Keywords: matrix means, degree of approximation, generalized Lipschitz class functions
Turkish Journal of Analysis and Number Theory, 2014 2 (6),
pp 198201.
DOI: 10.12691/tjant262
Received October 06, 2014; Revised November 15, 2014; Accepted November 23, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The Fourier series associate with f at point x of 2π periodic function in is given by
 (1.1) 
A function f ∈Lip if
, if
for 0 < α ≤ 1 and an integer .
if
provided is positive increasing function.
If then Lip coincide with Lip and if in Lip than Lip reduce to Lip α.
We observed that Lip for.
We define norm by
 (1.2) 
The degree of approximation E_{n} (f) is given by
 (1.3) 
where is a trigonometric polynomial of degree n.
Let be an infinite lower triangular matrix satisfying the condition(see, ^{[4]}) of regularity. Let infinite series such that whose partial sum s.
The sequencetosequence transformation defines the sequence of lower triangular matrix means of the sequence {s_{n}} generated by the sequence of coefficients (a_{n,k}).
The series is said to be summable to the sum s by lower triangular matrix method (see, ^{[1]}) if .
In this paper, we use following notations.
 (1.4) 
 (1.5) 
where is the greatest integer not greater than (1/t) and
 (1.6) 
2. Main Theorem
Chandra^{[5]} proved a theorem on the approximation of function belonging to Lip class by Nörlund and Riesz means. Mittal et. al. ^{[3]} extended the result of Chandra ^{[5]} by using the matrix means on same Lip class function. In ^{[6]}, Lal & Dhakal proved a theorem on approximation of a Lip class function by matrix means.
Aim of this paper is to extend the theorems of Chandra ^{[5]}, Mittal et.al. ^{[3]} and Lal & Dhakal ^{[6]} by using matrix means on Lip class functions as following way:
Theorem. Let T=(a_{n.k}) be an infinite regular lower triangular matrix such that the element (a_{n.k}) be nonnegative, nondecreasing with . If a function is 2πperiodic, Lebesgue integrable on [π, π] and belonging to Lip class then the degree of approximation of f by lower triangular matrix means of its Fourier series (1.1) satisfies, for n= 0,1,2,3…,
 (2.1) 
provided ξ (t) satisfies the following conditions:
 (2.2) 
 (2.3) 
where is an arbitrary number such that , q the conjugate index of p and conditions (2.2) & (2.3) hold uniformly in x.
3. Lemmas
For the proof of the theorem following lemmas are required.
Lemma 1. M_{n} (t) = O (n +1), if .
Proof. For , ,
Lemma 2. If (a_{n,k}) is non negative and non decreasing with , then,
, uniformly in .
Proof: Let Then
by Abel’s lemma,
and
therefore
Also, .
Thus
Lemma 3. , if
Proof: For , sin(t/2),we have
4. Proof of the Theorem
Following ^{[2]}, we have
then,
or
 (4.1) 
For, I_{1, }
Applying Holder’s inequality and fact that , we have
by condition (2.2) & Lemma 1.
Applying Second Mean Value Theorem for Integrals, we have
 (4.2) 
Applying Holder’s inequality, condition (2.3) and Lemma 2.
taking
 (4.3) 
Combining the conditions (4.1) – (4.3), we have
Now,
 (4.4) 
This completes the proof of the theorem.
5. Corollaries
Following corollaries can be derived from the theorem.
Corollary 1. If then degree of approximation of a function class is given by
for
Corollary 2. If ^{ }with , then degree of approximation of a function class is given by
References
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 In article  

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[6]  S. Lal and B. P. Dhakal, On Approximation of functions belonging to Lipschitz class by triangular matrix method of Fourier series, Int. Journal of Math. Analysis, 4(21), 2010, 10411047. 
 In article  
