IJSRP JUNE 2012 Online Print Version - IJSRP.ORG

International Journal of Scientific and Research Publications, Volume 2, Issue 6, June 2012 15
ISSN 2250-3153
Particular Cases : Two particular cases of (E, q) (N, pn) means are :
1) (E, q) (C, 1) if pn = 1, n
n1 pn
1
, 0
2) (E, q) (C, ) if
The following theorems are due to Binod Prasad Dhakal [1].
Theorem A : If
1
1
p
,
f : R R
is 2π – periodic function, Lebesgue integrable on [-π , π] and Lip(α, p) class function for
p 1
, then the degree of approximation of f by the (E,1) (N, pn) mean of its Fourier series is given by :
EN
n
t x f x
1
p
1
p
1 n 1
n 1
n k
EN
tn n pkrsr
2 k0 k P r 0
Where, k
(E, 1) (N, pn) means of Fourier series (1.1).
,
is
II. MAIN THEOREM
The degree of approximation of function belonging to the Lipschitz class by (E, q) (C, 1) and by (E, 1) (N, p n) mean has discussed
by a number of researchers like S.K. Tiwari and Chandrashekhar Bariwal [5] and Binod Prasad Dhakal [1]. But till now no work seem
to have been done to obtain the degree of approximation of the function belonging to
mean of its Fourier series.
p,
W L t
In an attempt to make study in this direction, one theorem on the degree of approximation of function of
product summability mean of the form (E, q) (N, pn) has been determined as following.
Theorem : If
q
f : R R
is 2π – periodic, Lebesgue integrable [-π, π] and belonging to the class
E N
tnx using on its Fourier series (1.1) is given by.
(2.1)
Provided
(2.2)
and
(2.3)
t
q
1
E N 1 P
tn f ( n 1)
p
n 1
satisfies the following conditions :
1
p
P
n1
t ()
t
p 1
sin t dt
(
t) n1
n1
1
p
P
t (
t) sin t
dt n
()
t
1
class by (E, q) (N, pn) product
p,
W L t
p,
W L t
,
class by
p 1
by
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