1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.5 Operations on Fourier Series 85

1.5 Operations on Fourier Series

In the course of this book we shall have to perform certain operations on
Fourier series. The purpose of this section is to find conditions under which
they are legitimate. Two things must be noted, however. First, the theorems
stated here are not the best possible: There are theorems with weaker hypothe-
ses and the same conclusions. Second, in applying mathematics, we often carry
out operations formally, legitimate or not. The results must then be checked
for correctness.
Throughout this section we shall state results about functions and Fourier
series with period 2π, for typographic convenience. The results remain true
when the period is 2ainstead. For functions defined only on a finite interval,
the periodic extension must fulfill the hypotheses. We shall refer to a function
f(x)with the series shown:

f(x)∼a 0 +

∑∞

n= 1

ancos(nx)+bnsin(nx). (1)

Theorem 1.The Fourier series of the function cf(x)has coefficients ca 0 ,can,and
cbn(cisconstant).

This theorem is a simple consequence of the fact that a constant passes
through an integral. The fact that the integral of a sum is the sum of the inte-
grals leads to the following.

Theorem 2.The Fourier coefficients of the sum f(x)+g(x)are the sums of the
corresponding coefficients of f(x)and g(x).

These two theorems are so natural that the reader has probably used them
already without thinking about it. The theorems that follow are much more
difficult to prove, but they are extremely important.

Theorem 3.If f(x)is periodic and sectionally continuous, then the Fourier series
of f may be integrated term by term:

∫b

a

f(x)dx=

∫b

a

a 0 dx+

∑∞

n= 1

∫b

a

(

ancos(nx)+bnsin(nx)

)

dx. (2)

Theorem 4.If f(x)is periodic and sectionally continuous and if g(x)is sectionally
continuous for a≤x≤b, then