Let's look at the sine series in
. Consider
for . From calculus its easy to see that
is orthonormal. We claim that it is in fact dense in
.
Take
such that
for all . Extend to
by
for . Consider also
the function defined (almost everywhere) on the unit circle
by
.
Notice
because is an odd
function and
for all .
So we have that
for all
because polynomials in and
are uniformly dense in
by the Stone-Weierstrass Theorem and
in
.
If a sequence
converges to uniformly, then
For any arc in
,
is the
-limit of a sequence of
continuous functions on
, so
.
For every measurable subset
we get a sequence
of
countable unions of open arcs with
and the measure of
converges to zero
by the regularity of Lebesque measure. So
for all implies
. Thus if is zero almost everywhere, then
is zero almost everywhere.
We have shown that
, and thus
for all . For any choice of that is readily integrable against each
, we get an amusing series summation. For instance, gives
Then
,
so
implies
and