Abstract
In this paper, two known theorems dealing with $|\bar{N},p_{n}|_{k}$ summability of infinite series and Fourier series have been generalized to ${\varphi}-|\bar{N},p_{n};\beta|_{k}$ summability.
Highlights
A sequence (An) is said to be δ-quasi-monotone if An → 0, An > 0 and ∆An ≥ −δn, where∆An=An − An+1 and δ = is a sequence of positive numbers
Two known theorems dealing with |N, pn|k summability of infinite series and Fourier series have been generalized to φ − |N, pn; β|k summability
In [3], the following theorem on δ-quasi-monotone sequences has been proved
Summary
A sequence (An) is said to be δ-quasi-monotone if An → 0, An > 0 and ∆An ≥ −δn, where. ∆An=An − An+1 and δ = (δn) is a sequence of positive numbers (see [1]). Let (φn) be a sequence of positive real numbers. In [3], the following theorem on δ-quasi-monotone sequences has been proved. Let (λn) → 0 as n → ∞ and (pn) be a sequence of positive numbers such that Pn = O(npn) as n → ∞. If all conditions of Theorem 1.1 are satisfied with the condition (1.1) replaced by (2.3) the series m φβnk−1|tn|k = O(Xm) as m → ∞, n=1 anλn is summable φ − |N , pn; β|k, k ≥ 1 and 0 ≤ β < 1/k. For the proof of Theorem 2.1, it is sufficient to show that φnβk+k−1 | In,r |k< ∞, f or r = 1, 2, 3, 4.
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