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μ -statistically convergent function sequences

Oktay Duman, Cihan Orhan (2004)

Czechoslovak Mathematical Journal

In the present paper we are concerned with convergence in μ -density and μ -statistical convergence of sequences of functions defined on a subset D of real numbers, where μ is a finitely additive measure. Particularly, we introduce the concepts of μ -statistical uniform convergence and μ -statistical pointwise convergence, and observe that μ -statistical uniform convergence inherits the basic properties of uniform convergence.

σ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş, Richard Patterson (2006)

Open Mathematics

This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 lim r 1 h r k ∈ I r : x σ k ( m ) y σ k ( m ) - L ⩾ ∈ = 0 uniformly in m = 1, 2, 3, ..., (denoted by x ∼ S σ , θ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent...

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