4 1. The Classical Modular Forms
plays a special role in the linear space of periodic functions. Any periodic
and piecewise continuous function / : R C has the Fourier series repre-
sentation
(1.3) f(x) =
y^2,ane{nx)
with coefficients given by
(1.4) an = f(x)e(—nx)dx.
Jo
If / is continuous, the Fourier series (1.3) converges absolutely, hence uni-
formly.
Exercise. Prove that the Fourier expansion for the fractional part of x is
given by the series
^
KX r
. 1 v-^ sin27rnx
(1-5)
W = ~ - 2^
2 ^— f nn
n=l
which is boundedly convergent (the partial sums are uniformly bounded),
but not absolutely. Precisely, we have
r ^ 1 \—vsin27rnx ^,,+ ,. .,*r\-i\
Z *—*( nn
n—l
where ||x|| denotes the distance from x to the nearest integer and the implied
constant is absolute.
Developing (1.1) into Fourier series, we find by unfolding the integral
(1.4) that
/
oo
f(x)e(—nx)dx = /(n)
-oo
where f(y) denotes the Fourier transform of f(x) on K;
/
oo
f(x)e(-xy)dx.
-OO
Therefore the Fourier expansion (1.3) appears as
(1.7) ^rf(x + n) = Ylf(n)e(nx).
n£Z n€Z
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