The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric

We investigate the approximation of functions by Bernstein polynomials. We prove that (1) $$^\tau [0,1]^{(f,B_n (f))} \leqslant \mu _f \left( {4\sqrt {\tfrac{{\ln n}}{n}} } \right) + \left( {4\sqrt {\tfrac{{\ln n}}{n}} } \right),$$ where r_[0,1](f, Bn(f)) is the Hausdorff distance between the functionsf(x) and B_n(f; x) in [0,1], is the modulus of nonmonotonicity off(x). The bound (1) is of better order than that obtained by Sendov. We show that the order of (1) cannot be improved.     

On the Hausdorff and packing measures of typical compact metric spaces

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.     

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces X and Y, let $\mathcal{M}_\flat \left( X \right)$ , $\mathfrak{M}\left( {V \to W} \right)$ and $\mathfrak{D}_\flat \left( {X,Y} \right)$ be the ballean of X (the family of the balls in X), the space of mappings from X to Y, and the space of mappings from the ballean of X to Y, respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $\hat \rho _u$ , $\hat \beta _{X,Y}^\lambda$ , $\hat \beta _{X,Y}^{ * \lambda }$ ) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, including some normed algebra structure. To some extent, the class $\hat \beta _{X,Y}^\lambda$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when X is compact and Y = K is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of K-valued measures on X.     

Hausdorff metric structure of the space of probability measures

The Hausdorff distance between two bounded functions on the real line was introduced by B. Sendov and B. Penkov. On the class of distribution functions it agrees with the Levi distance. The goal of this article is to analyze the Levi-Prokhorov metric and a new metric in terms of the relation between their structure and the Hausdorff distance.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 87–103, 1979.In conclusion I would like to take this opportunity to thank V. M. Zolotarev for posing the problem and discussing the results. I express my gratitude to V. Senatov for helpful advice which helped improve the work.     

Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .

     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

Entropy of the space of twice smooth curves in the Hausdorff metric

Bound from above and below for the entropy of the space of twice smooth curves on a plane in the Hausdorff metric are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 113–118, January, 1990.     

Best Hausdorff approximations by algebraic polynomials and porosity

For anyb (a, a), a sharp lower bound for the coefficient ofv-uniformity of the set of discontinuity points of a functionf on the interval (–b, b) is obtained.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 692–702, May, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236.     

Boundedness of trigonometric integrals and polynomials in the L metric

Conditions are derived for the uniform boundedness of special trigonometric integrals and sums in the L metric.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 811–822, December, 1970.     

On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H ^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variation fdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H ^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H ^L .Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.     

Hausdorff dimension 2 for Julia sets of quadratic polynomials

We consider families of quadratic polynomials which admit parameterisations in a neighbourhood of the boundary of the Mandelbrot set. We show how to find parameters such that the associated Julia sets are of Hausdorff dimension 2. Received October 11, 1999 / Published online April 12, 2001