Abstract
The Kλ-means were first introduced by Karamata. Vuckovic first studied the Kλ-summability of a Fourier series and later on Lal studied the Kλ-summability of a conjugate series. In the present paper, we have studied the |Kλ|-summability of Fourier series and conjugate series.
Highlights
Let f be a 2π-periodic function and integrable in the sense of Lebesgue over (−π, π)
Vuckovic [10] was the first to study the K λ-summability of Fourier series and his result reads as follows
Later Lal [4] obtained the following result for the conjugate series
Summary
If limn→∞ tn = s, we say that sequence {sn} (or the series un) is summable K λ to s. The series un (or the sequence {sn}) is said to be absolutely K λ-summable if {tn} ∈ BV ; i.e.,. We may derive the following useful identity (Proposition 1.1) which is similar to the Kogbetliantz identity [3] for the Cesàro mean, namely, where σnα is the (C, α) mean of n(σnα − σnα−1) = τnα, an and τn is the (C, α) mean of {nan}. Proposition 1.1 where tn is the K λ mean of (n − 1 + λ)(tn − tn−1) = ξn−1(u) un and ξn is the K λ mean of {un+1}; i.e., ξn(u) =. Proposition 1.2 The K λ-method is absolutely conservative; that is |C, 0| ⊂ |K λ|.
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