Extremal Problem For Polynomials Research Articles

An extremal problem for polynomials

For the polynomials F ( z ) = ∑ j = 1 N a j z j with real coefficients and normalization a 1 = 1 we solve the extremal problem sup a 2 , … , a N ⁡ ( inf z ∈ D ⁡ { Re ( F ( z ) ) : Im ( F ( z ) ) = 0 } ) . We show that the solution is − 1 4 sec 2 ⁡ π N + 2 , and the extremal polynomial 1 U N ′ ( cos ⁡ π N + 2 ) ∑ j = 1 N U N − j + 1 ′ ( cos ⁡ π N + 2 ) U j − 1 ( cos ⁡ π N + 2 ) z j is unique and univalent, where the U j ( x ) are the Chebyshev polynomials of the second kind, j = 1 , … , N . As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

Yudin–Hermite Extremal Problems for Polynomials

Yudin–Hermite Extremal Problems for Polynomials

Extremal problems for polynomials with real roots

We consider polynomials of degree d with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of such polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of dth powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The second family is related to generalized Jacobi (or Gegenbauer) polynomials, which helps us to find the associated discriminants. We also investigate the dual problem of maximizing the value of discriminant, while keeping the absolute value of polynomials at a point away from the real line fixed. Our results are then applied to problems on the largest disks contained in lemniscates, and to the minimum energy problems for discrete charges on the real line.

Dual mean value problem for complex polynomials

We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.

Effective computation of optimal stability polynomials

The construction of stable explicit multistage Runge-Kutta methods in 1950–1960 stumbled over a certain extremal problem for polynomials. The solution to this problem is known as the optimal stability polynomial and its computation is notoriously difficult. We propose a new method for the effective evaluation of optimal stability polynomials which is based on the explicit analytical representation of the solution. The main feature of the method is its independence of the computational complexity of the degree of the solution.

Certain extremal problems for polynomials

Extensions of two classical results about polynomials, one due to W. Markov and the other due to Duffin and Schaeffer, are obtained in this paper. An interesting result of S. Bernstein, which went unnoticed until it was rediscovered by P. Erdos, 34 years later, is also generalized. Our results are especially amenable to numerical calculations, and may, therefore, be of some practical importance.

An extremal problem for polynomials with a prescribed multiple zero

We obtain a sharp inequality for trigonometric polynomials which have a double zero at a specified point. A similar extremal problem has been studied by R.P. Boas Jr.

Convex Optimal Designs for Compound Polynomial Extrapolation

The extrapolation design problem for polynomial regression model on the design space [−1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs |ξ m * (z), ξ2m* (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.

Some extremal problems for polynomials in integral metrics

Some extremal problems for polynomials in integral metrics

An extremal problem for polynomials

Let f ( z ) = z n + ⋯ f(z) = {z^n} + \cdots be a polynomial such that the level set E = { z : | f ( z ) | ≤ 1 } E = \{ z:|f(z)| \leq 1\} is connected. Then max { | f ′ ( z ) | : z ∈ E } ≤ 2 ( 1 / n ) − 1 n 2 \max \{ |f’ (z)|:z \in E\} \leq {2^{(1/n) - 1}}{n^2} , and this estimate is the best possible.

An extremal problem for polynomials with a prescribed value at a given point

Let and denote by ωn(Z0,δ) the set of polynomials P of degree ≦n n for which (a) (b) We determine max .

An optimization principle applied to certain extremal problems for polynomials

An optimization principle applied to certain extremal problems for polynomials

On a class of extremal problems

In the paper we introduce a class of trigonometrical polynomial extremal problems depending on a continuous parameter 0≤r≤1. It turns out that the two border cases r=0 and r=1 are known problems investigated earlier by Kamae, Mendes-France, Ruzsa and the present author. We also introduce another set of extremal problems for measures with similar parametrization, and prove a duality relationship between the two type of extremal quantities. The proof relies on a minimax theorem proved earlier by the author. The known duality results are proved as corollaries.

An extremal problem for polynomials with a prescribed value at a given point of the real axis.

An extremal problem for polynomials with a prescribed value at a given point of the real axis.

An Extremal Problem for Polynomials with a Prescribed Zero

An Extremal Problem for Polynomials with a Prescribed Zero

An extremal problem for polynomials with a prescribed zero

In this paper a new elementary proof for solving one extremal problem of real polynomials with a given real zero is given.

An Extremal Problem for Polynomials with Nonnegative Coefficients

An Extremal Problem for Polynomials with Nonnegative Coefficients

An extremal problem for polynomials with nonnegative coefficients

Let W n {W_n} be the set of all algebraic polynomials of exact degree n n whose coefficients are all nonnegative. For the norm in L 2 [ 0 , ∞ ) {L^2}[0,\infty ) with generalized Laguerre weight function w ( x ) = x α e − x ( α > − 1 ) w(x) = {x^\alpha }{e^{ - x}}\quad (\alpha > - 1) , the extremal problem C n ( α ) = sup P ∈ W n ( ‖ P ′ ‖ / ‖ P ‖ ) 2 {C_n}(\alpha ) = {\sup _{P \in {W_n}}}{(\left \| {P’} \right \|/\left \| P \right \|)^2} is solved, which completes a result of A. K. Varma [7].

Extremal Problems for Polynomials with Exponential Weights

Extremal Problems for Polynomials with Exponential Weights

An extremal problem for polynomials

An extremal problem for polynomials