Equivalence of stability properties for ultradifferentiable function classes

Rainer, Armin; Schindl, Gerhard

doi:10.1007/s13398-014-0216-0

Cited by 34 publications

(48 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…More precisely, if E [M] is stable under derivation, then stability under composition is in turn equivalent to being holomorphically closed, being inverse closed, (M ♭(c) k ) 1 k being almost increasing, and M ♭(c) having the (FdB)-property. For further equivalent stability properties we refer to [33]. Inverse closedness has been studied intensively, e.g.…”

Section: Introductionmentioning

confidence: 99%

Composition in ultradifferentiable classes

2014

Self Cite

View full text Add to dashboard Cite

Abstract. We characterize stability under composition of ultradifferentiable classes defined by weight sequences M , by weight functions ω, and, more generally, by weight matrices M, and investigate continuity of composition (g, f ) → f • g. In addition, we represent the Beurling space E (ω) and the Roumieu space E {ω} as intersection and union of spaces E (M ) and E {M } for associated weight sequences, respectively.

show abstract

Section: Introductionmentioning

confidence: 99%

Composition in ultradifferentiable classes

2014

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, it is shown in Lemma 2.2 of Ref. [57] (see also [58,39]) that log-convexity implies (FdB)-stability with an (FdB)-stability constant C FdB = max{1, M 1 }. Since the class C{M} is invariant under multiplication by a constant, we can normalize the sequence M = {M k } k≥0 by an arbitrary positive constant.…”

Section: Remark 1 Using the Leibniz Differentiation Rules Log-superlmentioning

confidence: 99%

“…In other words log-convexity implies (FdB)-stability (4). Let us note that usually in the literature [8,60,50,37,66,39,40,41,57,58], ultradifferentiable classes assume log-convexity in order to achieve stability under composition. We refer the reader to [57,58] and references therein for a discussion of the role of log-convexity in achieving stability under composition.…”

Section: Remark 1 Using the Leibniz Differentiation Rules Log-superlmentioning

confidence: 99%

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Besse¹,

Frisch²

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries.For a good understanding, it is crucial to carry out, besides mathematical studies, highaccuracy and well-resolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved first, by using Cauchy's Lagrangian formulation of the Euler equations and second, by taking advantages of analyticity results of the Lagrangian trajectories for flows whose initial vorticity is Hölder-continuous. The latter has been known for about twenty years (Serfati, J. Math. Pures Appl. 74: 95-104, 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor series of the Lagrangian map, has been achieved only recently (Frisch and Zheligovsky, Commun. Math. Phys. 326: 499-505, 2014; Podvigina et al., J. Comput. Phys. 306: 320-342, 2016 and references therein).Here we extend this methodology to incompressible Euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Non-constructive regularity results for these cases have already been obtained by Glass et al. (Ann. Scient.Éc. Norm. Sup. 4 e série 45: 1-51, 2012). Using the invariance of the boundary under the Lagrangian flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy-Lagrangian method.

show abstract

“…If iii) in a) is satisfied, it follows from the hypotheses on ω and [RaSc2, Theorem 4] that 1 w ∈ E (ω) (X) and that for any m ∈ N the components of the smooth function (ψ m ) −1 : X → X belong to E (ω) (X). Therefore, applying again [RaSc2,Theorem 4] it follows that for every open subset Y of ψ m (X) and any f ∈ E (ω) ((ψ m ) −1 (Y )) the functionf…”

Section: Spaces Of Ultradifferentiable Functionsmentioning

confidence: 99%

Dynamics of weighted composition operators on function spaces defined by local properties

Kalmes

2019

Studia Math.

View full text Add to dashboard Cite

We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of R d , as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of R d .

show abstract

Equivalence of stability properties for ultradifferentiable function classes

Abstract: Abstract. We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.

Cited by 34 publications

References 18 publications

Composition in ultradifferentiable classes

Composition in ultradifferentiable classes

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Dynamics of weighted composition operators on function spaces defined by local properties

Contact Info

Product

Resources

About