Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 6, 198-201
Available online at http://pubs.sciepub.com/tjant/2/6/2
© Science and Education Publishing
DOI:10.12691/tjant-2-6-2
Using the Matrix Summability Method to Approximate
the Lip (ξ(t), p) Class Functions
Binod Prasad Dhakal
*
Central Department of Education (Mathematics), Tribhuvan University, Nepal
*Corresponding author: binod_dhakal2004@yahoo.com
Received October 06, 2014; Revised November 15, 2014; Accepted November 23, 2014
Abstract Most of the summability methods are derived from the matrix means. In this paper, author has been
determined the degree of approximation of certain trigonometric functions belonging to the Lip (ξ(t), p) class by
matrix method.
Keywords: matrix means, degree of approximation, generalized Lipschitz class functions
Cite This Article: Binod Prasad Dhakal, “Using the Matrix Summability Method to Approximate the Lip
(ξ(t), p) Class Functions.” Turkish Journal of Analysis and Number Theory, vol. 2, no. 6 (2014): 198-201.
doi: 10.12691/tjant-2-6-2.
1. Introduction
The Fourier series associate with f at point x of 2π
periodic function in (-π, π) is given by
( )
0 n n
1
1
( )~ a cos nx b sin nx .
2
n
fx a
∞
=
+ +
∑
(1.1)
A function f ∈Lip α if
( )
( ) () , 0 1. f x t f x O t for
α
α + − = < ≤
( , ), 0 2 f Lip p for x α π ∈ ≤ ≤ , if
( )
1
2
0
( ) () ,
p
p
f x t fx dx O t
π
α
+ − =
∫
for 0 < α ≤ 1
and an integer 1 p ≥ .
( ) ( ), f Lip t p ξ ∈ if
1
( ) () ( ( ))
b
p
p
a
fx t fx dx O t ξ
+ − =
∫
provided () t ξ is positive increasing function.
If ξ(t) = t
α
then Lip (ξ(t), p) coincide with Lip(α, p) and
if p → ∞ in Lip(α, p) than Lip(α, p) reduce to Lip α.
We observed that Lip α ⊆ Lip (α, p) ⊆ Lip (ξ(t), p)
for 0 < α ≤ 1.
We define norm .
p
by
1
2
0
() , 1.
p
p
p
f f x dx p
π
= ≥
∫
(1.2)
The degree of approximation E
n
(f) is given by
( ) min
n n
p
E f t f = − (1.3)
where t
n
(x) is a trigonometric polynomial of degree n.
Let T=(a
n,k
) be an infinite lower triangular matrix
satisfying the condition(see, [4]) of regularity. Let
0
n
m
u be an
∞
=
∑
infinite series such that whose
th
n partial
sum s
n
=
0
n
k
k
u
=
∑
.
The sequence-to-sequence transformation
,
0
n
n nk k
k
t a s
=
=
∑
defines the sequence { }
n
t of lower
triangular matrix means of the sequence {s
n
} generated by
the sequence of coefficients (a
n,k
).
The series
0
n
n
u
∞
=
∑
is said to be summable to the sum s
by lower triangular matrix method (see, [1]) if lim
n
n
t s
→∞
= .
In this paper, we use following notations.
() ( ) ( ) ( ) 2 t f x t f x t f x ϕ = + + − − (1.4)
, ,
,
n
n nk
k n
A a
τ
τ =−
=
∑
(1.5)
where
1
t
τ
=
is the greatest integer not greater than (1/t)
and
( )
n
1
,
2
0
M (t)
sin 1/2
sin( / 2)
n
nk
k
k t
a
t
π
=
+
=
∑
. (1.6)