Vol. 33, No. 1, 1970

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ISSN: 0030-8730
On C, 1 summability factors of Fourier series at a given point

Fu Cheng Hsiang

Vol. 33 (1970), No. 1, 139–147
Abstract

Let f(αj) be a function integrable in the sense of Lebesgue over the interval (π,π) and periodic with period 2π. Let its Fourier series be

f(x) a0
2 + n=1(a n cosnx + bn sinnx)
n=0A n(x).
Whittaker proved that the series
∑∞
An (x)∕na  (α > 0)
n=1

is summable |A| almost everywhere. Prasad improved this result by showing that the series

 ∞         k−1
∑         ∏    μ      k  1+𝜖     k
n=n An (x)∕(   log  n)(log n)     (log n0 > 0)
0       μ=1

is summable |A| almost everywhere.

In this note, the author is interested particularly in the |C,1| summability factors of the Fourier series at a given point x0.

Write

φ(t) = f(x0 + t) + f(x0 t) 2f(x0),Φ(t) = 0t|φ(u)|du.
The author establishes the following theorems.

Theorem 1. If

Φ(t) = O(t) (t → +0),

then the series

∞∑
An(x0)∕nα
n=1

is summable |C,1| for every α > 0.

Theorem 2. If

Φ(t) = O{∏k---t--μ 1-}
μ=1log -t

as t +0, then the series

 ∞∑  --------An(x0)-------
(∏k −1logμn)(logkn )1+𝜖
n=n0   μ=1

is summable |C,1| for every 𝜖 > 0.

Mathematical Subject Classification
Primary: 42.20
Milestones
Received: 28 February 1969
Published: 1 April 1970
Authors
Fu Cheng Hsiang