1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.6 Mean Error and Convergence in Mean 91

is a finite number. Let

f(x)∼a 0 +

∑∞

n= 1

ancos

(nπx

a

)

+bnsin

(nπx

a

)

and letg(x)have a finite Fourier series

g(x)=A 0 +

∑N

1

Ancos

(

nπx

a

)

+Bnsin

(

nπx

a

)

.

Then we may perform the following operations:

∫a

−a

f(x)g(x)dx=

∫a

−a

f(x)

[

A 0 +

∑N

1

Ancos

(nπx

a

)

+Bnsin

(nπx

a

)]

dx

=A 0

∫a

−a

f(x)dx+

∑N

1

An

∫a

−a

f(x)cos

(

nπx

a

)

dx

+

∑N

1

Bn

∫a

−a

f(x)sin

(nπx

a

)

dx.

We recognize the integrals as multiples of the Fourier coefficients off and
rewrite

1

a

∫a

−a

f(x)g(x)dx= 2 a 0 A 0 +

∑N

1

(anAn+bnBn). (1)

Now suppose we wish to approximatef(x)by afiniteFourier series. The
difficulty here is deciding what “approximate” means. Of the many ways we
can measure approximation, the one that is easiest to use is the following:

EN=

∫a

−a

(

f(x)−g(x)

) 2

dx. (2)

(Heregis the function with a Fourier series containing terms up to and in-
cluding cos(Nπx/a).) Clearly,ENcan never be negative, and iffandgare
“close,” thenENwill be small. Thus our problem is to choose the coefficients
ofgso as to minimizeEN.(WeassumeNfixed.)
To c o m p u t eEN, we first expand the integrand:

EN=

∫a

−a

f^2 (x)dx− 2

∫a

−a

f(x)g(x)dx+

∫a

−a

g^2 (x)dx. (3)

The first integral has nothing to do withg; the other two integrals clearly de-
pend on the choice ofgand can be manipulated so as to minimizeEN.We