Absolute Summability Factors Research Articles

On Absolute Summability Factors of Infinite Series

In this paper using δ-quasi-monotone sequences a theorem on\(\left| {\bar N,p_n ;\delta } \right|_k \) summability factors of infinite series, which generalizes a theorem of Bor [4] on\(\left| {\bar N,p_n } \right|_k \) summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|k summability factors.

On absolute summability factors

In this paper a theorem on | N ¯ , p n | k |\overline N ,{p_n}{|_k} summability factors, which generalizes a theorem of Mazhar (Indian J. Math. 14 (1972), 45-48) on | C , 1 | k |C,1{|_k} summability factors, has been proved.

On two absolute Riesz summability factors of infinite series

This paper gives a necessary and sufficient condition in order that a series ∑ a n ε n \sum {{a_n}} {\varepsilon _n} should be summable | R , q n | |R,{q_n}| whenever ∑ a n \sum {{a_n}} is summable | R , p n | k , k ⩾ 1 |R,{p_n}{|_k},\;k \geqslant 1 , and so extends the known result of Bosanquet to the case k > 1 k > 1 .

Absolute Summability Factors in a Sequence

AbstractLet α≥0 and β>— 1. The main result gives necessary and sufficient conditions for the sequence (εn) in order that the sequence (εnUn) will be absolutely summable by the Cesàro method Cβ for each sequence (Un) which is bounded or summable by the method CαAnother theorem is proven when Cα and Cβ are replaced by triangular methods A = (ank) and B=(bnk) satisfying , where (ξnk) = (ank)-1.

Absolute Riesz summability factors of Fourier series

Absolute Riesz summability factors of Fourier series

On the absolute summability factors of infinite series

On the absolute summability factors of infinite series

On the absolute summability factor of Fourier series

The purpose of this paper is to give a general theorem on the absolute Riesz summability factor of Fourier series which implies Matsumoto's Theorem [Tôhoku Math. J. 8 (1956), 114–124] and to deduce some results from the theorem.

On absolute Riesz summability factors of Fourier series

On absolute Riesz summability factors of Fourier series

Summability factors for lower-semi-matrix transformations

In view of Kogbetliantz's identity [7] the absolute Cesaro summability of orderk (k)>−1) of an infinite seriesΣ a n is the same as the absolute convergence ofΣ(τ n k )n−1 whereτ n k is then-th Cesaro mean of orderk of sequence {na n }.Das [5] has shown that similar dependence is true for certain classes of Norlund means. The object of this paper is to establish two theorems on absolute summability factors involving two lower-semimatrix transformations and thereby to generalise a result ofChow [3] on absolute Cesaro summability factors and a result ofBosanquet andDas [1] on absolute Harmonic summability factors.

A theorem on the absolute Riesz summability factors of a Fourier series

A theorem on the absolute Riesz summability factors of a Fourier series

The absolute harmonic summability factors of infinite series

The absolute harmonic summability factors of infinite series

On the absolute summability factors of infinite series

On the absolute summability factors of infinite series

Absolute Summability Factors of Infinite Series

In this paper we have proved the follovving theorem which generalizes a the­ orem of Pati [M.Z. 78 (1962), 293-297] and also a theorem of Prasad and Bhatt [Duke Math. J. 24 (1957), 103-117]

Absolute summability factors of Fourier series

1. Let f(t) be a perodic function with period 2π and integrable in the sense of Lebesgue over (–π, π), then its Fourier series is

Absolute Riesz summability factors for Fourier series

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,and, for r>0, we write is known as the Riesz sum of “ type ” λn and “ order ” r, andis called the Riesz mean of type λn and order r.

On the absolute Nörlund summability factors of infinite series

Taking the start from an earlier result on the absolute harmonic summability factors due to S. N. Lal, we obtain in this paper suitable factors $\{ {\epsilon _n}\}$ so that the series $\sum {{a_n}} {\epsilon _n}$ may be summable $|N,\;{p_n}|$, whenever the series $\sum {{a_n}}$ is summable $|C,\;1|$.

On the absolute Cesaro summability factors of trigonometric series

Let be any given infinite series with sn as its n-th partial sum.We writeandwhere

Correction: ``Absolute summability factors, I''

Correction: ``Absolute summability factors, I''

Absolute summability factors for series bounded $$(\bar N,p_n )$$

Absolute summability factors for series bounded $$(\bar N,p_n )$$

Absolute harmonic summability factors of a Fourier series

Absolute harmonic summability factors of a Fourier series