In this paper, we introduce two new general theorems on ϕ − A, p n k summability factors of infinite and Fourier series. By using these theorems, we obtain some new results regarding other important summability methods and investigate conversions between them.

Communicated by: T. Qian MOS Classification: 26D15; 42A24; 40F05; 40G99Bor has recently obtained a main theorem dealing with absolute weighted mean summability of Fourier series. In this paper, we generalized that theorem for |A, n | k summability method. Also, some new and known results are obtained dealing with some basic summability methods.

The notion of a lacunary statistical δ 2 -quasi-Cauchyness of sequence of real numbers is introduce and investigated. In this work, we present interesting theorems related to lacunary statistically δ 2 -ward continuity. A function f, whose domain is included in R, and whose range included in R is called lacunary statistical δ 2 ward continuous if it preserves lacunary statistical δ 2 quasi-Cauchy sequences, i.e. (f(xk)) is a lacunary statistically δ 2 quasi-Cauchy sequence whenever (xk) is a lacunary statistically δ 2 quasi-Cauchy sequence, where a sequence (xk) is called lacunary statistically δ 2 quasi-Cauchy if (∆ 2 xk) is a lacunary statistically quasi-Cauchy sequence. We find out that the set of lacunary statistical δ 2 ward continuous functions is closed as a subset of the set of continuous functions. İstatistiksel boşluklu delta 2 quasi Cauchy dizileri ÖZBu makalede istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi kavramı tanımlanmış ve araştırılmıştır. Bu araştırmada istatistiksel boşluklu δ 2 -süreklilik ile ilgili ilgi çekici teoremler ispatlanmıştır. (∆ 2 xk) istatistiksel boşluklu quasi Cauchy dizisi olduğunda (xk) dizisine istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi dendiğine göre, reel sayılar kümesinin bir alt kümesi üzerinde tanımlı reel değerli bir f fonksiyonuna eğer terimleri A da olan istsatistiksel boşluklu δ 2 -quasi-Cauchy dizilerini koruyor ise, yani (xk) dizisi terimleri A da olan istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi olduğunda (f(xk)) dizisi de istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi oluyor ise istatistiksel boşluklu δ 2 -ward süreklidir denir. İstatistiksel boşluklu δ 2 -ward sürekli fonksiyonların kümesinin sürekli fonksiyonlar uzayının kapalı bir alt kümesi olduğu ortaya çıkarılmıştır.Anahtar Kelimeler: toplanabilme, quasi Cauchy dizisi, istatistiksel boşluklu yakınsaklık, süreklilik 1 Ahi Evran Üniversitesi Fen Edebiyat Fakültesi / Matematik sebnemyildiz@ahievran.edu.tr

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In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (α k ) is lacunary statistically p-quasi-Cauchy if limr→∞ 1 hr |{k ∈ Ir : |α k+p − α k | ≥ ε}| = 0 for each ε > 0. A function f is called lacunary statistically p-ward continuous on a subset A of the set of real numbers R if it preserves lacunary statistically p-quasi-Cauchy sequences, i.e. the sequence (f (αn)) is lacunary statistically p-quasi-Cauchy whenever α = (αn) is a lacunary statistically pquasi-Cauchy sequence of points in A. It turns out that a real valued function f is uniformly continuous on a bounded subset A of R if there exists a positive integer p such that f preserves lacunary statistically p-quasi-Cauchy sequences of points in A.