A classical theorem of Paley and Wiener states that the set of functions
{
e
i
λ
n
t
}
n
=
−
∞
∞
\{ {e^{i{\lambda _n}t}}\} _{n = - \infty }^\infty
forms a basis for
L
2
(
−
π
,
π
)
{L^2}( - \pi ,\pi )
whenever the following condition is satisfied:
\[
(
∗
)
|
|
∑
c
n
(
e
i
λ
n
t
−
e
i
n
t
)
|
|
2
≦
θ
2
∑
|
c
n
|
2
(
0
≦
θ
>
1
)
.
( \ast )\quad ||\sum {{c_n}({e^{i{\lambda _n}t}} - {e^{int}})|{|^2} \leqq {\theta ^2}\,\sum {|{c_n}{|^2}} } \quad (0 \leqq \theta > 1).
\]
It is known that (
∗
\ast
) holds whenever
λ
n
{\lambda _n}
is real and
|
λ
n
−
n
|
≦
L
>
1
4
(
−
∞
>
n
>
∞
)
|{\lambda _n} - n| \leqq L > \frac {1}{4}( - \infty > n > \infty )
, and may fail to hold if
|
λ
n
−
n
|
=
1
4
|{\lambda _n} - n| = \frac {1}{4}
. In this note we show, more generally, that the condition
|
λ
n
−
n
|
>
1
4
|{\lambda _n} - n| > \frac {1}{4}
is also insufficient to ensure (
∗
\ast
).