Let $K$ be an absolutely convex infinite-dimensional compact in a Banach space $\mathcal{X}$. The set of all bounded linear operators $T$ on $\mathcal{X}$ satisfying $TK\supset K$ is denoted by $G(K)$. Our starting point is the study of the closure $WG(K)$ of $G(K)$ in the weak operator topology. We prove that $WG(K)$ contains the algebra of all operators leaving $\overline{\lin(K)}$ invariant. More precise results are obtained in terms of the Kolmogorov $n$-widths of the compact $K$. The obtained results are used in the study of operator ranges and operator equations.
In this paper we obtain asymptotic estimates of Kolmogorov and linear widths of the weighted Besov classes with singularity at the origin. In addition, estimates of Kolmogorov and linear widths of finite-dimensional balls in a mixed norm are obtained.
Order estimates for the Kolmogorov widths of an intersection of two finite-dimensional balls in a mixed norm under some conditions on the parameters are obtained.
In this paper we obtain order estimates for the Kolmogorov widths of the set $B_{p_0}^m\cap νB_{p_1}^m$ in $l_q^m$; here $0\le n\le m/2$, $p_0>p_1$, $q<\infty$.
In this article, we obtain the order estimates for the Kolmogorov widths of sets with conditions on the norm in the weighted Sobolev space $W^r_{p_1}$ and in the weighted space $L_{p_0}$.
In this paper order estimates for the Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain are obtained. The classes are defined by conditions on the highest-order derivatives and the derivative of order zero.
We consider the family of integral operators $(K_αf)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_αf)(x)=\int_0^1(1-xy)^{α-1}\,f(y)\,\operatorname{d}\!y, \qquad 0<α<1.$$ The main objective is to find upper bounds for the Kolmogorov widths, where the $n$th Kolmogorov width is the infimum of the deviation of $(K_αf)$ from an $n$-dimensional subspaces of $L^p[0,1]$ (with the infimum taken over all $n$-dimensional subspaces), and is therefore a measure of how well $K_α$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $\exp(-κ\sqrt{n})$ f…
Direct estimates between linear or nonlinear Kolmogorov widths and entropy numbers are presented. These estimates are derived using the recently introduced Lipschitz widths. Applications for m-term approximation are obtained.
Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. Thi…
Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.
In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,θ}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max(\frac12,1-\frac{1}θ)}n,\quad 2<q<\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\ell_q^N$.
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficient…
Kolmogorov widths of an intersection of Besov classes with dominating mixed smoothness in a Besov space
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University（莫斯科国立大学力学与数学系）；Moscow Center for Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）
We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor $L^2$-approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the $L^2$-topology.
Let $P(D)$ be the differential operator induced by a polynomial $P$, and let ${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that $\|P(D)(f)\|_2\leq 1$. The problem of computing the asymptotic order of the Kolmogorov $n$-width $d_n({U^{[P]}_2},L_2)$ in the general case when ${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when $P(D)$ is non-degenerate.
In this paper we completely solve the problem of finding the (upper) approximation order with respect to the Kolmogorov, Gel'fand, and linear widths for the embedding of the Sobolev spaces $W^{α,p}$ and $W_{0}^{α,p}$ in the euclidian measure spaces $L_ν^{q}$ for an arbitrary Borel probability measure $ν$ with support contained in the open $m$-dimensional unit cube and for all possible choices of $1\leq p,q\leq\infty$. We will determine the exact values for the various upper approximation orders in terms of the $L^{q}$-spectrum of $ν$ only and finally give sufficient conditions imposed on the r…
Let $X$ be a Banach space and $\mathcal{K}$ be a compact subset in $X$. We consider a greedy algorithm for finding an $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $\mathcal{K}$. We are interested in how well the space $V_n$ approximates the elements of $\mathcal{K}$. For this purpose we compare the performance of greedy algorithm measured by $σ_n(\mathcal{K})_X:=\text{dist}(\mathcal{K},V_n)_X$ with the Kolmogorov width $d_n(\mathcal{K})_X$ which is the best possible error one can achieve when approximating $\mathcal{K}$ by $n$-dimensional subspaces. …
Order estimates for Kolmogorov, Gelfand and linear widths of a weighted Sobolev class on a domain with a peak in a weighted Lebesgue space are obtained for some special weights.
In this paper, estimates for Kolmogorov, Gelfand and linear widths of function classes on sets with a tree-like structure are obtained. As examples we consider weighted Sobolev classes on a John domain, as well as some function classes on a metric and combinatorial tree.
Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $ε$-entropy of compact sets in the 1950s. In …
In this work we investigate $n$-widths of multiplier operators $Λ_*$ and $Λ$, defined for functions on the complex sphere $Ω_d$ of $\mathbb{C}^d$, associated with sequences of multipliers of the type $\{λ_{m,n}^*\}_{m,n\in \mathbb{N}}$, $λ_{m,n}^*=λ(m+n)$ and $\{λ_{m,n}\}_{m,n\in \mathbb{N}}$, $λ_{m,n}=λ(\max\{m,n\})$, respectively, for a bounded function $λ$ defined on $[0,\infty)$. If the operators $Λ_{*}$ and $Λ$ are bounded from $L^p(Ω_d)$ into $L^q(Ω_d)$, $1\leq p,q\leq\infty$, and $U_p$ is the closed unit ball of $L^p(Ω_d)$, we study lower and upper estimates for the $n$-widths of Kolmog…
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.
In this paper we prove that Kolmogorov widths of weighted Sobolev classes with restrictions $f(a)=\dots=f^{(k-1)}(a)=f^{(k)}(b)=\dots=f^{(r-1)}(b)=0$ in a weighted Lebesgue space and spectral numbers of some non-linear differential equation coincide. Here we suppose that the embedding is compact.
We prove a characterization for the Peetre type $K$-functional on $\mathbb{M}$, a compact two-point homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well known results on the spherical setting. The characterization is employed to show that an abstract Hölder condition or finite order of differentiability condition imposed on kernels generating certain operators implies a sharp decay rates for their eigenvalues sequences. The latest is employed to obtain estimates for the Kolmogorov $n$-width of unit balls in Rep…
