On application of matrix summability to Fourier series

Yıldız, Şebnem

doi:10.1002/mma.4635

Cited by 12 publications

(5 citation statements)

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“…Thus this contradiction completes the proof of the theorem. If a closed subset of R is lacunary statistically p-ward compact for a positive integer p, then any sequence of points in A has a (P n , s)-absolutely almost convergent subsequence (see [27], [37], [52], [62], [2], [65], and [63]).…”

Section: Definition 22mentioning

confidence: 99%

Variations on lacunary statistical quasi Cauchy sequences

Yıldız¹

2019

AIP Conference Proceedings

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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.

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Section: Definition 22mentioning

confidence: 99%

Variations on lacunary statistical quasi Cauchy sequences

Yıldız¹

2019

AIP Conference Proceedings

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“…Conversely, suppose that E is not bounded above. Pick an element It follows from the above theorem that if a closed subset E of R is ρ-statistically downward compact, and −A is ρ-statistically downward compact, then any sequence of points in E has a (P n , s)-absolutely almost convergent subsequence (see [36], [52], [62], [3], [64], [63], and [65]…”

Section: Now We Give Some Interesting Examples That Show Importance Omentioning

confidence: 99%

A Variation on Statistical Ward Continuity

Cakalli

2015

Bull. Malays. Math. Sci. Soc.

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A real valued function defined on a subset E of R, the set of real numbers, is ρ-statistically downward continuous if it preserves ρ-statistical downward quasi-Cauchy sequences of points in E, where a sequence (α k ) of real numbers is called ρ-statistically downward quasi-Cauchy if limn→∞ 1 ρn |{k ≤ n : ∆α k ≥ ε}| = 0 for every ε > 0, in which (ρn) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim sup n ρn n < ∞, ∆ρn = O(1), and ∆α k = α k+1 − α k for each positive integer k. It turns out that a function is uniformly continuous if it is ρ-statistical downward continuous on an above bounded set. 1 hr k∈Ir |α k − L| = 0, where I r = (k r−1 , k r ], and k 0 = 0, h r := k r − k r−1 → ∞ as r → ∞ and θ = (k r )is an increasing sequence of positive integers (see also [45], and [46]). In [43] Fridy and Orhan introduced the concept of lacunary statistically convergence in the sense that a sequence (α k ) of points in R is called lacunary statistically convergent, or Date: September 27, 2018. 2010 Mathematics Subject Classification. Primary: 40A05; Secondaries: 40D25; 40A35; 40A30; 26A15.

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“…Özarslan [9], [10] used the definition of almost increasing sequence for absolute summability. Yildiz [17], [18] determined theorems on generalized absolute matrix summability factors. Mishra et al [7], [8] provide interesting result on matrix summability and absolute summability.…”

Section: Introductionmentioning

confidence: 99%