Lower bounds for quasianalytic functions, II The Bernstein quasianalytic functions.

Borichev, Alexander; Nazarov, Fëdor; Sodin, Mikhail

doi:10.7146/math.scand.a-14448

Cited by 3 publications

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“…In [6], [28]- [30] analytic and quasi-analytic functions have been studied from a similar point of view.…”

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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“…In [6], [28]- [30] analytic and quasi-analytic functions have been studied from a similar point of view.…”

Section: )mentioning

confidence: 99%

Remez-type inequality for discrete sets

Yomdin

2011

Isr. J. Math.

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“…The theory for Carleman classes was developed in order to understand for which classes of functions the (formal) Taylor series at a point uniquely determines the function. Denjoy [12] provided an answer under regularity assumptions on the weight sequence, and Carleman [8][9][10] proved what has since become known as the Denjoy- [2], together with some further developments in the theory of quasianalytic functions, we refer to the work of Borichev, Nazarov, Sodin, and Volberg [6,17]. The present work is devoted to the study of the analogous classes defined in terms of L p -norms, mainly for 0 < p < 1.…”

Section: The L P -Carleman Spaces and Classesmentioning

confidence: 99%

“…As this result was contained in Bang's thesis [2], written in Danish, the result appears not to be known to a wider audience. For an account of several of the interesting results in [2], as well as of some further developments in the theory of quasianalytic functions, we refer to the work of Borichev, Nazarov, Sodin, and Volberg [17,6]. The present work is devoted to the study of the analogous classes defined in terms of L pnorms, mainly for 0 < p < 1.…”

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confidence: 99%

A critical topology for $$L^p$$ L p -Carleman classes with $$0<p<1$$ 0 < p < 1

Hedenmalm

Wennman

2018

Math. Ann.

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In this paper, we explore a sharp phase transition phenomenon which occurs for L p -Carleman classes with exponents 0 < p < 1. These classes are defined as for the standard Carleman classes, only the L ∞ -bounds are replaced by corresponding L p -bounds. We study the quasinormsfor some weight sequence M = {M n } n of positive real numbers, and consider as the corresponding L p -Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the L p -Carleman class. A particular degenerate instance is when M n = 1 for 0 ≤ n ≤ k and M n = +∞ for n > k. This would give the L p -Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these L p -Sobolev spaces are highly degenerate for 0 < p < 1. Indeed, the canonical map W k, p → L p fails to be injective, and there is even an isomorphismCommunicated by Loukas Grafakos. , . . . , f (k) ) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they "disconnect"). Here, we analyze this degeneracy for the more general L p -Carleman classes defined by a weight sequence M. If M has some regularity properties, and if the given collection of test functions is what we call ( p, θ)-tame, then we find that there is a sharp boundary, defined in terms of the weight M: on the one side, we get Douady-Peetre's phenomenon of "disconnexion", while on the other, the completion of the test functions consists of C ∞ -smooth functions and the canonical map f → ( f, f , f , . . .) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the L p setting, with 0 < p < 1.

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Identity principles for Bernstein quasianalytic functions

Pleśniak

2013

commath

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Lower bounds for quasianalytic functions, II The Bernstein quasianalytic functions.

Cited by 3 publications

References 6 publications

Remez-type inequality for discrete sets

Remez-type inequality for discrete sets

A critical topology for $$L^p$$ L p -Carleman classes with $$0<p<1$$ 0 < p < 1

Identity principles for Bernstein quasianalytic functions

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