ON AN EXTREMAL PROBLEM FOR POLYNOMIALS IN<i>n</i>VARIABLES

Let F be a class of functions with the uniqueness property: if f ∈ F vanishes on a set E of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. the estimate from below for the norm of the restriction operator f → f | E or, equivalently, a lower bound for |f | outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex-and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions.In the first part, we consider quasianalytically smooth functions. This part relies upon Bang's approach and includes the proofs of relevant results of Bang. In the second part, which is to be published separately, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence.The proofs are based on the elementary calculus technique.

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“…The second one is a rough version of the classical Remez inequality [19] (cf. [8] and [4]). Different proofs of (1.1) are known.…”

Section: Motivation and The Resultsmentioning

confidence: 99%

Lower bounds for quasianalytic functions, I. How to control smooth functions

Nazarov¹,

Sodin²,

Volberg³

2004

MATH. SCAND.

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“…It is worth pointing out that in recent years an essential progress was done in studying Bernstein-and Markovtype inequalities for algebraic and analytic functions. Such inequalities proved to be important in different areas of modern analysis, see, e.g., [B,BG,BMLT,Br1,Br2,FN1,FN2,FN3,G,KY,RY,S]. We hope that the inequalities established in this paper would also have various applications in the fields that make use of the classical polynomial inequalities (approximation theory, trace and embedding theorems, signal processing, PDE, etc).…”

mentioning

confidence: 89%

On Local Behavior of Analytic Functions

Brudnyi

1999

Journal of Functional Analysis

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“…And it is the case -a version of Remez inequality for convex sets is due to Brudnyi and Ganzburg (cf. [28], see also Corollary 4.2). The situation is completely different if we consider nonconvex sets.…”

Section: Introductionmentioning

confidence: 99%