Remez-Type Inequality for Smooth Functions

Abstract

The classical Remez inequality bounds the maximum of the absolute value of a polynomial P (x) of degree d on [−1, 1] through the maximum of its absolute value on any subset Z of positive measure in [−1, 1]. Similarly, in several variables the maximum of the absolute value of a polynomial P (x) of degree d on the unit ball B n ⊂ R n can be bounded through the maximum of its absolute value on any subset Z ⊂ Q n 1 of positive n-measure m n (Z). In [12] a stronger version of Remez inequality was obtained: the Lebesgue n-measure m n was replaced by a certain geometric quantity ω n,d (Z) satisfying ω n,d (Z) ≥ m n (Z) for any measurable Z. The quantity ω n,d (Z) can be effectively estimated in terms of the metric entropy of Z and it may be nonzero for discrete and even finite sets Z.In the present paper we extend Remez inequality to functions of finite smoothness. This is done by combining the result of [12] with the Taylor polynomial approximation of smooth functions. As a consequence we obtain explicit lower bounds in some examples in the Whitney problem of a C k -smooth extrapolation from a given set Z, in terms of the geometry of Z.----------------

Norming Sets and Related Remez-Type Inequalities

“…Its solution would clarify the interconnection of topological and analytical properties of polynomials. It would also clarify some 'rigidity' properties of smooth functions appearing in the framework of the approach developed in [41] (see also references therein). This approach transfers to several variables the classical Rolle lemma and some of its important consequences.…”

“…It is well known that inequalities of the form (1.2) may be true also for some sets Z of Lebesgue measure zero and even for certain finite sets Z; see, for example, [5,11,12,16,19,22,24,33,[40][41][42].…”

Remez-Type and Nikol’skii-Type Inequalities: General Relations and the Hyperbolic Cross Polynomials

“…are sometimes called the Turán inequality after Paul Turán [28] who studied related inequalities for algebraic complex-valued polynomials. In [17] Here, A > 0 is an absolute constant, independent of n. Many different applications of Remez type inequalities include extension theorems (see, e.g., [3,30]) and polynomial inequalities (see, e.g., [11,6,15]). Moreover, Remez inequalities were used to obtain the uncertainty principle relations of the type…”

“…Many different applications of Remez type inequalities include extension theorems (see, e.g., [3,30]) and polynomial inequalities (see, e.g., [11,6,15]). Moreover, Remez inequalities were used to obtain the uncertainty principle relations of the type…”