2010
DOI: 10.1016/j.jat.2009.03.005
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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 .

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Cited by 10 publications
(16 citation 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%
See 1 more Smart Citation
“…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%
“…
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.
…”
mentioning
confidence: 99%
“…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%
“…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%