Generalizations of the Kolmogorov-Barzdin embedding estimates

We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.Comment: 45 page

Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov $n$-widths $d_n(B_p^r, L^q)$ and linear $n$-widths \da_n(B_p^r, L^q), ($1\leq q\leq \infty$) of Sobolev's classes $B_p^r$, ($r>0$, $1\leq p\leq \infty$) on compact two-point homogeneous spaces (CTPHS) are established. For part of $(p, q)\in[1,\infty]\times[1,\infty]$, sharp orders of $d_n(B_p^r, L^q)$ or \da_n (B_p^r, L^q) were obtained by Bordin, Kushpel, Levesley and Tozoni in a recent paper `` J. Funct. Anal. 202 (2) (2003), 307--326''. In this paper, we obtain the sharp orders of $d_n(B_p^r, L^q)$ and \da_n (B_p^r, L^q) for all the remaining $(p,q)$. Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund type inequalities on CTPHS

Estimates of n-widths of Sobolev's classes on compact globally symmetric spaces of rank one

AbstractEstimates of Kolmogorov's and linear n-widths of Sobolev's classes on compact globally symmetric spaces of rank 1 (i.e. on Sd, Pd(R), Pd(C), Pd(H), P16(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. New estimates for the (p,q)-norms of multiplier operators Λ={λk}k∈N are given. We apply our results to get sharp orders of best polynomial approximation and n-widths