Smooth solutions of quasianalytic or ultraholomorphic equations

Thilliez, Vincent

doi:10.1007/s00605-009-0108-0

Cited by 17 publications

(20 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, S. Malek [37] has studied some singularly perturbed small step size difference-differential nonlinear equations whose formal solutions with respect to the perturbation parameter can be decomposed as sums of two formal series, one with Gevrey order 1, the other of 1 + level, a phenomenon already observed for difference equations [12]. In a different context, V. Thilliez [56] has proven some stability results for algebraic equations whose coefficients belong to a general ultraholomorphic class defined by means of a so-called strongly regular sequence (comprising, but not limiting to, Gevrey classes), stating that the solutions will remain in the corresponding class. All these examples made it interesting for us to provide the tools for a general, common treatment of summability in ultraholomorphic classes in sectors, extending the powerful theory of k−summability.…”

Section: Introductionmentioning

confidence: 95%

Strongly regular sequences and proximate orders

Jiménez-Garrido

Sanz

2016

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence (M p ) p∈N0 , have been put forward by A. Lastra, S. Malek and the second author [27]. We study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders. In particular, we give a simple condition that allows us to associate a proximate order with a strongly regular sequence. Under this assumption, and through the characterization of strongly regular sequences in terms of so-called regular variation, we show that the growth index γ(M) defined by V.Thilliez [55] and the order of quasianalyticity ω(M) introduced by the second author [50] are the same.

show abstract

Section: Introductionmentioning

confidence: 95%

Strongly regular sequences and proximate orders

Jiménez-Garrido

Sanz

2016

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…In a similar way as in the proof of Theorem 2.22 (see [ 21 ]), it is easy to deduce that, given a bounded holomorphic function f in a sector that admits a continuous extension to the boundary the fact that and f is -flat amounts to the existence of constants such that ( 3.5 ) holds.…”

Section: Watson’s Lemmasmentioning

confidence: 71%

“…([ 21 ], Proposition 4) Given the following are equivalent: and f is flat. For every bounded proper subsector T of G , there exist with …”

Section: Preliminariesmentioning

confidence: 99%

A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

2019

View full text Add to dashboard Cite

We prove that, for asymptotically bounded holomorphic functions in a sector in C, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén-Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindelöf and G. Valiron.AMS Classification: 30E15, 30C80, 26A12, 30H50.We warn the reader that there is no agreement about the terminology in this respect: while most authors adhere, as we will do, to the convention that the asymptotics in (1.1) is Gevrey of order 1/k, others (for example, Fruchard and Zhang or W. Balser in [1]) say this is of order k. Moreover, the notion of type is not standard, compare with the definition by M. Canalis-Durand [2] for whom the type in case one has (1.1) is (1/R + δ) k . It should also be mentioned that the factor Γ(1 + p/k) could be changed into (p!) 1/k without changing the asymptotics, but this would affect the base of the geometric factor providing the type (by Stirling's formula, see [2, pp. 3-4]) in any case. As it will be explained below, our interest in the type will be limited, and so we will choose a simple approach in this respect, see Definitions 2.2 and 2.11.The proof of this result is based on the classical Phragmén-Lindelöf theorem and on the so-called Borel-Ritt-Gevrey theorem. This last statement provides the surjectivity, as long as the opening of the sector is at most π/k, of the Borel map sending a function with Gevrey asymptotic expansion of order 1/k in a sector to its series of asymptotic expansion, whose coefficients will necessarily satisfy Gevrey-like estimates. Also, the injectivity of the Borel map in sectors of opening greater than π/k (known as Watson's lemma) plays an important role when specifying conditions that guarantee the uniqueness of a function with a prescribed Gevrey asymptotic expansion of order 1/k in a direction.The main aim of this paper is to extend these results for other types of asymptotic expansions available in the literature. This possibility was already mentioned in [11], where A. Lastra, J. Mozo-Fernández and the second author of this paper generalized the results of Fruchard and Zhang for holomorphic functions of several variables in a polysector (cartesian product of sectors) admitting strong asymptotic expansion in the sense of H. Majima [12,13], considering also the Gevrey case as introduced by Y. Haraoka in [5].The asymptotics we will consider are those associated to the consideration of general ultraholomorphic classes of functions defined by constraining the growth of the sequence of their successive derivatives in a sector in terms of a sequence M = (M p ) p∈N0 of positive numbers (N 0 ...

show abstract

“…It follows that P •c is a polynomial, whose coefficients are C M germs at 0 ∈ R and which admits a C ∞ parameterization pr i (g.c) (for 1 ≤ i ≤ m and g ∈ G) of its roots. By [17,Theorem 2], each pr i (g.c) is actually C M , hence,c is C M .…”

Section: Differentiable Liftsmentioning

confidence: 98%

A generalization of Puiseux’s theorem and lifting curves over invariants

2011

View full text Add to dashboard Cite

Let ρ : G → GL(V ) be a rational representation of a reductive linear algebraic group G defined over C on a finite dimensional complex vector space V . We show that, for any generic smooth (resp. C M ) curve c : R → V //G in the categorical quotient V //G (viewed as affine variety in some C n ) and for any t 0 ∈ R, there exists a positive integer N such that t → c(t 0 ± (t − t 0 ) N ) allows a smooth (resp. C M ) lift to the representation space near t 0 . (C M denotes the Denjoy-Carleman class associated with M = (M k ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V //G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ∞ curve which represents a lift of a germ of a quasianalytic C M curve in V //G is actually C M . There are applications to polar representations.

show abstract

Smooth solutions of quasianalytic or ultraholomorphic equations

Cited by 17 publications

References 12 publications

Strongly regular sequences and proximate orders

Strongly regular sequences and proximate orders

A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

A generalization of Puiseux’s theorem and lifting curves over invariants

Contact Info

Product

Resources

About