Extremal Trigonometric and Power Polynomials in Several Variables

We study extremal nonnegative polynomials in several variables. Our approach makes substantial use of block Toeplitz matrices. Note that the blocks of these matrices are themselves Toeplitz matrices.     

On Extremal Properties of Interpolation Polynomials for Functions of Several Variables

By using earlier results, we find solutions of two extremal problems on sets of interpolation polynomials for functions of several variables.     

Extremal Polynomials on Discrete Sets

We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15.     

Polynomials with Critical Values on Intervals

For polynomials P(z) with real coefficients having a fixed leading coefficient and satisfying the conditions P(z) ∈ [?1, 1] for z ∈ [?1, 1] and P(z) ∈ [?1, 1] if P′(z) = 0, we obtain new covering theorems, a Bernshtein-type inequality, and inequalities for the coefficients. The proofs are based on the use of univalent conformal mappings.     

Deformation of Minimal Polynomials and Approximation of Several Intervals by an Inverse Polynomial Mapping

In this paper we show that for a given set of l real disjoint intervals E_l=^l_j=1 [a_2j−1, a_2j] and given >0 there exists a real polynomial and a set of l disjoint intervals E_l=^l_j=1 [ã_2j−1, ã_2j] with E_lE_l and (ã₁, …, ã_2l)−(a₁, …, a_2l)_max<, such that ⁻¹([−1, 1])=E_l. The statement follows by showing how to get in a constructive way by a continuous deformation procedure from a minimal polynomial on E_l with respect to the maximum norm a polynomial mapping of E_l.     

Extremal Homogeneous Polynomials on Real Normed Spaces

IfPis a continuousm-homogeneous polynomial on a real normed space andPis the associated symmetricm-linear form, the ratio P/P always lies between 1 andm^m/m!. We show that, as in the complex case investigated by Sarantopoulos (1987,Proc. Amer. Math. Soc.99, 340–346), there areP's for which P/P=m^m/m! and for whichPachieves norm if and only if the normed space contains an isometric copy of ℓ^m₁. However, unlike the complex case, we find a plentiful supply of such polynomials providedm4.     

Extremal Values of Moments of Nonnegative Polynomials

Mathematical Notes -     

A Class of Extremal Functions and Trigonometric Polynomials

In this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL¹-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials.     

Extremal Polynomials with Preassigned Zeros and Rational Approximants

Let p _n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g _Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g _Ω (z) . We prove corresponding results when we restrict the freedom of p _n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.     

Bounds and Inequalities for General Orthogonal Polynomials on Finite Intervals

In this paper some characterizations of the regularity of (0, 2) interpolation are given.     

On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

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, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].     

Extremal subspaces and their submanifolds

It was proved in the paper [KM1] that the properties of almost all points of being not very well (multiplicatively) approximable are inherited by nondegenerate in (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper a.ne subspaces, and prove that the aforementioned Diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces. Submitted: March 2002; Revision: August 2002     

On Extremal Problems on a Set of Interpolation Polynomials in a Hilbert Space

We give solutions of two extremal problems, a generalization of a result of W. Porter on an interpolation polynomial in L₂(a,b) with minimal norm, which we get here in an abstract Hilbert space with a measure, and an analogue of a theorem of M. Golomb and H. Weinberger for a bounded set of operator interpolants.     

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

We consider the asymptotic zero behavior of polynomials that are extremal with respect to slowly decaying weights on [0, ∈fty) , such as the log-normal weight \exp(-γ ² log  ² x) . The zeros are contracted by taking the appropriate d _n th roots with d _n →∈fty . The limiting distribution of the contracted zeros is described in terms of the solution of an extremal problem in logarithmic potential theory with a circular symmetric external field. November 23, 1998. Date revised: February 8, 1999. Date accepted: March 2, 1999.     

On the Computation of Powers of Sparse Polynomials

This paper examines the time for computing integer powers of a class of polynomials in one or several variables. A member f of the class of “sparse” polynomials can be characterized by the number of nonzero terms, t, in the representation of f. Where, for power 4, the most obvious methods result in multiplications, a method is presented which requires fewer than multiplications. This algorithm requires, for t » n » 1 about tⁿ/n! + O(t^{n ? 1} /(2(n ? 2)!)) multiplications, which is proved to be a lower-bound on the computation.     

Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded.