Yuri Brudnyi

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C1·ω (Rn)to an arbitrary closed subset ofRn.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.

Methods of geometric analysis in extension and trace problems

Part 3. Lipschitz Extensions from Subsets of Metric Spaces.- Chapter 6. Extensions of Lipschitz Maps.- Chapter 7. Simultaneous Lipschitz Extensions.- Chapter 8. Linearity and Nonlinearity.- Part 4. Smooth Extension and Trace Problems for Functions on Subsets of Rn.- Chapter 9. Traces to Closed Subsets: Criteria, Applications.- Chapter 10. Whitney Problems.- Bibliography.- Index.

Whitney’s extension problem for multivariate ^{1,}-functions

We prove that the trace of the space C1,ω(Rn) to an arbitrary closed subset X ⊂ Rn is characterized by the following “finiteness” property. A function f : X → R belongs to the trace space if and only if the restriction f |Y to an arbitrary subset Y ⊂ X consisting of at most 3·2n−1 can be extended to a function fY ∈ C1,ω(Rn) such that sup{‖fY ‖C1,ω : Y ⊂ X, cardY ≤ 3 · 2n−1} <∞. The constant 3 · 2n−1 is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right. 1. Main results The results of the paper are concerned with the following problem having its origin in two classical papers of Hassler Whitney [W1], [W2] which appeared in 1934. Let C(R) be the space of k-times continuously differentiable functions f satisfying

Metric Spaces with Linear Extensions Preserving Lipschitz Condition

We study a new bi-Lipschitz invariant λ(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are expanded by a factor controlled by λ(M). We prove that λ(M) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which λ(M) = ∞.

Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach

The paper presents a survey of the theory of local polynomial approximation and its applications to the study of the classical spaces of smooth functions. The study includes such topics as embeddings and extensions, pointwise differentiability and Luzin type theorems, nonlinear approximation by piecewise polynomials and splines, and the real interpolation.

LINEAR AND NONLINEAR EXTENSIONS OF LIPSCHITZ FUNCTIONS FROM SUBSETS OF METRIC SPACES

A relationship is established between the linear and nonlinear extension constants for Lipschitz functions defined on subsets of metric spaces. Proofs of several results announced in our earlier paper are presented.

Timan's type result on approximation by algebraic polynomials

The paper presents two approximation results of the Jackson-Timan type, which were included in my lecture at the All-Union Conference in Approximation Theory (Dniepropetrovsk, June 26–28, 1990).y1 The conference was dedicated to the seventieth birthday of Professor Alexander Timan (26.6.1920–13.8.1988). Unfortunately, proceedings of the conference never appeared because of the well-known events in the former Soviet Union, and only the abstracts were published (see [B3]). Here we present the approximation method which has been used in the proof of these results. In fact, it can be applied to many other approximation problems of such a kind.

A universal Lipschitz extension property of Gromov hyperbolic spaces

A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.

Extensions of Lipschitz Maps

The main part of this chapter deals with the theory of Lang and Schlichenmaier [LSchl-2005] devoted to Lipschitz extensions of maps acting between metric spaces. The theory gives a uni_ed approach to most previously established results of this kind which now become consequences of the main extension theorem established in [LSchl-2005]. All of these results are not sharp in the sense that the extensions do not preserve Lipschitz constants; in particular, the classical Kirszbraun and Valentine results cannot be proved in this way. Moreover, the estimates of Lipschitz extension constants in the main theorem contain unspeci_ed quantities whose dependence on the basic parameters are either implicit or too rough. More precise estimates require, for every special case, new approaches and methods. Sparse results of this kind are discussed in the _nal part of the chapter.

Interpolation of several multiparametric approximation spaces

We prove a general interpolation theorem for linear operators acting simultaneously in several approximation spaces which are defined by multiparametric approximation families. As a consequence, we obtain interpolation results for finite families of Besov spaces of various types including those determined by a given set of mixed differences.