On the |Kλ|-summability of Fourier series and its conjugate series

The Kλ-means were first introduced by Karamata. Vučković 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.


Definitions and notations
For n = 0, 1, 2, . . . , define the numbers [ n k ] for 0 ≤ k ≤ n by n−1 where n−1 ν=0 (x + ν) = x(x + 1) · · · (x + n − 1) = (x+n) (x) . Clearly, [ n k ] = 0 when k < 1 and k > n. We shall use the convention that [ 0 0 ] = 1. The numbers of [ n k ] are known as Stirling's number of first kind. We know [8, p. 43] the following recursion formula Let ∞ n=0 u n be an infinite series with sequence of partial sums {s n } i.e., s n = n k=0 u k . Let λ > 0, the K λ -mean (t n ) of the sequence {s n } is defined by [2,5] t n = (λ) (n + λ) n k=0 [ n k ]λ k s k . (1.3) If lim n→∞ t n = s, then we say that sequence {s n } (or the series u n ) is summable K λ to s. The series u n (or the sequence {s n }) is said to be absolutely K λ -summable if {t n } ∈ 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, n is the (C, α) mean of a n and τ n is the (C, α) mean of {na n }. Proposition 1.1 where t n is the K λ mean of u n and ξ n is the K λ mean of {u n+1 }; i.e., (1.10) which shows that the K λ -method is absolutely conservative.

Application to trigonometric Fourier series
Let f be a 2π-periodic function and integrable in the sense of Lebesgue over (−π, π). Let the trigonometric Fourier series of f at x be given by The conjugate series of (2.1) is given by f (x; ), whenever the limit exists.
The K λ -means were first introduced by Karamata [2]. Lototsky [5] reintroduced the special case λ = 1. Vučković [10] was the first to study the K λ -summability of Fourier series and his result reads as follows.
then the trigonometric Fourier series of f at x = t is K λ -summable to f (x).
Later Lal [4] obtained the following result for the conjugate series.
In the present paper, we study the absolute K λ -summability of a Fourier series and its conjugate series. We prove

Notations and lemmas
For the proofs of the theorems, we need the following additional notations: We need the following lemmas to prove our theorems: from which the lemma follows.
Lemma 3.2 [9, Chapter 5, Lemma 5.5] Let R(n, t), K λ n (t) and K λ n (t) be defined as in Sect. 3. Then for some positive constant A and all t ∈ (0, π) Proof Note that R(n, t) attains its maximum value for t = 0, and it is easy to see that R(n, 0) = 1. This ensures the first estimates of (i) Now We observe that where A is a positive constant. Using (3.2) in (3.1), we obtain the second estimate of Lemma 3.2 (i). The proof of Lemma 3.2(ii) and (iii) follows from Lemma 3.2(i).
Next, we note that and so, by logarithmic differentiation, since from which the lemma follows. Proof Integrating by parts, we have say.
and hence α n,2 is absolutely convergent. And, since |K λ |-method is absolutely conservative It remains to show that α n,1 ∈ |K λ |.
By definition the series α n, Using the notation of Sect. 3 and Lemma 3.2(iii), we get which implies that α n,1 ∈ |K λ |. As α n ∈ |K λ |, and collecting the above results, it follows that that is, |g(n,δ)| n < ∞. This completes the proof of the lemma.
Proof of (i) Integrating by parts we have, from which (i) follows.
Proof of (ii) Integrating by parts, we get from which the result follows. Then Proof of (i) Integrating by parts, we get since the last integral is negative, and this completes the proof of (i).
Proof of (ii) Let 0 < β < 2. By the simple computation, we get d du e −Au 2 log n (log 1/u) 2 = 2e −Au 2 log n (log 1/u) 3 The expression e −Au 2 log n (log 1 u ) 2 is monotonic decreasing in u whenever 1 − Au 2 log 1 u log n < 0. It is easy to see that that is, In view of this inequality, we get which is same as the first integral in the first case discussed above. Lastly, in case (A log n) −β < t, L(n, t) is majorized by the second integral δ (A log n) −β in the first case and this completes the proof of (ii).

Lemma 3.8
Let 0 < δ < e −2 and > 1, however large. Then Proof of (i) As K λ n (u) = O(1) by Lemma 3.2(iii), the result follows. Proof of (ii) First using Lemma 3.1(ii) and thereafter applying Lemma 3.3(ii), we obtain Integrating by parts and using Lemma 3.4 and Lemma 3.2(i), we get Using Lemma 3.2 and Lemma 3.6(ii) , we have t). (3.8) Collecting the results from (3.6) to (3.8) and using the estimate for G(n, t) from Lemma 3.6(i), we obtain the desired estimate for h(n, t).
For n ≥ 1 and 0 < δ < e −2 , we write Let ξ n (Q) be the nth K λ -mean of the sequence {Q n+1 }. The series Q n ∈ |K λ |, if and only if By simple computation and an appeal to Lemma 3.2(iii) This ensures (4.2) and consequently vindicates that the |K λ |-summability of trigonometric Fourier series is a local property. Writing g(t) = φ(t) log t −1 , α n = δ 0 g(u) cos nu du log u −1 and integrating by parts, we obtain say. (4.3) As α n ∈ |K λ | by Lemma 3.5 it remains to prove that β n ∈ |K λ |. Let ξ n (β) be the nth K λ -mean of the sequence {β n+1 }. It is easily seen that By definition β n ∈ |K λ |, if and only if By Lemma 3.8(i) By Lemma 3.5 and Lemma 3.
L(n,t) n log n and employing Lemma 3.7(ii) and Lemma 3.7(i), respectively, for the first and second sums, we get (4.9) Collecting the results from (4.6)-(4.9), we get (4.5) and this completes the proof of Theorem 2.3.

Proof of Theorem 2.4
We need the following additional lemmas for the proof of Theorem 2.4.