffers from the drawback that their selection is mostly based on the relative values of ϵ and μ, otherwise they fail to provide monotonic solutions. We also investigate a number of issues associated with a variety of NSFDMs and finally provide some comparative numerical results.

On convergence of greedy algorithm by Walsh system in the space C(0, 1)

The paper studies convergence of the greedy algorithm by the Walsh system in the space C(0, 1). Some sufficient conditions for uniform convergence are given. It is proved that there exists a function satisfying more restrictive conditions, for which the sequence of the partial sums of the Fourier-Walsh series diverges at the point 0.

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization.

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.

Series expansions for analytic systems linear in control

This paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields. We present a series expansion that can be computed via explicit recursive expressions, and we derive sufficient conditions for uniform convergence over the finite- and infinite-time horizon. Furthermore, we present a simplified series and convergence analysis for the setting of second-order polynomial vector fields. The treatment only relies on elementary notions on analytic functions, number theory, and operator norms.

Uniform Convergence of Sequence of Holomorphic Functions in the Unite Disk

In this paper, suppose that ƒ m (z)(m = 1,2,…) are of infinite order. Applying the notion of infinite type [4], we obtain the sufficient conditions for uniform convergence of sequence of holomorphic functions in the unite disk.