ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26, 41, 42 Babenko, Vladislav F., Dnepropetrovsk State University, Ukraine Inequalities of Landau–Kolmogorov–Nagy Type We shall discuss connections of exact inequalities of Landau–Kolmogorov– Nagy type with various problems of Analysis, present new inequalities for univariate and multivariate functions and describe some of their applications. As an example we present here the inequality for having zeros univariate periodic functions x such that x^{(r−1)} is locally absolutely continuous and x^{(r)} ∈ L_{∞}:

∥x∥_{q} ≤ sup_{0≤c≤∥φ_{r}∥_{∞}} \begin{matrix}∥φ_{r} + c∥ & q \\ ∥φ_{r} + c∥ & p^{\frac{r+1/q}{r+1/p}} \end{matrix} ∥x∥p^{\frac{r+1/q}{r+1/p}} ∥x^{(r)}∥∞^{\frac{1/p−1/q}{r+1/p}} , 0 < p < q ≤ ∞,

(1)

(Here, φ_{r} is the r^{th} periodic primitive with zero mean value over a period for φ_{0}(t) = sgn sin t.) The following result illustrates applications of (1):

sup_{n∈N} sup_{{Tn∈Tn \atop Tn≠0}} \frac{∥T_{n}∥_{q}}{n^{1/p−1/q} ∥T_{n}∥_{p}} = sup_{c∈[0,1]} \begin{matrix}∥cos(·) + c∥ & q \\ ∥cos(·) + c∥ & p \end{matrix} , 0 < p < q ≤ ∞, (2)

(where T_{n} is the set of all real-valued trigonometric polynomials of order n.) Relations (1) and (2) are joint work with V. A. Kofanov and S. A. Pichugov.