Abstract
The standard well-known Remez inequality gives an upper estimate of the values of polynomials on [- 1 , 1] if they are bounded by 1 on a subset of [- 1 , 1] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.
| Original language | English |
|---|---|
| Pages (from-to) | 529-554 |
| Number of pages | 26 |
| Journal | Constructive Approximation |
| Volume | 54 |
| Issue number | 3 |
| DOIs | |
| State | Published - Dec 2021 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2021, The Author(s).
Funding
P. Y. was supported by the Austrian Science Fund FWF, project no: P32855. B. E. was supported by the Austrian Science Fund FWF, project no: J4138 and project no: P33885
| Funders | Funder number |
|---|---|
| Austrian Science Fund | P32855, J4138, P33885 |
Keywords
- Chebyshev and Akhiezer polynomials
- Comb domains
- Remez inequality
- Totik–Widom bounds