Best approximation of functions in generalized Hölder class

Here, for the first time, error estimation of the functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g\in H_{z}^{(w)}$\end{document}g∈Hz(w) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{g}\in H_{z}^{(w)}$\end{document}g˜∈Hz(w) classes using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$TC^{1}$\end{document}TC1 method of F. S. (Fourier Series) and C. F. S. (Conjugate Fourier Series), respectively, are determined. The results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013) become the particular cases of our Theorem 2.1. Some important corollaries are also deduced from our main theorems.

Our motivation for this work is to consider a more advanced class of functions that can provide best approximation by a trigonometric polynomial of degree not more than r. Therefore, in this work, we generalize the results of Kushwaha and Dhakal [3] and Dhakal [1,2]. In fact, we obtain the results on the error estimation for the function f ∈ H (w) z (z ≥ 1) by T.C 1 method by F. S. Thus, the results of Kushwaha and Dhakal [3] and Dhakal [1,2] become the particulars cases of our Theorem 2.1.
We also obtain the results on the error estimation of the functiong ∈ H (w) z (z ≥ 1) by T.C 1 method of C. F. S.
Let "T = (a r,m ) be an infinite triangular matrix satisfying the conditions of regularity [13], i.e.,  The sequence-to-sequence transformation t T r := r m=0 a r,m s m = r m=0 a r,r-m s r-m (2) defines the sequence t T r of triangular matrix means of the sequence {s r } generated by the sequence of coefficients (a r,m ).
If t T r → s as r → ∞, then the infinite series ∞ r=0 h r or the sequence {s r } is summable to s by a triangular matrix (T-method) [14]. " "Let C 1 r = s 0 + s 1 + · · · + s r r + 1 If C 1 r → s as r → ∞, then the infinite series ∞ r=0 h r is summable to s by C 1 means [14]. " The TC 1 means (T-means of C 1 means) is given by If t T.C 1 r → s as r → ∞, then the series ∞ r=0 h r or the sequence {s r } is summable to s by T.C 1 means.
The regularity of T and C 1 methods implies the regularity of T.C 1 method.
Remark 1 (Example) Consider an infinite series The nth partial sum of (5) is given by Therefore, series (5) is not summable by (C, 1) means. If we take a n,k = 1 n+1 , then series (5) is also not summable by T means. But series (5) is summable by T.C 1 means. So, the product means is more powerful than the individual means.

Note 1 w(l)
and v(l) denote "Zygmund moduli of continuity [14]. " If we consider w(l) v(l) as positive and non-decreasing, Thus, Remark 4 We are not representing here the F. S. and C. F. S. as these trigonometric series are well known and the detailed work on these series can be found in [14]. We denote the rth partial sum of the F. S. as The rth partial sum of C. F. S. is defined as "The error estimation of function g is given by where t r is a trigonometric polynomial of degree r [14]. " We write .

Main theorems
where T = (a r,m ) is an infinite triangular matrix satisfying (1) and w, v are defined as in Note 1 provided class; z ≥ 1 and w(l) v(l) are positive and non-decreasing, then the error estimation ofg by TC 1 means of C. F. S. is where T = (a r,m ) is an infinite triangular matrix satisfying (1), (6) and w, v are defined as in Note 1.
Proof This lemma can be proved along the same lines as the proof of Lemma 3.5(iii).

Proof of the main theorems 4.1 Proof of Theorem 2.1
Proof Following Titchmarsh [17], s r (g; x) of F. S. is given by dl. dl, Let Then "Using generalized Minkowski's inequality Chui [18], " we get Using Lemmas 3.1 and 3.5(iii), we have Also, using Lemmas 3.2 and 3.5(iii), we get By (9), (10), and (11), we have Again applying Minkowski's inequality, Lemma 3.1, Lemma 3.2, and φ(·, l) z = O(w(l)), we obtain Now, we have Using (12) and (13), we get By the monotonicity of v(l), Since w and v are moduli of continuity such that w(l) v(l) is positive and non-decreasing, therefore . Then From (16) and (17), we get

Proof of Theorem 2.2
Proof The integral representation of s r (g; x) is given by dl.

Conclusion
Approximation by trigonometric polynomials is at the heart of approximation theory. Much of the advances in the theory of trigonometric approximation are due to the periodicity of the functions. The study of error approximation of periodic functions in Lipschitz and Hölder classes has been of great interest among the researchers [1][2][3][4][5][6][7][8][9][10][11], and [12] in recent past. The trigonometric Fourier approximation (TFA) is of great importance due to its wide applications in different branches of engineering such as electronics and communication engineering, electrical and electronics engineering, computer science engineering, etc. Several elegant results on TFA can be found in a monograph [14].
In this paper, we, for the first time, obtain the best approximation of the functions g and g in a generalized Hölder class H (w) r (r ≥ 1) using Matrix-C 1 (T.C 1 ) method of F. S. and C. F. S. respectively. Since, in view of Remark 2, the product summability means H.C 1 , N p C 1 , N p,q C 1 , andN p C 1 are the particular cases of Matrix-C 1 method, so our results also hold for these methods, which are represented in a form of corollaries. In view of Remark 1, it has been shown that (TC 1 ) method is more powerful than the individual T method and C 1 method. Moreover, in view of Remark 5, some previous results (see Sect. 6) become the particular cases of our Theorem 2.1. We also deduce a corollary for the H α,r class (r ≥ 1).
Some other studies regarding the modulus of continuity (smoothness) of functions using more generalized functional spaces may be addressed as a future work.