Approximation of function using generalized Zygmund class

In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g′, where g′ is a derived function of a 2π -periodic function g, in the generalized Zygmund class X (η) z , z ≥ 1, using matrix-Cesàro (TC ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.


Introduction
In the past few decades, researchers have been greatly interested in studying the error estimation of functions in different function spaces using summability operators due to their various applications in science and engineering. In this direction, several researchers [11][12][13][14][15][16][17][18][19][20][21] have obtained results on error estimation of functions in different Lipschitz and Hölder classes using different single summability operators. Taking a view point that a product summability operator is more effective than the individual single summability operator, researchers [5,9,[21][22][23][24] have obtained the degree of approximation of functions in different Lipschitz and Hölder classes by different product summability operators.
After reviewing the above mentioned work, we observe that all of the above works cannot provide the best approximation of a function in the function spaces considered. This fact strongly motivated us to consider a more advanced class of functions, which can provide the best approximation of a function.
Therefore, in the present work, we establish a theorem on the best error approximation of a function g in the generalized Zygmund class X (η) z (z ≥ 1) by using the matrix-Cesàro (TC δ ) product operator of its Fourier series.
In the recent past, researchers have also been greatly interested in studying the derived Fourier series by single or product means. In this direction, Chandra and Dikshit [25] have studied |B| and |E, q| means of derived Fourier series. Lal and Nigam [26] have studied Karamata means of derived Fourier series. Lal and Yadav [27] have considered the (N, p, q)(C, 1) product means of derived Fourier series. However, no one has studied the degree of approximation of functions in any function space by using single or product means of its derived Fourier series. This fact has also motivated us to pursue a study of the degree of approximation of a function in a generalized Zygmund class by the matrix-Cesàro (TC δ ) method of its derived Fourier series. Therefore, in this paper, we also establish a theorem on the best error approximation of a function in the generalized Zygmund class X (η) z (z ≥ 1) by the matrix-Cesàro (TC δ ) product of its derived Fourier series. It is important to note that we have considered the TC δ product operator, which is the most general product operator developed for matrix-Cesàro means.
A separate study of derived Fourier series in the present direction of work is justified, due to its important applications in science and engineering.
In the last six years, investigators [1,2] were working on the error estimation of a function in the X (η) z (z ≥ 1) space using different summability operators. In both works [1,2], the second theorem has been proved by considering η(l) lξ (l) as a nonincreasing function l, in addition to the condition that η(l) ξ (l) is nondecreasing, which was considered in their first theorems.
Since the modulus of continuity η is a subadditive function, η(l) l is a nonincreasing function of l, and the second theorem in each of the above works follows from the first theorem without any additional condition.
Let ∞ j=0 d j be an infinite series having jth partial sum s j = j ν=0 d ν . Under usual assumptions on the function g, the Fourier series of g is given by with the jth partial sums s j (g; y), and the conjugate Fourier series of g is given bỹ (a j sin jyb j cos jy).
The series which is obtained by differentiating (1) term-by-term, is called the first derived Fourier series of g(y) (Zygmund,[28]).
Let T ≡ (a j,r ) be an infinite triangular matrix satisfying the conditions of regularity [29], i.e., j r=0 a j,r = 1 as j → ∞; each r ≥ 0 is such that a j,r = 0 as j → ∞; ( 4 ) The sequence-to-sequence transformation a j,r s r = j r=0 a j,j-r s j-r defines the sequence t T j of triangular matrix means of the sequence {s j }, generated by the sequence of coefficients (a j,r ).
If t T j → s as j → ∞, then the infinite series ∞ j=0 d j or the sequence {s j } is summable to s by the triangular matrix (T) [28]. We , and E δ j for the value of A δ j when a 0 = 1 and a j = 0 for j > 0, i.e., when A j = 1. If and E δ j = j+δ δ , then we say that ∞ j=0 d j or the sequence {s j } is summable to the sum s by C δ (the Cesàro means of order δ) [30].
Superimposing the T-method on the C δ method, TC δ is obtained. That is, the TC δ mean of the sequence {s j } is given by If t TC δ j → s as j → ∞, then {s j } is summable by the TC δ means to the limit s.
Since the T and C δ methods are regular, the TC δ method is also regular. This can be shown as follows: Remark 1 The TC δ means reduces to: where p j and q j have their usual meanings; (iii) (N, p j )C δ or N p C δ if a j,r = p j-r P j , P j = j kr=0 p j = 0, q j = 1 ∀j, where p j has its usual meaning; (iv) (Ñ, p j )C δ orÑ p C δ if a j,r = p r P j , q j = 1 ∀j, where q j has its usual meaning; Remark 2 In view of Remark 1, TC δ (δ = 1) mean also reduces to HC 1 , N p,q C 1 , N p C 1 , Then {s j } of (5) is given by Take Then t T j = n r=0 a j,r = a j,0 s 0 + a j,1 s 1 + · · · + a j,j s j = 1 5 j j 0 4 j · 1 -j 1 4 j-1 · 9 + j 2 4 j-2 · 9 2 + · · · + (-1) n n n 4 n-n · 9 n = 1 5 j (4 -9) j = (-1) j . Here, We observe that (5) is not summable by C 1 means. If a j,r = 1 5 j j r 4 j-r , then (5) is also not summable by T means. But (5) is summable by TC 1 means as (6) is summable by C 1 means. This shows the effectiveness of the product means as compared to single means.
Let C 2π denote the Banach space of all 2π -periodic and continuous functions defined on [0, 2π] under the supremum norm [28].
The jth order error approximation of a function g ∈ C 2π is defined by E j (g) := inf t j gt j where t j is a trigonometric polynomial of degree j [28].
If E j (g) → 0 as j → ∞, the E j (g) is said to be the best approximation of g [28].
The L z space is given by The norm · r is defined by Let η : [0, 2π] → R be an arbitrary function with η(l) > 0 for 0 < l ≤ 2π and lim l→0 + η(l) = η(0) = 0. We define z . The completeness of L z (z ≥ 1) implies the completeness of the space [28]. One can also refer to the papers [31,32] for more details on the Zygmund space.
Remark 3 Throughout the paper, η and ξ denote the second order moduli of continuity such that η(l) ξ (l) is positive and nondecreasing in l. Then Thus, Remark 4 Necessary and sufficient conditions for a function to be a modulus of continuity of the first order were pointed out by Lebesgue [33] and Nikol'skii [34]. Any modulus of continuity of the first order, η = η(g, ·), satisfies the following conditions: (iv) the function η is semiadditive, i.e., the inequality η(l 1 + l 2 ) ≤ η(l 1 ) + η(l 2 ) holds for any l 1 ≥ 0 and l 2 ≥ 0.
Conversely, if a function η satisfies (i)-(iv), then it is the first-order modulus of continuity of the function g(y) = η(|y|). Moreover, it can be easily seen that η is the second order modulus of continuity of the function g(y) = η(|y|) 2 . If a function η satisfies conditions (i)-(iii) and the function η(l) l is nonincreasing on (0, +∞), then the semiadditivity condition (iv) also holds, and so η is the modulus of continuity of the first and second order for some continuous functions.
The second order modulus of continuity satisfies conditions (i)-(iii) and a further condition, given as follows: (v) the inequality η(jl) ≤ j 2 η(l) holds for any l ≥ 0 and j ∈ N.
Geit [35] constructed a wide class of functions that are second-order moduli of continuity of 2π -periodic functions. It can be easily shown that condition (v) for nonnegative functions follows from the following condition: (vi) the function η(l) l 2 is nonincreasing on (0, +∞).

Note 1
Readers may refer to the paper of Konyagin [36] in support of Remark 4. Readers may also refer to the paper of Weiss and Zygmund [37], which dealt with conditions on the second-order modulus of smoothness, sufficient to force absolute continuity of a function. The technique employed in [37] is nearly identical to that of [38].
Remark 5 Therefore, in view of Remark 4 and Note 1, we drop the second theorem established in the papers [1,2], etc., where the condition that η(l) lξ (l) is a nonincreasing function of l, in addition to the condition of their first theorem, is used.
We write

Theorems
Theorem 2.1 If g is a 2π -periodic function belonging to the class X (η) z , z ≥ 1, then the best error estimate of g by the TC δ method of its F.S. is given by where η(l) and ξ (l) are as defined in Remark 3, provided

Theorem 2.2 If g is a 2π -periodic function belonging to the class X
(η) z , then the best error estimate of g by the TC δ method of its D.F.S. is given by where η(l) and ξ (l) are as defined in Remark 3, provided

Lemma 3.2
Under the conditions of regularity of matrix T ≡ (a j,r ) for 1 j+1 ≤ l ≤ π , Proof For 1 j+1 ≤ l ≤ π , by applying Jordan's lemma and the facts that sin l 2 ≥ l π , sin jl ≤ 1, we get
Again, due to the monotonicity of the function ξ (l), Thus Using Remark 3, we have , which gives lξ (l) dl log π(j + 1) .

Corollary 6.16
Following Remark 2, we obtain Remark 7 (i) Corollary 6.1 can be further reduced using TC 1 means in view of Remark 2.