Ultradifferentiable extension theorems: A survey

Rainer, Armin

doi:10.1016/j.exmath.2021.12.001

Cited by 7 publications

(10 citation statements)

References 66 publications

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“…The standard definition of ultra-differentiable functions involves Denjoy-Carleman sequences, that is, real sequences satisfying certain conditions which act as bounds on the successive derivatives of a given function. However, the above definition, introduced by Braun-Meise-Taylor ( [6]), can be linked to Denjoy-Carleman classes (see [12], Theorem 11.6). Since Fourier series appear naturally in the problem considered here, we chose to use Braun-Meise-Taylor classes as a starting point.…”

Section: Ultra-differentiabilitymentioning

confidence: 99%

Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

Chatal¹,

Chavaudret²

2022

Preprint

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Section: Ultra-differentiabilitymentioning

confidence: 99%

Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

Chatal¹,

Chavaudret²

2022

Preprint

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“…Here are some easy consequences of the definition: M j M k ≤ M j+k , (M k ) 1/k ≤ µ k , and (M k ) 1/k → ∞ if and only if µ k → ∞ (cf. [27,Lemma 2.3]).…”

Section: Ultradifferentiable Classes and Weightsmentioning

confidence: 99%

“…see e.g. [7], [35,Section 4], or [27,Theorem 11.17]. Then the class and the weight function ω are called quasianalytic, and non-quasianalytic otherwise.…”

Section: Weight Functionsmentioning

confidence: 99%

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The Borel map in the mixed Beurling setting

Nenning¹,

Rainer²,

Schindl³

2022

Preprint

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The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical Denjoy-Carleman and Braun-Meise-Taylor classes. More precisely, we characterize when the Borel image of one class covers the sequence space of another class in terms of the two weights that define the classes. We present two independent solutions to this problem, one by reduction to the Roumieu case and the other by dualization of the involved Fréchet spaces, a Phragmén-Lindelöf theorem, and Hörmander's solution of the ∂-problem.

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“…where N is given by ( 16) and µ |N , M |N are obtained from µ |∞ , M |∞ simply by setting all elements with index ≥ N + 1 equal to ∞. In particular, the bound on the number of zeros in (17) depends only on the derivatives up order N of f .…”

Section: 2mentioning

confidence: 99%

Quantitative tame properties of differentiable functions with controlled derivatives

Rainer¹

2022

Preprint

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We show that differentiable functions, defined on a convex body K ⊆ R d , whose derivatives do not exceed a suitable given sequence of real numbers share many properties with polynomials. The role of the degree of a polynomial is hereby replaced by an integer associated with the given sequence of reals, the diameter of K, and a real parameter linked to the C 0 -norm of the function. We give quantitative information on the size of the zero set, show that it admits a local parameterization by Sobolev functions, and prove an inequality of Remez-type. From the latter, we deduce several consequences, for instance, a bound on the volume of sublevel sets and a comparison of L pnorms reversing Hölder's inequality. The validity of many of the results only depends on the derivatives up to some finite order; the order can be specified in terms of the given data.

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Ultradifferentiable extension theorems: A survey

Cited by 7 publications

References 66 publications

Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

The Borel map in the mixed Beurling setting

Quantitative tame properties of differentiable functions with controlled derivatives

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