Polynomials Of Least Deviation Research Articles

Polynomials of Least Deviation from Zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<infty ) for the case p=1. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n-mathbf {d}^* zeros on the convex hull of the support of the measure, where mathbf {d}^* denotes the number of terms in the discrete part.

Polynomials of least deviation from zero

Polynomials of least deviation from zero

Trigonometric polynomials of least deviation from zero in measure and related problems

We give a solution of the problem on trigonometric polynomials f n with the given leading harmonic y cos n t that deviate the least from zero in measure, more precisely, with respect to the functional μ ( f n ) = mes { t ∈ [ 0 , 2 π ] : | f n ( t ) | ≥ 1 } . For trigonometric polynomials with a fixed leading harmonic, we consider the least uniform deviation from zero on a compact set and find the minimal value of the deviation over compact subsets of the torus that have a given measure. We give a solution of a similar problem on the unit circle for algebraic polynomials with zeros on the circle.

On polynomials of least deviation from zero

A new family of polynomials of least deviation from zero is defined on the unit disk B. Lower bounds for best approximations in the space L p (B), p ≥ 1, are given.

Polynomials of least deviation from zero in the L[−1, 1] metric with five fixed coefficients

We study properties of polynomials R n+5(x) of least deviation from zero in the L[−1, 1] metric, with five given leading coefficients whose forms were calculated previously. Theorems 1 and 2 together with Theorem A contain, in particular, a final classification of polynomials R n+5(x) that have exactly (n + 1) sign changes in (−1, 1).

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.

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.

Polynomials of Least Deviation from Zero

A new family of polynomials of least deviation from zero is defined on the unit disk B. Lower bounds for best approximations in the space Lp(B), p ≥ 1, are given.

On Polynomials of Least Deviation from Zero in Several Variables

A polynomial of the form xa – p(x), where the degree of p is less than the total degree of xa , is said to be least deviation from zero if it has the smallest uniform norm among all such polynomials. We study polynomialsof least deviation from zero over the unit ball, the unit sphere, and the standard simplex. For d = 3, extremal polynomial for (x 1 x 2 x 3) k on the ball and the sphere is found for k = 2 and 4. For d ≥ 3, a family of polynomials of the form (x l… xd )2 – p(x) is explicitly given and proved to be the least deviation from zero for d = 3, 4, 5, and it is conjectured to be the least deviation for all d.

Bivariate Polynomials of Least Deviation from Zero

AbstractBivariate polynomials with a fixed leading term xmyn, which deviate least fromzero in the uniform or L2-norm on the unit disk D (resp. a triangle) are given explicitly. A similar problem in Lp, 1 ≤ p ≤ ∞, is studied on D in the set of products of linear polynomials.

Effective computation of Chebyshev polynomials for several intervals

A cell decomposition of the space of polynomials of least deviation from zero on a system of several closed intervals of the real axis is discussed. An effective method for calculating the in each cell making use of automorphic functions is put forward.

On the Polynomials of A. O. Gelfond Having Minimal Deviation from Zero together with Their Derivatives

Some results of A.O. Gelfond concerning monic polynomials of least deviation (uniform) from zero together with their derivatives are extended to certain Lp - norms and to several variables. As an application we extend a recent result of Xiaoming Huang to functions f∈C(n)n)([−1, 1]) which satisfy f(n)(x)≥1.

A remark on complex polynomials of least deviation

A remark on complex polynomials of least deviation

A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part I

Article A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part I was published on January 1, 1993 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 8, issue 3).

A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part II

Article A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part II was published on January 1, 1993 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 8, issue 5).

On multivariate polynomials of least deviation from zero on the unit cube

In the family of all r-variable real polynomials with total degree not exceeding μ and with maximum norm on the unit-cube not exceeding 1, any of the leading coefficients is maximum for a special product of one-variable Chebyshev polynomials of the first kind. This is a consequence of an even more general result on polynomials of least deviation from zero on the unit cube.

On multivariate polynomials of least deviation from zero on the unit ball

We introduce a new family of multivariate polynomials by means of a very simple generating function which reduces to the generating function of the Chebyshev polynomials of the first kind in case of one single variable. The family solves several problems of best approximation on the unit ball. Let IP~, / ~ N o , denote the space of all real polynomials in r variables (x 1 . . . . , x r ) = x of degree not exceeding/~. For convenience, define IP" 1:= [01 and x m : ~ XTI mr : ...x~ , Ira1 = m l + ' " + m , . ,