Whitney’s jets for Sobolev functions

We present two fundamental facts from the jet theory for Sobolev spaces W ^{m, p} . One of these facts is that the formal differentiation of the k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of the Euclidean spaces ℝⁿ. The second result describes the Sobolev imbedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 345–358, March, 2007.     

On the extent of the (non) quasi-analytic classes

The authors thank the referee for giving them a reference for Lemma 2.1.     

Absolutely continuous measures on non quasi-analytic curves with independent powers

We prove that every non quasi-analytic Carleman class contains functions whose graph supports measures that are absolutely continuous with respect to arc length measure and yet they have independent convolution powers in the measure algebra . The proof relies on conditions which ensure that the canonical map between two Cantor sets can be extended to a function in an arbitrary prescribed non quasi-analytic Carleman class.

     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

Almost periodic ultradistributions of Beurling and of Roumieu type

We characterize the almost periodic ultradistributions of Beurling and of Roumieu type in terms of classical Bohr almost periodicity. Then we study the Fourier series associated with such an ultradistribution.

     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.     

The structure of spaces of quasianalytic functions of Roumieu type

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ_{w}(Ω), on an open convex set , satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the spaces are derived. In particular, Fréchet quotients of ℰ_{w}(Ω) have property (), while dual Fréchet quotients have property () of Vogt. The work of P. Domański was supported by Committee of Scientific Research (KBN), Poland, grant P03A 022 25.     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Analytic methods for treating unsteady axisymmetric jets

Summary The steady axisymmetric jet and Hill's spherical vortex are two rare examples of closed form solutions of the Navier-Stokes equation. The work of Proudman and Pearson showed an infinite series based on Legendre polynomials which leads to a simplification of the steady Navier-Stokes equation. These earlier discoveries are combined by reformulating the axisymmetric jet solution into an infinite series based on Legendre polynomials. This new solution form is then used as a guide in developing an unsteady ansatz or template to which the steady solution will satisfy in the steady limit. This ansatz when inserted into the unsteady Navier-Stokes equation reduces the equation into a single linear ordinary differential equation. This new differential equation not only describes the axisymmetric jet of Landau and Squire but also Hill's spherical vortex and an infinite number of other jets. Through use of this new ordinary differential equation, the unsteady axisymmetric jet is solved to second order. In addition, a complete formulation of Hill's spherical vortex is developed to first order.

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Zusammenfassung Der stationäre achsensymmetrische Freistrahl und Hill's spherical vortex sind zwei der seltenen Beispiele für geschlossen angebbare Lösungen der Navier-Stokes-Gleichungen. Proudman und Pearson zeigten, daß eine auf Legendre-Polyonomen aufbauende Reihenentwicklung zu einer Vereinfachung der zeitunabhängigen Navier-Stokes-Gleichungen führt. Diese früheren Arbeiten werden hier verknüpft, in dem die Lösung für den achsensymmetrischen Freistrahl umgeformt wird in eine Reihendarstellung, die auf Legendre-Polynomen aufbaut. Diese Lösungsform liefert die Vorlage für die Entwicklung eines Lösungsansatzes im nichtstationären Fall, der im Grenzfall der stationären Lösung genügt. Mit diesem Ansatz reduzieren sich die Navier-Stokes-Gleichungen auf eine gewöhnliche lineare Differentialgleichung. Sie beschreibt nicht nur den stationären Freistrahl nach Landau und Squire, sondern auch Hill's spherical vortex und eine unbegrenzte Anzahl anderer Freistrahlen. Für den zeitabhängigen achsensymmetrischen Freistrahl werden Lösungen zweiter Ordnung dieser Differentialgleichung hergeleitet und für Hill's spherical vortex solche erster Ordnung.

     

A constructive proof of a Whitney extension theorem in one variable

We prove a new stability property of the classical Hermite interpolation scheme in one variable. This stability property refines the usual convergence property of this interpolation scheme. Consequently, we obtain a natural proof of a Whitney extension theorem in one variable.     

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤p≤∞. Here Q is a d-parallelepiped in R^d with sides parallel to the coordinate axes. We consider the error of best approximation of a function f by algebraic polynomials of fixed degree at most r_i−1 in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.     

Fourier transform and quasi-analytic classes of functions of bounded type on tubular domains

A condition for a function of bounded type to belong to the Hardy class H ¹ in terms of the Fourier transform of the boundary values of this function on R ⁿ is found. Applications of the obtained result to the theories of Hardy classes and of quasi-analytic classes of functions are given.