A Note on the |N,p,q| Summability of a Factored Fourier Series

Abstract

In this paper I have proved a theorem on |N,p,q| summability of a
factored Fourier series, which generalizes various known results. However
the theorem is as follows.
Theorem: If F(t) is a function of bounded variation in (0,p ) then the
factored Fourier series.
( * )
( )
p q n n A t n
n
l
Σ a
is | N, p, q | summable at t = x where the sequences {pn} and {qn} are
non-negative non-increasing such that
(i) ( 1)
( * )
n pn
p q n
 + 
 
 
is of bounded variation
(ii) ( 1)
( * )
n qn
p q n
 + 
 
 
is of bounded variation
(iii) ( * ) p q n
na
 
 
 
is bounded
(iv) 1
p
n
pn
   + 
 
is non-decreasing
(v) 1
q
n
qn
   + 
 
is non-decreasing
and {ln}is a convex sequence such that n
n
l
Σ < ¥ .