Tamás Erdélyi

Polynomials and Polynomial Inequalities

Chaptern 1 Introduction and Basic Properties.- 2 Some Special Polynomials.- 3 Chebyshev and Descartes Systems.- 4 Denseness Questions.- 5 Basic Inequalities.- 6 Inequalities in Muntz Spaces.- Inequalities for Rational Function Spaces.- Appendix A1 Algorithms and Computational Concerns.- Appendix A2 Orthogonality and Irrationality.- Appendix A3 An Interpolation Theorem.- Appendix A5 Inequalities for Polynomials with Constraints.- Notation.

Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials

The authors obtain upper bounds for Jacobi polynominals which are uniform in all the parameters involved and which contain explicit constants. This is done by a combination of some results on generalized Christoffel functions and some estimates of Jacobi polynomials in terms of Christoffel functions.

Sharp extensions of Bernstein's inequality to rational spaces

Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K = r (mod 2 π), on [-1, 1 ] and on ℝ. The key result is the establishment of the inequality for every rational function f = p n / q n , where p n is a polynomial of degree at most n with complex coefficients and with | a j | ≠ 1 for each j and for every z o ∈ δ D , where δ D ,= { z ∈ ℂ: | z | = l}. The above inequality is sharp at every z 0 ∈δ D .

Littlewood-Type Problems on [0,1]

We consider the problem of minimizing the uniform norm on

Remez-type inequalities and their applications

[0, 1]

The integer Chebyshev problem

over non-zero polynomials

Extremal Properties of Polynomials

p

Remez- and Nikolskii-type inequalities for logarithmic potentials

of the form

Inequalities for generalized nonnegative polynomials

p(x) = \sum_{j=0}^n a_jx^j \quad\text{with } |a_j| \le 1,\, a_j \in {\Bbb C},