About Widths of Wiener Space in theLq-Norm

We calculate the average Kolmogorov and linearn-widths of the Wiener space in theL_q-norm. For the case 1 ≤q< ∞, then-widthsd_ndecrease asymptotically asn^-1/2.     

Closed embeddings of Hilbert spaces

Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L₂(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators.     

The Gelfand and Bernsteinn-widths of some classes of analytic functions

This paper contains generalizations of a well-known theorem of Ismagilov on Kolmogorovn-widths in a Hilbert space for Bernstein and Gelfandn-widths. Some examples are considered. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 112–129.     

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c₀ of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X−X has finite Assouad dimension.     

Optimal Subspaces for n -Widths of Bounded Subsets in Hilbert Spaces

In this article, we consider the problem of proving the optimality of several approximation spaces by means of n-widths. Specifically, they are optimal subspaces for approximating bounded subsets in some Hilbert spaces with mesh-dependent norms. We prove that finite element spaces and newly developed generalized L-spline spaces are optimal subspaces for n-widths.     

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.     

Equivariant Embeddings of the n-Dimensional Space into a Hilbert Space

The present paper is devoted to the study of equivariant embeddings of the n-dimensional space into a Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle for this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of the additive group of real numbers and construct an embedding of the n-dimensional space with metric d(x,y)=|x-y| into the Hilbert space. Bibliography: 5 titles.     

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L₂‐small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of “tensor product”. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper 6 , where the regular behavior at infinity of marginal eigenvalues was assumed.     

On the widths of Sobolev classes

Optimal estimates of Kolmogorov’s n-widths, linear n-widths and Gelfand’s n-widths of the weighted Sobolev classes on the unit sphere S^d are established. Similar results are also established on the unit ball B^d and on the simplex T^d.     

Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α?1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ?_p, and the particular case of the hypercube.     

Best polynomial approximations in L 2 of classes of 2π-periodic functions and exact values of their widths

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L ₂ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L ₂.     

On Exact Values of the Kolmogorov Width of Compact Sets in Hilbert Space

A two-sided bound for the Kolmogorov width of compact sets in Hilbert space is established. The Kolmogorov width of a set of equidistant points in real Hilbert space and the 1-width of the continuous Wiener spiral are computed.     

Optimal Approximation on Sd

The asymptotic behavior of the n-widths of a wide range of sets of smooth functions on a d-dimensional sphere in L_q(S^d) is studied. Upper and lower bounds for the n-widths are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples.     

Approximation of eigenfunctions in kernel-based spaces

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the “native” Hilbert space \(\mathcal {H}\) in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L ₂-orthonormal and orthogonal in \(\mathcal {H}\) (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of \(\mathcal {H}\). These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.     

Uniform embeddability and exactness of free products

Let A and B be countable discrete groups and let Γ=A∗B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is Γ. We give two different proofs: the first directly constructs a uniform embedding of Γ from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is Γ.     

Sharp asymptotics of the Kolmogorov entropy for Gaussian measures

Let U denote the unit ball of the Cameron-Martin space of a Gaussian measure on a Hilbert space. The sharp asymptotics for the Kolmogorov (metric) entropy numbers of U is derived. The condition imposed is regular variation of the eigenvalues of the covariance operator. A consequence is a precise link including constants to the functional quantization problem.     

A concrete estimate for the weak Poincar�� inequality on loop space

The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with a Hilbert space structure and the Ornstein?CUhlenbeck operator d*d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of the underlying manifold. The order of the rate function is s ^??? for any ?? >?0.     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.