Sequence Of Real-valued Functions Research Articles

On ρ−statistical convergence of order α of sequences of function

Abstract In this study, by using definition of ρ-statistical convergence which was defined by Çakallı [5], we introduce the concepts of pointwise w ρ α (f)-summability, pointwise ρ-statistical convergence of order α and uniform ρ-statistical convergence of order α sequences of real-valued functions. Also, we give some inclusion relations between these concepts.

A characterization of the uniform convergence points set of some convergent sequence of functions

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

Convergence in measure and in category, similarities and differences

Convergence in category of a sequence of real-valued functions has been introduced by E. Wagner as an analogue of convergence in measure. In the paper it is shown that in some circumstances both kinds of convergence behave similarly, but sometimes the behaviour is different.

A Remark on the Uniform Convergence of Some Sequences of Functions

We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.

Effective Dini's Theorem on Effectively Compact Metric Spaces

We show that if a computable sequence of real-valued functions on an effectively compact metric space converges pointwise monotonically to a computable function, then the sequence converges effectively uniformly to the function. This is an effectivized version of Dini's Theorem.

A limit theorem for the separable statistic in a random assignment scheme

One considers the following scheme of random assignment of n particles in an infinite sequence of cells. Each particle is. assigned to the k-th cell with probability pk and one assumes that pk⩾ pk+1 and pk > 0 for each k. Let Xk(n) be the number of particles in the k-th cell and let f1(x), f2(x), ... be a sequence of real-valued functions defined for x=0, 1,2, ... Under certain conditions on the distribution of the probabilities and on the sequence f1(x), f2(x), ..., one investigates the asymptotic normality of the random variable . (The random variable Zn is proper since )

On an Asymptotic Integral

1. This note gives an asymptotic evaluation of an integral of the formas n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.