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).

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