Abstract
1.1. In the theorems which follow we are concerned with functions
f
(
x
) real for real
x
and integrable in the sense of Lebesgue. We do not, however, remain in the field of the real variable, for we suppose, in 4
et seq
., that
f
(
x
), or a function associated with
f
(
x
), is analytic, or, at any rate, harmonic, in a region of the complex plane associated with the particular real value of
x
considered. The Fourier’s series considered are those associated with the interval (0, 2π). If
a
is a point of the interval, we write
ϕ(u)
= ½ {
f
(
a
+
u
) +
f
(
a
-
u
) - 2
s
} (0 <
a
< 2π), (1.11)
ϕ(u)
= ½ {
f
(
u
) +
f
(2π -
u
) - 2
s
} (
a
= 0,
a
= 2π), (1.12) where
s
is a constant.
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