ON DEGREE OF APPROXIMATION BY PRODUCT MEANS OF CONJUGATE SERIES OF A FOURIER SERIES

In this paper a theorem on degree of approximation of a function f ∈ Lip(α , r) by product summability (E, q )(N , pn) of conjugate series of Fourier series associated with f has been established.


Introduction
Let Σ a n be a given infinite series with the sequence of partial sums {s n }. Let  The sequence -to-sequence transformation (1.2) defines the sequence { t n } of the (N , p n ) -mean of the sequence {s n } generated by the sequence of coefficient {p n }. If (1.3) then the series Σ an is said to be (N , p n ) summable to s. The conditions for regularity of (N , p n )-summability are easily seen to be [1] (1.4) The sequence -to-sequence transformation, [1] (1.5) defines the sequence {T n } of the (E , q ) mean of the sequence {s n }. If (1.6) then the series Σ a n is said to be (E , q ) summable to s . Clearly (E , q ) method is regular. Further, the (E , q ) transform of the (N , p n ) transform of {s n } is defined by then Σ a n is said to be (E, q)(N , p n )-summable to s .
Let f (t ) be a periodic function with period 2 and L-integrable over ) . The Fourier series associated with f at any point x is defined by (1.9) and the conjugate series of the Fourier series (1.9) is (1.10) Let s n ( f ; x) be the n-th partial sum of (1.10).The L -norm of a function f : R  R is defined by (1.11) and the L v -norm is defined by The degree of approximation of a function f : R  R by a trigonometric polynomial P n ( x) of degree n under norm . is defined by [5]. BSMaSS Volume 4 We use the following notation throughout this paper : ( 1.17) and (1.18) Further, the method (E, q )(N , p n ) is assumed to be regular .

Known Theorem
Dealing with The degree of approximation by the product (E, q ) (C ,1) -mean of Fourier series, Nigam et al [3] proved the following theorem: If a function f is 2 -periodic and belonging to class Lipa , then its degree of approximation by (E, q ) (C ,1) summability mean on its Fourier series is given by where represents the (E , q )transform of (C ,1)

transform of s n ( f ; x).
Recently, Misra et al [2] proved the following theorem using (E, q )(N , p n ) mean of conjugate series of the Fourier series : given where  n is as defined in 1.7) .

Main theorem
In this paper, we have proved a theorem on degree of approximation by the product mean (E, q )(N , p ) of conjugate series of Fourier series of a function of class Lip (a , r ) . We prove: where  is as defined in (1.7) .

Required Lemmas
We require the following Lemmas to prove the theorem. This proves the lemma.

Proof of theorem 3.1
Using Riemann -Lebesgue theorem, we have for the n-th partial sum s n ( f ; x) of the conjugate