On covering numbers of sublevel sets of analytic functions

Brudnyi, Alexander

doi:10.1016/j.jat.2009.03.005

Cited by 10 publications

(16 citation statements)

References 17 publications

(24 reference statements)

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“…For similar results for spaces V of real analytic functions, see [11]. Still, subsets Z ⊂ Q n 1 with ω d,n (Z) > 0 are (in a certain discrete sense) 'massive in dimension n − 1'.…”

Section: 3mentioning

confidence: 81%

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mentioning

confidence: 99%

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Norming Sets and Related Remez-Type Inequalities

Brudnyi¹,

Yomdin²

2015

J. Aust. Math. Soc.

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The classical Remez inequality ([33]) bounds the maximum of the absolute value of a real polynomial P of degree d on [−1, 1] through the maximum of its absolute value on any subset Z ⊂ [−1, 1] of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, e.g., [14,39,8]-norming sets are exactly those not contained in any algebraic hypersurface of degree d in R n , there are many apparently unrelated reasons for Z ⊂ [−1, 1] n to have this property.)In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces V of continuous functions on [−1, 1] n , remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for Z to be V -norming, partly known, partly new, restricting ourselves to the simplest non-trivial examples. Next, we extend the Turan-Nazarov inequality for exponential polynomials to several variables, and on this base prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants N V (Z) in the Remez-type inequalities for V , as the function of the set Z, showing that it is Lipschitz in the Hausdorff metric.

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“…For similar results for spaces V of real analytic functions, see [11]. Still, subsets Z ⊂ Q n 1 with ω d,n (Z) > 0 are (in a certain discrete sense) 'massive in dimension n − 1'.…”

Section: 3mentioning

confidence: 81%

“…

…”

mentioning

confidence: 99%

Norming Sets and Related Remez-Type Inequalities

Brudnyi¹,

Yomdin²

2015

J. Aust. Math. Soc.

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“…In [7] estimates have been obtained for covering numbers of sub-level sets of families of analytic functions depending analytically on a parameter. Using these estimates strong Remez type inequalities have been proved for the restrictions of analytic functions to certain fractal sets.…”

Section: )mentioning

confidence: 99%

Remez-type inequality for discrete sets

Yomdin

2011

Isr. J. Math.

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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 cube Q n 1 ⊂ R n can be bounded through the maximum of its absolute value on any subset Z ⊂ Q n 1 of positive n-measure. The main result of this paper is that the n-measure in the Remez inequality can be replaced by a certain geometric invariant ω d (Z) which can be effectively estimated in terms of the metric entropy of Z and which may be nonzero for discrete and even finite sets Z.

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“…Some other examples and a more detailed discussion can be found in [1,2,17]. In particular, we will use a result of [17] (Theorem 3.2 below),…”

Section: "Remez-type" Inequalitiesmentioning

confidence: 99%

Smooth rigidity and Remez-type inequalities

Yomdin¹

2020

Preprint

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This follows from the polynomial interpolation of f at its zeroes, with Lagrange's remainder formula. This is one of the simplest examples of what we call "smooth rigidity": certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f . In dimensions greater than one, the powerful one-dimension tools, like Lagrange's remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables ([15]).In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of [15], including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a truly multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that a "smooth rigidity" of a zeroes set Z is approximately the "inverse Remez constant" of Z.

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On covering numbers of sublevel sets of analytic functions

Abstract: In this paper we estimate covering numbers of sublevel sets of families of analytic functions depending analytically on a parameter. We use these estimates to study the local behavior of these families restricted to certain fractal subsets of R N .

Cited by 10 publications

References 17 publications

Norming Sets and Related Remez-Type Inequalities

Norming Sets and Related Remez-Type Inequalities

Remez-type inequality for discrete sets

Smooth rigidity and Remez-type inequalities

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