Vol. 48, No. 2, 1973

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ISSN: 0030-8730
Matrix summability of a class of derived Fourier series

Aribindi Satyanarayan Rao

Vol. 48 (1973), No. 2, 481–484
Abstract

Let f be L-integrable and periodic with period 2π, and let

∑∞
n (bn cosnx − ansinnx)
n=1
(1.1)

be the derived Fourier series of the function f with partial sums sn(x). We write

ψx(t) = f(x+ t)− f(x− t);

      -ψx-(t)-
gx(t) = 4 sint∕2.

In this paper, the following theorems are established.

Theorem 1. Let A = (amn) be a regular infinite matrix of real numbers. Then, for every x [π,π] for which gx(t) is of bounded variation on [0],

    ∑∞      ′
lmi→m∞    amns n(x) = gx(0+ )
n=1
(1.2)

if and only if

     ∞∑
lim     amnsin(n + 1∕2)t = 0 for all t ∈ [0,π].
m→ ∞n=1
(1.3)

Theorem 2. Let A = (amn) be an almost regular infinite matrix of real numbers. Then, for each x [π,π] for which gx(t) is of bounded variation on [0],

      p− 1
lim  1 ∑  t′   (x) = g (0+ )
p→ ∞ p j=0 m+j       x

uniformly in m if and only if

    1p∑−1 ∞∑
lp→im∞ p       am+j,n sin(n+ 1∕2)t = 0 for all t ∈ [0,π],
j=0n=1

uniformly in m, where

 ′      ∞∑      ′
tm(x) =    amnsn(x).
n=1

Mathematical Subject Classification 2000
Primary: 42A24
Milestones
Received: 26 May 1972
Published: 1 October 1973
Authors
Aribindi Satyanarayan Rao