On mean approximation of function and its derivatives

In the first section the best approximations of periodic functions of one real variable by trigonometric polynomials are studied. Estimates of these approximations in terms of averaged differences are obtained. A multidimensional generalization of these estimates is presented in the second section. As a consequence. The multidimensional Jackson's theorem is proved.

Asymptotic properties of particle filter-based maximum likelihood estimators for state space models

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces Lp, 1 < p < +∞. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator ψ, sufficient constructive ψ-smoothness conditions, estimates of best approximations of ψ-derivatives, estimates of best approximations of ψ-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a &gt; 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

Rational approximations of the conjugate function on the segment $[-1,~1]$ by Abel--Poisson sums of conjugate rational integral Fourier-Chebyshev operators with restrictions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established.Rational approximations on the segment $[-1,~1]$ of the conjugate function with density $(1-x)^\gamma,$ $\gamma\in (1/2,~1),$ by Abel-Poisson sums are studied. An integral representation of approximations and estimates of approximations taking into account the position of a point on the segment $[-1,~1]$ are obtained. An asymptotic expression as $r\to 1$ for the majorant of approximations, depending on the parameters of the approximating function is established. In the final part, the optimal values of parameters which provide the highest rate of decrease of this majorant are found. As a corollary we give some asymptotic estimates of approximations on the segment $[-1,~1]$ of the conjugate function by Abel-Poisson sums of conjugate polynomial Fourier-Chebyshev series.

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.

We prove Jackson type estimates for the approximation of monotone nondecreasing functions by monotone nondecreasing splines with equally spaced knots. Our results are of the same order as the Jackson type estimates for unconstrained approximation by splines with equally spaced knots.

If the operator A is positive, i.e., K(t) _> 0, then it is known that in the case of p = i, estimate (3) is good enough, in the sense that its application leads to the precise value of the deviation on various classes of functions. However, for the values p ~ i, = estimate (3) is already rough, which is related to the strict convexity of the corresponding spaces Lp. Our goal is to obtain a more precise than (3) estimate of approximation by positive operators, which would reduce to the equality for a sufficiently wide class of functions. In doing so, we shall use the following variant of the Riesz-Thorin theorem on complex interpolation of operators in the spaces Lp with mixed norm, contained in [i]. Let LDr (i < p, r < ~) be the space of 2~-periodic in each argument complex functiorls g(x, t) such that

We find necessary and sufficient conditions for the uniform approximation of continuous functions by Lagrange–Sturm–Liouville processes in an arbitrary interval and obtain estimates for approximation errors.

Some properties of the functions integrable on the segment were considered in this article. Estimates for approximation are obtained.

We show that the validity of Jackson-type estimates in comonotone and coconvex approximations of continuous $2\pi$-periodic functions by trigonometric polynomials is equivalent to the validity of the corresponding estimates of approximation by continuous $2\pi$-periodic piecewise algebraic polynomials with equidistant knots, having a minor additional property.

This study intends to heighten the training expression of radial basis function neural network (RbfN) via artificial neural network (ANN) and metaheuristic (MH) algorithms. Further, the self-organizing map neural network (SOMN), artificial immune system (AIS) and ant colony optimization (ACO)-based algorithms are employed to train RbfN for function approximation and demand prediction estimation. The proposed hybrid of SOMN with AIS-based and ACO-based (HSIA) algorithm incorporates the complementarity of exploitation and exploration potentialities to attain problem settle. The experimental consequences have evidenced that AIS-based and ACO-based algorithms can be integrated cleverly and propose a hybrid algorithm which attempts for receiving the optimal training expression among corresponding algorithms in this study. Additionally, method appraisal consequences for two benchmark nonlinear test functions illustrate that the proposed HSIA algorithm surpasses corresponding algorithms in accuracy of function approximation issue and the industrial computer (IC) demand prediction exercise.

The article considers the LP(T2) Lebesque space of periodec functions of two variables. The problems of approximation of functions of two variables by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses are stydied. Value EQγn(f)p=inft∈(Qγn)⌈f−t⌉p,i≤p≤∞ the best approximation of the function f(x) by trigonometric polynomials with “numbers” of harmonics from a step hyperbolic cross of Qγn The article consists of two sections. The first section contains some well-known statements necessary to prove the main results. In the second section, exact estimates of the best approximations of certain functions are established. These estimates make it possible to estimate the upper bounds of the best approximations for certain classes of functions. As approximation apparatuses, trigonometric polynomials with a spector from a stepwise hyperbolic cross are used. The questions considered in this work belong to the circle of questions studied in the works of K. I. Babenko, S. A.Telyakovsky, I. S.Bugrova, N.S.Nikolsky.