Interpolation of bounded sequences

This paper deals with an interpolation problem in the open unit disc ⅅ of the complex plane. We characterize the sequences in a Stolz angle of ⅅ, verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on ⅅ, but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.     

Polynomial sequences of bounded operators

The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator T,T²,…,Tⁿ,…. In this paper, we replace the powers Tⁿ by certain polynomials p_n(T), and make use of special properties of the polynomial sequence to derive some new results concerning operators. For example, using an arbitrary polynomial sequence , we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.     

Rearrangements of bounded variation sequences

Letbv be the set of all bounded variation sequences. In the present paper we deduce from a theorem of Mears a necessary and sufficient condition for the rearrangement (a _p(k)) to be of bounded variation whenevera _k)∈bv; interestingly it coincides with Pleasants’ criterion for convergence-preserving.     

Kernels of sequences of complex numbers and their regular transformations

It is proved that , where U(a, r) is the ball of radius r with center at the pointa, is the smallest closed convex set containing the kernel of any sequence {y_n} obtained from the sequence {x_n} by means of a regular transformation (c_nk) satisfying the condition , where x, x_n, c_nk (n, k=1, 2,...) are complex numbers.Traslated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 815–823, December, 1977.     

Summation of bounded sequences by regular positive matrices

In terms of elements of regular positive discrete and semicontinuous matrices sufficient conditions are obtained that these matrices do not sum a given real,divergent, bounded sequence and, in particular, that the core of a given sequence coincides with the core of the sequence (function) transformed with the aid of a matrix. These conditions permit certain facts, some well-known and some new, on the theory of inefficient matrices to be obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 179–188, February, 1973.     

Selection principle for pointwise bounded sequences of functions

For a number ? > 0 and a real function f on an interval [a, b], denote by N(?, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a _i , b _i ], i = 1, …, n, on [a, b] such that |f(a _i ) ? f(b _i )| > ? for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f _j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(?) ≡ lim sup _j→∞ N(?, f _j , [a, b]) < ∞ for any ? > 0, then the sequence {f _j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(?, f, [a, b]) ≤ n(?) for any ? > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f _j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.     

Spreading-invariant sequences and processes on bounded index sets

We say that a random sequence is spreadable if all subsequences of equal length have the same distribution. For infinite sequences the notion is equivalent to exchangeability but for finite sequences it is more general. The present paper is devoted to a systematic study of finite spreadable sequences and of processes on [0, 1] with spreadable increments. In particular, we show how many basic results in the exchangeable case—notably the predictable sampling theorem, the Wald-type identities, and various martingale and weak convergence results—admit extensions to a spreadable setting. We also identify some additional conditions that ensure the exchangeability of a spreadable sequence or process. Received: 9 November 1999 / Revised version: 16 March 2000 / Published online: 18 September 2000     

Moderate deviations for stationary sequences of Hilbert-valued bounded random variables

In this paper, we derive the Moderate Deviation Principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non-adapted sequences of Hilbert-valued random variables. Applications to Cramér-Von Mises statistics, functions of linear processes and stable Markov chains are given.     

On sequences of Dirichlet polynomials uniformly bounded on half planes

Given and a sequence of Dirichlet polynomials estimates for the coefficientsa _n are proved if {_n} is uniformly bounded on a region containing a half plane. Thereby a result is obtained which is an analogue of a known result for polynomials, that is for theA-transforms of the geometric sequence; moreover a Jentzsch type theorem for {_n(z)} is derived.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Convergence rates in the strong law for bounded mixing sequences

Summary Speed of convergence is studied for a Marcinkiewicz-Zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate. Though -mixing is also considered, the most interesting result concerns absolutely regular sequences. The results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible. Another application sharpens a result for averaging a function along a random walk.     

Recursive Kernels

This paper is an extension of earlier papers [8, 9] on the “native” Hilbert spaces of functions on some domain Ω ⊂ R ^d in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recursively defined reproducing kernels. As an application, we get a recursive Neville-Aitken-type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.