On approximation in generalized Zygmund class

Abstract: Here, we estimate the degree of approximation of a conjugate function g̃ and a derived conjugate function g̃′, of a 2π-periodic function g ∈ Zλ r , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced fromourmain theorems.We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

The study of error estimates of a conjugate functiong of a 2π-periodic function g belonging to different Lipschitz classes using the products of Cesàro, Hölder and Euler has also been of great interest among the researchers like Lal and Singh [13,14], Dhakal [15], Mishra et al. [16], Nigam and Sharma [17][18][19][20] and Padhy et al. [21] in the recent past.
It can be noted that the matrices involved in Cesàro, Hölder, Euler and their product are Hausdorff matrices 1 . Thus, considering this view point, Singh and Srivastava [25] studied error estimates of a conjugate functiong of a 2π-periodic function g ∈ W(Lr , ξ (t)) using Hausdorff means.
Lal and Mishra [26] have obtained results on approximations of a conjugate functiong of a 2π-periodic function g ∈ Lipα and g ∈ Lip(ξ (t), r) using Euler-Hausdorff means.
The review of the above research clearly suggests that the studies of error estimation of a conjugate functiong of 2π-periodic function g ∈ Z λ r , r ≥ 1, using Hausdorff means of CFS 2 have not been initiated so far. Note 1: The CFS is not necessarily a FS 3 , for example, the series ∑︀ ∞ n=2 sin(nx) log n conjugate to the Fourier series ∑︀ ∞ n=2 cos(nx) log n is not a Fourier series (Zygmund [27,p. 186]).
In view of the above example, a separate study in case of CFS in the present direction of work is so required. The study of DFS 4 by single and product summability means has been of great interest among researchers.
A study of CDFS 5 was initiated for the first time by Moursund [28] in 1935. Moursund [28] established the results on Nörlund and Cesàro means of CDFS.
Thus, in the same direction, Chandra and Dikshit [29] have studied |B| and |E, q| summabilities of DFS and CDFS. Lal and Nigam [30] have studied K λ -summability of DFS and Lal and Yadav [31] have studied (N, p, q)(C, 1) product summability of DFS.
Since the studies of error estimates of a conjugate derived functiong ′ of a 2π-periodic function g either in the Lipschitz space or in the Zygmund space have not been initiated so far. Therefore, in this paper, we, for the very first time, also study the error estimates of a conjugate derived functiong ′ of a 2π-periodic function g ∈ Z λ r , using Hausdorff means of CDFS. A separate study in case of CDFS in the present direction is justified due to its importance in applications to science and engineering.
As the trigonometric series FS, CFS, DFS and CDFS are well known, we will not present them here. The detailed work on FS, CFS and DFS can be found in [27] and that on CDFS in [28].
We denote the j th partial sum of C F S ass j (g; y), which is given bỹ whereg is the conjugate function of 2π-periodic function g, and is expressed as Now, we denote the j th partial sum of C D F S ass ′ j (g; y), which is given bỹ whereg ′ is the conjugate derived function of 2π-periodic function g, which is expressed as g ′ (y) = 1 4π π ∫︁ 0 cosec 2 l 2 ρ (y) (l)dl (Moursund [28]) .
We mention here some standard inequalities which are used in the paper: sin l ≤ l, l ≥ 0; (6) | sin l| ≤ 1, | cos l| ≤ 1, for all l.
In 1921, Hausdorff [33] gave the following definition: A Hausdorff matrix H ≡ ( j,a ) is an infinite lower triangular matrix with nonzero entries where ∆ is a forward difference operator defined by ∆µ j = µ j − µ j+1 and ∆ a+1 µ j = ∆ a (∆µ j ). If H is regular, then µ j is known as a moment sequence. µ j has the representation µ j = is known as the mass function. ζ (v) is continuous at v = 0 and it belongs to BV[0, 1] such that ζ (0) = 0, ζ (1) = 1; and for The Hausdorff means H of CFS is given bỹ IfM H j (g; y) → s as j → ∞, then the CFS is said to be summable to s by the H method [34]. Note 3: A detailed study of Hausdorff matrices can be found in [23,35].

Particular cases of Hausdorff means:
Hausdorff means have following particular cases: (iii) For the mass function where b = 1/(1 + q), q > 0, the Hausdorff method H ≡ ( j,a ) is the Euler means (E, q), q > 0.

Remark 1.
As per the above discussion of particular cases of Hausdorff means, our main results also hold for Cesàro, Euler and Hölder means, as well as their product methods. In view of Remark 1, the products of Euler and Cesàro are again Hausdorff matrices [9,[22][23][24]. Note that we need to check the order of the product of Euler and Cesàro means. Now a question arises: Is the (E, q) method generated from the Hausdorff moment sequence ζ j = 1 (q+1) j a right or left shifting sequence of moments. Similarly, is the (C, 1) method generated from the Hausdorff moment sequence ζ j = 1 j+1 a right or left shifting sequence of moments.
Lal and Mishra [26] have considered the (E, q)-Hausdorff product transform and have reduced the Hausdorff matrix to a (C,1) matrix as a right shifting sequence (see corollary 8.3 of [26, p. 12]), which appears false because a (C, 1) matrix is a left shifting sequence as defined in Keska [36, p. 68]. Infact, Keska [36] gave the following: Definition 1. [36, p. 67] A moment sequence ζ j is known as a right-shifting sequence of moments if there exists ζ * 0 ∈ R such that the sequence is a moment sequence where χ is a real, bounded variation function defined on the interval [0, 1] satisfying the conditions χ(0+) = χ(0) = 0 and χ(1) = 1.
be the space of 2π-periodic and integrable functions. We define ‖ · ‖r by where λ : [0, 2π] → R is an arbitrary function with λ(l) > 0 for 0 < l ≤ 2π and lim The completeness of the space Z (λ) r can be discussed by considering the completeness of L r , r ≥ 1. Hence the space Z λ r is a Banach space under the norm ‖ · ‖ (λ) r .

Remark 3.
Throughout the paper, λ and ν denote moduli of continuity of order two such that λ(l) ν(l) is positive and non-decreasing in l, then Thus,
Conversely, if a function λ satisfies conditions (i)-(iv), then it is the first order modulus of continuity of the function g(y) = λ(|y|). Moreover, it can be easily seen that λ is the second order modulus of continuity of the function g(y) = λ(|y|) 2 . If a function λ satisfies conditions (i)-(iii) and the function λ(l) l is non-increasing on (0, +∞), then the semiadditivity condition (iv) also holds, and so λ is the modulus of continuity of the first and second order for some continuous functions.
The modulus of continuity of second order satisfies conditions (i)-(iii) and a further condition, given as follows: (v) the inequality λ(nl) ≤ n 2 λ(l) holds for any l ≥ 0 and n ∈ N.
Geit [37] constructed a wide class of the functions that are second-order moduli of continuity of 2πperiodic functions. It can be easily shown that condition (v) for non-negative functions follows from the following condition: (vi) the function λ(l) l 2 is non-increasing on (0, +∞). Note 4: Readers may refer to the paper of Konyagin [38] in support of Remark 4. Readers may also refer to the paper of Weiss and Zygmund [39] which dealt with conditions on the second order modulus of smoothness sufficient to force absolute continuity of a function. The technique employed in [39] is nearly identical to that of [40]. Remark 5.
We write

The Main Theorems
Theorem 1. Error approximation of a conjugate functiong of 2π-periodic function g ∈ Z (λ) r , r ≥ 1, using H ≡ ( j,a ) of CFS is given by where λ(l) and ν(l) are as defined in Remark 3, provided Theorem 2. Error approximation of a conjugate derived conjugate functiong ′ of a 2π-periodic function g ∈ Z (λ) r , r ≥ 1, using H ≡ ( j,a ) of CDFS is given by where λ(l) and ν(l) are as defined in Remark 3, provided

Lemmas
We use the following lemmas: Lemmas 1 and 2 are proved using similar arguments to those used in the proofs of Singh and Srivastava ( [25], Lemma 1).
Since ν(l) is positive and increasing and for l ≤ z, we have )︂)︂ .