Polynomials Of Best Approximation Research Articles

On derivatives of an algebraic polynomial of best approximation with weight

On derivatives of an algebraic polynomial of best approximation with weight

Towards the Sign Function Best Approximation for Secure Outsourced Computations and Control

Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for building lightweight, low-cost, smarter robots with intelligent brains consisting of data centers, knowledge bases, task planners, deep learning, information processing, environment models, communication support, synchronous map construction and positioning, etc. It enables robots to be endowed with secure, powerful capabilities while reducing sizes and costs. Processing encrypted information using homomorphic ciphers uses the sign function polynomial approximation, which is a widely studied research field with many practical results. State-of-the-art works are mainly focused on finding the polynomial of best approximation of the sign function (PBAS) with the improved errors on the union of the intervals [−1,−ϵ]∪[ϵ,1]. However, even though the existence of the single PBAS with the minimum deviation is well known, its construction method on the complete interval [−1,1] is still an open problem. In this paper, we provide the PBAS construction method on the interval [−1,1], using as a norm the area between the sign function and the polynomial and showing that for a polynomial degree n≥1, there is (1) unique PBAS of the odd sign function, (2) no PBAS of the general form sign function if n is odd, and (3) an uncountable set of PBAS, if n is even.

Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

Abstract The relation between derivatives of a polynomial of best approximation and the best approximation of the function is investigated in generalized Lebesgue spaces with variable exponent. In addition, the relationship between the fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied.

Mathematical modelling and computational algorithms in radio engineering systems

We developed computer software for determining polynomials of best approximation to match functional dependencies, calibration curves of measuring systems and sensors with enhanced precision and performance. We ensured that the components of the output error cancel each other out. We optimised a set of algorithms for computing the arcsin( x ) function

Improving Polynomial Methods of Reconstruction of Functional Dependences in Information-Measuring Systems

Automated methods and algorithms for use in finding polynomials of best approximation for approximating functional dependences and calibration characteristics from the standpoint of optimization of the criteria of a computational process are improved. Methods of mutual compensation of the errors of specialized computational measuring systems are considered.

On the norms of Favard Kernels

For Favard kernels, which are used to construct polynomials of best approximation for classes of periodic differentiable functions W r , we obtain exact values of their norms as simple finite sums for odd r and lower bounds for the norms for even r. Similar results are also obtained for the norms of the adjoint Favard kernels.

A note on uniform approximation of functions having a double pole

AbstractWe consider the classical problem of finding the best uniform approximation by polynomials of$1/(x-a)^2,$where$a>1$is given, on the interval$[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight \(u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}\), α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.

Lagrange interpolation with exponential weights on [formula omitted

In order to approximate functions defined on (−1,1) with exponential growth for |x|→1, we consider interpolation processes based on the zeros of orthonormal polynomials with respect to exponential weights. Convergence results and error estimates in weighted Lp metric and uniform metric are given. In particular, in some function spaces, the related interpolating polynomials behave essentially like the polynomial of best approximation.

Polynomial of Best Uniform Approximation to and Smoothing in Two-level Methods

AbstractWe derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

On uniform constants of strong uniqueness in Chebyshev approximations and fundamental results of N. G. Chebotarev

In the problem of the best uniform approximation of a continuous real-valued function in a finite-dimensional Chebyshev subspace , where is a compactum, one studies the positivity of the uniform strong uniqueness constant . Here stands for the strong uniqueness constant of an element of best approximation of , that is, the largest constant such that the strong uniqueness inequality holds for any . We obtain a characterization of the subsets for which there is a neighbourhood of satisfying the condition . The pioneering results of N. G. Chebotarev were published in 1943 and concerned the sharpness of the minimum in minimax problems and the strong uniqueness of algebraic polynomials of best approximation. They seem to have been neglected by the specialists, and we discuss them in detail.

Polynomial approximation with an exponential weight in [−1, 1] (revisiting some of Lubinsky’s results)

Revisiting the results in [7], [8], we consider the polynomial approximation on (−1, 1) with the weight \(w(x) = e^{-(1-x^2)^{-\alpha}}\), α > 0. We introduce new moduli of smoothness, equivalent to suitable K-functionals, and we prove the Jackson theorem, also in its weaker form. Moreover, we state a new Bernstein inequality, which allows us to prove the Salem-Stechkin inequality. Finally, also the behaviour of the derivatives of the polynomials of best approximation is discussed.

On Complex (Non-Analytic) Chebyshev Polynomials in ℂ2

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in ℂ2 by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation for the original problem are given. Furthermore, we derive asymptotics for the minimal deviation.

Best approximation to monomials on a cube

The paper considers a multivariate analogue of the Chebyshev problem on the cube concerning the construction of polynomials of least deviation from zero. A classification of monomials possessing a unique polynomial of best approximation in the space of continuous functions on the unit cube in is given. Precise solutions in some weighted spaces are found. Bibliography: 11 titles.

Finding polynomials of best approximation with weight

A new iterative method for finding the parameters of polynomials of best approximation with weight in is presented. It is based on the representation of the error in the trigonometric form in terms of the phase function. The iterative method of finding the corrections to the phase functions that determine the joint motion of the zeros and the e-points of the error is based on inverse analysis, perturbation theory, and asymptotic formulae for extremal polynomials.Bibliography: 24 titles.

On polynomials of best approximation

In the space of continuous functions defined on the ball from ℝn, n ≥ 2, we study the set of algebraic polynomials of least deviation from zero. Its width and dimension are estimated.

On the boundedness below of trigonometric polynomials of best approximation

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α n } n=0 ∞ , are uniformly bounded below if the conditions a n = O(n −1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent −1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L 2π 1 .

On the Bernstein Constants of Polynomial Approximation

Assume \(\alpha >0\) is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit $\Lambda _{\infty ,\alpha }^{\ast }=\lim_{n\rightarrow \infty }n^{\alpha}E_{n}[ \vert x\vert ^{\alpha };L_{\infty }[ {-}1,1]] .$ Here \(E_{n}[ \vert x\vert ^{\alpha };L_{\infty }[ -1,1] ] \) denotes the error in best uniform approximation of \(\left\vert x\right\vert ^{\alpha }$ on $\left[ {-}1,1\right] \) by polynomials of degree \(\leq n\). Bernstein proved that \(\Lambda _{\infty ,\alpha }^{\ast} \) is itself the error in best uniform approximation of \(\left\vert x\right\vert ^{\alpha }\) by entire functions of exponential type at most 1, on the whole real line. We prove that the best approximating entire function is unique, and satisfies an alternation property. We show that the scaled polynomials of best approximation converge to this unique entire function. We derive a representation for \(\Lambda _{\alpha ,\infty }^{\ast }\), as well as its \(L_{p}\) analogue for \(1\leq p<\infty \).

Weighted Convergence of Lagrange Interpolation Based on Gauss-Kronrod Nodes

The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendrepolynomials and Stieltjes polynomials, is a standard rule for automaticnumerical integration in mathematical software libraries. For a long time,very little was known about the underlying Lagrange interpolationprocesses. Recently, the authors proved new bounds and asymptoticproperties for the Stieltjes polynomials and, subsequently, appliedthese results to investigate the associated interpolation processes. Thepurpose of this paper is to survey the quality of these interpolationprocesses, with additional results that extend and complete the existingones. The principal new results in this paper are necessary and sufficientconditions for weighted convergence. In particular, we show that theLagrange interpolation polynomials associated with the above interpolationprocesses have the same speed of convergence as the polynomials of bestapproximation in certain weighted Besov spaces.

On the Zeros of Polynomials of Best Approximation

Given a function f, uniform limit of analytic polynomials on a compact, regular set E⊂CN, we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate Lq norms. These results give multivariable versions of one-variable results due to Blatt–Saff, Pleśniak and Wójcik.