�
�
87
LECTURE
NOTES
FOR
18.102,
SPRING
2009
Lecture
14.
Tuesday,
March
31:
Fourier
series
and
L
2
(0
,
2
π
)
.
Fourier
series.
Let
us
now
try
applying
our
knowledge
of
Hilbert
space
to
a
concrete
Hilbert
space
such
as
L
2
(
a, b
)
for
a
finite
interval
(
a, b
)
⊂
R
.
You
showed
that
this
is
indeed
a
Hilbert
space.
One
of
the
reasons
for
developing
Hilbert
space
techniques
originally
was
precisely
the
following
result.
Theorem
12.
If
u
∈
L
2
(0
,
2
π
)
then
the
Fourier
series
of
u,
(14.1)
1
�
c
k
e
ikx
, c
k
=
�
u
(
x
)
e
−
ikx
dx
2
π
k
∈
Z
(0
,
2
π
)
converges
in
L
2
(0
,
2
π
)
to
u.
Notice
that
this
does
not
say
the
series
converges
pointwise,
or
pointwise
almost
everywhere
since
this
need
not
be
true
–
depending
on
u.
We
are
just
claiming
that
1
�
(14.2)
lim

u
(
x
)
−
2
π
c
k
e
ikx

2
= 0
n
→∞

k
≤
n
for
any
u
∈
L
2
(0
,
2
π
)
.
First
let’s
see
that
our
abstract
Hilbert
space
theory
has
put
us
quite
close
to
proving
this.
First
observe
that
if
e
�
k
(
x
)
=
exp(
ikx
)
then
these
elements
of
L
2
(0
,
2
π
)
satisfy
�
�
2
π
�
0
if
k
=
j
(14.3)
e
�
k
e
�
j
=
exp(
i
(
k
−
j
)
x
) =
�
0
2
π
if
k
=
j.
Thus
the
functions
e
�
k
1
ikx
(14.4)
e
k
=
=
e
�
e
�
k
�
√
2
π
form
an
orthonormal
set
in
L
2
(0
,
2
π
)
.
It
follows
that
(14.1)
is
just
the
FourierBessel
series
for
u
with
respect
to
this
orthonormal
set:
(14.5)
c
k
=
√
2
π
�
u, e
k
�
=
1
c
k
e
ikx
=
�
u, e
k
�
e
k
.
⇒
2
π
So,
we
alreay
know
that
this
series
converges
in
L
2
(0
,
2
π
)
thanks
to
Bessel’s
identity.
So
‘all’
we
need
to
show
is
Proposition
21.
The
e
k
, k
∈
Z
,
form
an
orthonormal
basis
of
L
2
(0
,
2
π
)
,
i.e.
are
complete:
(14.6)
ue
ikx
= 0
∀
k
=
⇒
u
= 0
in
L
2
(0
,
2
π
)
.
This
however,
is
not
so
trivial
to
prove.
An
equivalent
statement
is
that
the
finite
linear
span
of
the
e
k
is
dense
in
L
2
(0
,
2
π
)
.
I
will
prove
this
using
Fej´
er’s
method.
In
this
approach,
we
check
that
any
continuous
function
on
[0
,
2
π
]
satisfying
the
additional
condition
that
u
(0)
=
u
(2
π
)
is
the
uniform
limit
on
[0
,
2
π
]
of
a
sequence
in
the
finite
span
of
the
e
k
.
Since
uniform
convergence
of
continuous
functions
certainly
implies
convergence
in
L
2
(0
,
2
π
)
and
we
already
know
that
the
continuous
functions
which
vanish
near
0
and
2
π
are
dense
in
L
2
(0
,
2
π
)
(I
will
recall
why
later)
this
is
enough
to
prove
Proposition
21.
However
the
proof
is
a
serious
piece
of
analysis,
at
least
it
is
to
me!