Estimates of derivatives of random functions, II

Approximate formulas for the derivatives of functions of several variables contaminated by noise are given. Error estimates are derived. The formulas are optimal in some sense.     

Estimates of derivatives ofn-convex functions

In the form of nonlinear norm estimates, a quantitative refinement of the following fact is given: for n-convex functions L¹-convergence implies Cⁿ⁻²-convergence. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 123–133. Translated by I. A. Fedortsova.     

Estimates of regression parameters of random fields. I

Asymptotic properties of the estimates of the modified method of least squares of regression parameters of random fields are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 497–502, April, 1992.     

On random arithmetical functions. I

Let A _Q be the group of complex unit roots of an integer order \( Q \geqslant 2 \). Let \( {\xi_p}\left( {p \in \mathcal{P}} \right) \) be independent random variables distributed uniformly on the set A _Q , where \( \mathcal{P} \)is the set of primes. Let f be a completely multiplicative function defined on \( \mathcal{P} \) by f(p) = ξ _p . We investigate the summatory function of f(n) and the density of those n for which f(n + j) = κ _j (j = 0, …, t), where κ _j ∈ A _Q .     

Estimates for derivatives of holomorphic functions in a hyperbolic domain

Let f(z) be a holomorphic function in a hyperbolic domain Ω. For 2?n?8, the sharp estimate of |f(n)(z)/f^′(z)| associated with the Poincaré density λΩ(z) and the radius of convexity ρΩc(z) at z∈Ω is established for f(z) univalent or convex in each Δc(z) and z∈Ω. The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev-Wirths conjecture is also discussed.     

Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ _θ ), θ ∈ Θ}. Here {χ _θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ?₊ such that sup _θ∈Θ{Ψ(χ _θ ) ? U(θ)}₊ is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.     

Estimates of derivatives in a mixed Lp-norm on a domain and an extension of functions

We extend to the case of a mixed L_p-norm the estimates of derivatives that are well known for the ordinary L_p-norm.Translated from Matematicheskie Zametki. Vol. 7, No. 2, pp. 147–154, February, 1970.     

Inverting random functions

In this paper we study how to invert random functions under different criteria. The motivation for this study is phylogeny reconstruction, since the evolution of biomolecular sequences may be considered as a random function from the set of possible phylogenetic trees to the set of collections of biomolecular sequences of observed species. Our results may affect how we think about maximum likelihood estimation (MLE) in phylogeny. For inverting random functions, MLE is optimal under a first criterion, although it is not optimal under a second criterion which is at least equally natural but more conservative. Furthemore, MLE has to be used differently from the way it has been used in the phylogeny literature, if we have a prior distribution on trees and mutation mechanisms and want to keep MLE optimal under the same first criterion. Some of the results of this paper have been known in the setting of statistical decision theory, but have never been discussed in the context of phylogeny.Supported by the New Zealand Marsden Fund.Supported by the National Science Foundation grant DMS 9701211, and the Hungarian National Science Fund contract T 016 358.     

Interpolation of random functions

Summary Error bounds for the interpolation of random functions are derived. On contrast to various results in the literature, no additional assumptions such as stationarity are required.Dedicated to the memory of Professor Lothar Collatz     

Estimates for the measures of exceptions of the exceptional values associated with logarithmic derivatives of meromorphic functions

In this paper certain estimates of type Σδ^α for the measures of exceptions of the exceptional values associated with logarithmic derivatives of meromorphic functions are obtained.     

Estimates of regularized solutions of convolution-type equations in the spaces of functions with derivatives of fractional orders

The paper deals with convolution type integral equations of the first kind. For exact solutions it is typical that the solutions are discontinuous or unbounded. The aim of the paper is to construct approximate solutions which may also be discontinuous or unbounded. A priori information about the solution and estimates for the approximate solutions, as well as for the differences between the approximate solution and the exact one, are given in terms of Nikol'skii-Besov spaces with differentiation of fractional orders. Bibliography: 8 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 170–194, 1995. This work was supported by the Russian Fundamental Research Foundation (Project No 93-01-00255).