Problems In Approximation Theory Research Articles

The second order modulus revisited: remarks, applications, problems

Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.

On Conic Equations Under Bernstein Operators

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approximation is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this paper belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circles, ellipses, hyperbolas and parabolas. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into.

Convergence rates for energies of interacting particles whose distribution spreads out as their number increases

We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.

Necessary conditions of extremality in one problem of approximation theory

Necessary conditions of extremality in one problem of approximation theory.

On extremal problems of approximation theory for classes of functions of two discrete variables in $l^2_{q,p}$

For classes of $2\pi$-periodic functions of two discrete variables, defined on grid set $\sigma_{q,p}$, we found exact values of Kolmogorov and linear quasiwidths of classes $W^{r,p}(l^2_{q,p})$ in $l^2_{q,p}$ space.

Some problems of approximation theory for powers of normal operators in Hilbert space

The best approximation of unbounded operator $A^k$ in class with $\| A^r x \| \leqslant 1$ and the best approximation of class with $\|A^k x \| \leqslant 1$ by class with $\| A^r x \| \leqslant N$, $N > 0$ for powers $k < r$ of normal operator $A$ in the Hilbert space $H$ are found.

Autonomous Eco-Driving with Traffic Light and Lead Vehicle Constraints: An Application of Best Constrained Interpolation

Abstract Eco-Driving is a critical technology for improving automotive transportation efficiency. It is achieved by modifying the driving trajectory over a particular route to minimize required propulsion energy. Eco-Driving can be approached as an optimal control problem subject to driving constraints such as traffic lights and positions of other vehicles. In this paper we demonstrate the connection between Eco-Driving and best interpolation in the strip which is a problem in approximation theory and optimal control. By exploiting this connection, we are able to generate optimal Eco-Driving trajectories that can be driven with an autonomous system and evaluate them using conventional, hybrid electric, and fully electric vehicle models from FASTSim software. Our results quantify the energy efficiency improvements that can be achieved with the proposed approach.

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

Piecewise polynomials with different smoothness degrees on polyhedral complexes

For a given d-dimensional polyhedral complex Δ and a given degree k, we consider the vector space of piecewise polynomial functions on Δ of degree at most k with a different smoothness condition on each pair of adjacent d-faces of Δ. This is a finite dimensional vector space. The fundamental problem in Approximation Theory is to compute the dimension of this vector space. It is known that the dimension is given by a polynomial for sufficiently large k via commutative algebra. By using the technique of McDonald and Schenck [3] and extending their result to a plane polyhedral complex Δ with varying smoothness conditions, we determine this polynomial. This gives a complete answer for the dimension. At the end we discuss some examples through this technique.

Approximation by Entire Functions on a Countable Union of Real-Axis Segments. 4. Inverse Theorem

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

Approximation by rational functions in Smirnov classes with variable exponent

In this article, we investigate the direct problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on a doubly connected domain bounded by two Dini-smooth curves.

On Best Polynomial Approximations in the Spaces <i>S<sup>p</sup></i> and Widths of Some Classes of Functions

In the article are studied some problems of approximation theory in the spaces Sp (1 ≤ p < ∞) introduced by A.I. Stepanets. It is obtained the exact values of extremal characteristics of a special form which connect the values of best polynomial approximations of functions en-1(f)Sp with expressions which contain modules of continuity of functions f(x) є Sp. We have obtained the asymptotically sharp inequalities of Jackson type that connect the best polynomial approximations en-1(f)Sp with modules of continuity of functions f(x) є Sp (1 ≤ p < ∞). Exact values of Kolmogorov, linear, Bernstein, Gelfand and projection n-widths in the spaces Sp are obtained for some classes of functions f(x) є Sp. The upper bound of the Fourier coefficients are found for some classes of functions.

On uniform approximation on subsets

1. The development of many areas of approximation theory was stimulated by the necessity of solving applied problems. In the present paper, we consider a problem in approximation theory, which is a formalization of the problem of “approximation of a document photograph’s background” considered in the theory of image processing [1]. Suppose we are given a set X with probability measure μ, a set of functions M ⊂ L∞(X), and a number u ∈ (0, 1). In image processing theory, it is assumed that, for X = [0, 1]2, the class M consisting of sufficiently smooth functions, and the functions f ∈ L∞(X) corresponding to the photograph of a document, the following representation is valid:

Approximation in Smirnov classes with variable exponent

In this work, the inverse problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on the Jordan domains with a Dini-smooth boundaries, is studied. First, for this purpose, an inverse theorem in the variable exponent Lebesgue spaces of periodic functions is obtained. Later, using the special linear operators, this inverse theorem to the variable exponent Smirnov classes of analytic functions is moved.

On the best mean square approximations by entire functions of exponential type in L 2(ℝ) and mean ν-widths of some functional classes

We consider some extremal problems of approximation theory of functions on the whole real axis ℝ by entire functions of the exponential type. In particular, we find the exact values of the mean ν-widths of classes of functions, defined by the modules of continuity of the mth order ωm and majorants ψ satisfying the special type of restriction.

Approximation in weighted Smirnov classes

In this work, the direct and inverse problems of approximation theory in the weighted Smirnov classes of analytic functions defined on the domains with a regular boundary are investigated. In particular, the constructive characterization of the generalized Lipschitz classes of functions are obtained.

Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schr?dinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions.

On the Application of Iterative Methods of Nondifferentiable Optimization to Some Problems of Approximation Theory

We consider the data fitting problem, that is, the problem of approximating a function of several variables, given by tabulated data, and the corresponding problem for inconsistent (overdetermined) systems of linear algebraic equations. Such problems, connected with measurement of physical quantities, arise, for example, in physics, engineering, and so forth. A traditional approach for solving these two problems is the discrete least squares data fitting method, which is based on discretel2-norm. In this paper, an alternative approach is proposed: with each of these problems, we associate a nondifferentiable (nonsmooth) unconstrained minimization problem with an objective function, based on discretel1- and/orl∞-norm, respectively; that is, these two norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residual using these two norms. Respective subgradients are calculated, and a subgradient method is used for solving these two problems. The emphasis is on implementation of the proposed approach. Some computational results, obtained by an appropriate iterative method, are given at the end of the paper. These results are compared with the results, obtained by the iterative gradient method for the corresponding “differentiable” discrete least squares problems, that is, approximation problems based on discretel2-norm.

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.

Relative Widths of Some Sets of l<sup>m</sup><sub>p</sub>

n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid) of