George Lorentz and inequalities in approximation

Abstract

George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the Introduction. 0. Introduction 1. Bernstein-type inequalities for exponential sums. 2. Remez-type inequalities for exponential sums. 3. Lorentz degree of polynomials. 4. Markovand Bernstein-type inequalities for constrained polynomials. 5. Muntz-type theorems. 6. Remez-type inequalities and Newman’s product problem. 7. Multivariate approximation. 8. Newman’s inequality. 9. Littlewood polynomials. 10. Inequalities for generalized polynomials. 11. Markovand Bernstein-type inequalities for rational functions. 12. Nikolskii-type inequalities for shift-invariant function spaces. 13. Inverse Markovand Bernstein-type inequalities. 14. Ultraflat sequences of unimodular polynomials. 15. Zeros of polynomials with coefficient constraints. 1. Bernstein-type Inequalities for Exponential Sums The results in this section were, in large measure, motivated by the letter of Lorentz below. “Dear Tamas: Feb. 27, 1988 I know you are interested in Bernstein-type inequalities and I am also. In some non-linear cases one has ‖P ‖X ≤ Φ(n)‖P‖Y , where n is the dimension of the set of the P ’s, and the norms are taken in different Banach spaces X and Y . For instance, inequalities of Dolzhenko and Pekarskii for rational 2000 Mathematics Subject Classifications: Primary: 41A17 Typeset by AMS-TEX 1 functions are of this type. I have proved an inequality of this type for exponential functions ∑n 1 aje λjx or the extended exponential sums