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.     

Widths of classes of finitely smooth functions in Sobolev spaces

We describe the weak asymptotics of the behavior of the Kolmogorov, Gelfand, linear, Aleksandrov, and entropy widths of the unit ball of the space W _p ^l H^w (I ^d) in the space W _q ^m (I ^d).Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 535–539.Original Russian Text Copyright © 2005 by S. N. Kudryavtsev.This revised version was published online in April 2005 with a corrected issue number.     

Average widths of anisotropic Besov-Wiener classes

This paper concerns the problem of average σ-K width and average σ-L width of some anisotropic Besov-Wiener classes Srp q θb(Rd) and Srp q θB(Rd) in Lq(Rd) (1≤q≤p＜∞). The weak asymptotic behavior is established for the corresponding quantities.     

Kolmogorov widths of Sobolev classes on an irregular domain

We establish asymptotic estimates for the Kolmogorov widths of Sobolev classes W _p ^s (K) in the metric of L _q (K) for a power-law peak K ? ? ^d . These estimates are sharp in the order and coincide with order estimates for the unit cube.     

Kolmogorov widths of weighted Sobolev classes on closed intervals

In this paper, we estimate the asymptotics of the Kolmogorov widths of weighted Sobolev classes in the metric of L _p . We establish the relationship between the width of the set W _∞,g ¹ and the approximation of the antiderivative function g by piecewise constant functions.     

Estimation of the widths of classes of smooth functions in the space Lq

Order-precise estimates of the widths of classes of infinitely differentiable functions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 279–280, February, 1990.     

Relative widths of Sobolev classes in the uniform and integral metrics

Let W_p^r be the Sobolev class consisting of 2π-periodic functions f such that ‖f^(r)‖_p ≤ 1. We consider the relative widths d_n(W_p^r, MW_p^r, L_p), which characterize the best approximation of the class W_p^r in the space L_p by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions g should lie in MW_p^r, i.e., ‖g^(r)‖_p ≤ M. We establish estimates for the relative widths in the cases of p = 1 and p = ∞; it follows from these estimates that for almost optimal (with error at most Cn^?r, where C is an absolute constant) approximations of the class W_p^r by linear 2n-dimensional spaces, the norms of the rth derivatives of some approximating functions are not less than cln min(n, r) for large n and r.     

Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W _p ¹ on the “zero” cusp with a Hölder singularity at the vertex.     

Estimates of the widths of classes of analytic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 4, pp. 567–570, April, 1989.     

Estimates for widths of classes of periodic multivariable functions

Doklady Mathematics -     

Lower bound for quantum integration error on anisotropic Sobolev classes

We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev class R（Wpr（[0, 1]d）） （r R＋d）. Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Hblder- Nikolskii class R（H∞r（[0,1]d）） and Sobolev class R（W∞r（[0,1]d））. The results show that for these two types of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.     

Smallest constant of extension of functions of Sobolev classes

The problem is posed on the class-preserving extension of functions of Sobolev class is a finite domain in the in-dimensional space) onto the whole of such that the supports of the extended functions would lie in a specified finite domain and such that the so-called constant of extension would be minimal. The existence of such an extension is proved under constraints on of the type of a certain minimal smoothness; an algorithm for the approximate computation of the minimal constant of extension is indicated forp=2. Ifp=2 andS=I while are concentric balls, then the exact value of the constant of extension has been computed under the usual definition of a norm in; it is expressed in terms of Bessel functions of an imaginary argument. The exact value mentioned permits the estimation from above and from below of the constant of extension in the case when the domain is diffeomorphic to a sphere.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 150–185, 1979.     

Exact values of relative widths of classes of differentialbe functions

We study relative widths in the spacesC andL of classes of periodic differentiable functionsW ^r,r=1,2,…, when in contrast to the Kolmogorov widths it is additionally required that the approximating functions belong to the classMW ^r with a given majorantM of the norm of the derivative of orderr. It is proved that ifM satisfies the estimate