Nondecreasing Sequence Of Positive Numbers Research Articles

On λ statistical upward compactness and continuity

A sequence (?k) of real numbers is called ?-statistically upward quasi-Cauchy if for every ? > 0 limn?? 1/?n |{k?In:?k-?k+1 ? ?}| = 0, where (?n) is a non-decreasing sequence of positive numbers tending to 1 such that ?n+1 ? ?n + 1, ?1 = 1, and In = [n-?n+1,n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is ?-statistically upward continuous if it preserves ?-statistical upward quasi-Cauchy sequences. ?-statistically upward compactness of a subset in real numbers is also introduced and some properties of functions preserving such quasi Cauchy sequences are investigated. It turns out that a function is uniformly continuous if it is ?-statistical upward continuous on a ?-statistical upward compact subset of R.

The $\lambda ^{+r}(\mu )$ -statistical convergence

Let λ=(λn)n≥1 be a nondecreasing sequence of positive numbers tending to infinity such that λ1=1 and λn+1≤λn+1 for all n, and let In=[n−λn+1,n] for n=1,2,…. Then for any given nonzero sequence μ, we define by Δ+(μ) the operator that generalizes the operator of the first difference and is defined by Δ+(μ)xk=μk(xk−xk+1). In this article, for any given integer r≥1, we deal with the λ+r(μ) -statistical convergence that generalizes in a certain sense the well-known λEr-statistical convergence. The main results consist in determining sets of sequences χ and χ′ of the form sξ0 satisfying χ⊂[V,λ]0(Δ+r(μ))⊂χ′ and sets κ and κ′ of the form sξ satisfying κ≤[V,λ]∞(λ+r(μ))≤κ′. This study is justified since the infinite matrix associated with the operator Δ+r(μ) cannot be explicitly calculated for all r.

Generalized statistical convergence in 2-normed spaces

In this paper we introduce the concept of λ-statistical convergence in 2-normed spaces. Some inclusion relations between the sets of statistically convergent, λ-statistically convergent and statistically λ-convergent sequences in 2-normed spaces are established, where λ = (λm) is a non-decreasing sequence of positive numbers tending to infinity such that λm+1 ≤ λm + 1, λ1 = 1.

A note on estimates for the growth order of sequences of multiple rectangular Fourier sums

Let $$\left\{ {S_{n_k } \left( {f,x} \right)} \right\}_{k = 1}^\infty$$ be some sequence of rectangular partial sums of the multiple trigonometric Fourier series of a function f, and let {λ k } =1 ∞ be a nondecreasing sequence of positive numbers. We investigate conditions on a class φ(L) under which any function f from this class satisfies estimates of the form $$S_{n_k } \left( {f,x} \right) = o\left( {\lambda _k } \right)$$ a.e., where the right-hand side depends on k only.

On Λ-statistical convergence in random 2-normed space

Recently, Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of � -statistical convergence and � -statistical Cauchy sequences in random 2-normed spaces , where λ = (λm) be a non-decreasing sequence of positive numbers tending to infinity such that λm+1 ≤ λm + 1, λ1 = 1 and prove some theorems. In last section we will give the definition of the � − limit and cluster points and we will show their relation between those classes.

The Intersection Conditions For 〈<em>s</em>〉-Density Systems of Paths

AbstractWe investigate the intersection conditions for an hsi-density systemof paths. We show, for example, that for every unbounded and nonde-creasing sequence of positive numbers hsi such that liminf n!1s n s n+1 =0, there exists a system of paths connected with hsi-density points whichdoes not satisfy the intersection conditions. Moreover, we show that afunction f : R ! R is hsi-approximately continuous if and only if f iscontinuous with respect to some hsi-density system of paths. 1 Introduction. The notion of a density point was introduced by Lebesgue at the beginningof the 20th century. Together with his fundamental theorem that almost allpoints of a Lebesgue measurable set are density points of that set, they turnedout to play an important role in the theory of real functions.The density topology connected with the notion of a density point wasdiscovered by Haupt and Pauc [5] and was subsequently studied in detailby Go man, Neugebauer, Nishiura, Waterman and Tall. It turned out thata function is approximately continuous if and only if it is continuous withrespect to the density topology.Some generalizations of the notion of a Lebesgue density point on the realline were introduced by Taylor [8]. Wilczynski presented the concept of adensity point with respect to category [10]. Further, Filipczak and Hejduk [3],generalized the notion of a density point using unbounded and nondecreasingsequences of positive numbers. Such an approach to density points gave riseto hsi-density topologies and hsi-approximately continuous functions. Some

Strong law of large numbers for multilinear forms

Let $m \geq 2$ be a nonnegative integer and let ${X^{(l)}, X_i^{(l)}}_{i \epsilon \mathbb{N}}, l = 1, \dots, m$, be $m$ independent sequences of independent and identically distributed symmetric random variables. Define $S_n = \Sigma_{1 \leq i_1, \dots, i_m \leq n} X_{i_1}^{(l)} \dots X_{i_m}^{(m)}$, and let ${\gamma_n}_{n \epsilon \mathbb{N}}$ be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For $m = 2$ necessary and sufficient conditions are obtained for the strong law of large numbers $\gamma_n^{-1} S_n \to 0$ a.s. to hold, and for $m > 2$ the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the $X^{(l)}$ .

Univalent functions with univalent Gelfond-Leontev derivatives

Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),(iv) if D corresponds to the shift operator (dn ≡ l), then .Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

Estimating functions by partial sums of their Fourier series

For classes of functions with convergent Fourier series. the problem of estimating the rate of convergence of the Fourier series has always been of interest. A classical theorem like that of Dirichlet and Jordan for functions of bounded variation (BV) assures the convergence of the Fourier series but gives no estimate of the rate of convergence. Such an estimate was recently provided by Bojanic [ 11. For certain classes of functions of generalized bounded variation the conclusions of the Dirichlet-Jordan theorem also hold. Waterman 15, 7 1 has shown that the class of functions of harmonic bounded variation (HBV) is, in a certain sense, the largest such class. An estimate of the rate of convergence of the Fourier series has been made for certain classes which lie between BV and HBV [2]. Here we consider this problem in greater generality for /IBV classes in that range to obtain a result which includes the previous estimates and allows us to make an estimate for a particular class which is closer to HBV than the classes previously con sidered. If f is a real valued function on the interval (a. 61 and .I = {A,,/ is a nondecreasing sequence of positive numbers such that x l/i,, diverges. we say that f is of /i-bounded variation (,4BV) if the sums

UNIQUENESS THEOREMS FOR ANALYTIC FUNCTIONS ASYMPTOTICALLY REPRESENTABLE BY DIRICHLET-TAYLOR SERIES

Dirichlet-Taylor series are considered, where is a sequence of complex numbers, is a nondecreasing sequence of positive numbers, and is the number of times ?j occurs in the segment {?1,?,?j}. An adherence principle is established for these series. As applications of this principle, uniqueness theorems are proved for analytic functions which are asymptotically representable by partial sums of Dirichlet-Taylor series in strips. Bibliography: 16 items.