6
MARIA CRISTINA PEREYRA
[Hu].
The case p
=
1 had been ruled out by Kolmogorov's famous example of an
integrable function whose Fourier series diverges everywhere, see
[Kol].
By a limiting procedure on the unit disc, similar to the one described in the
upper half plane, one can conclude that the boundary values of the harmonic conju-
gate of the harmonic extension of a periodic, real-valued, continuously differentiable,
function
f
on
TI'
are given by
(1.2)
- 111
f(t)
Hf(x)
=
p.v.- ( ( )
dt;
1r
0
tan
1r
x - t)
here we are identifying x with z
=
e21rix.
The singularity at the diagonal is comparable to that of the Hilbert transform;
so this would be the appropriate analogue of the Hilbert transform on the unit circle.
On Fourier side, one can check that a similar identity holds, namely: (
fi
fY' (
n)
=
-isgn(n)](n).
Note that the Fourier transform of the partial Fourier sum of a nice function is
also given by a similar Fourier multiplier:
(SN ft(n)
=
Xiki::;N(n)}(n).
EXERCISE
1.4. Check that
Xiki::;N(n)
=
~(sgn(n-
N)-
sgn(n
+
N)).
Further-
more remember that the Fourier transform maps modulations into translations,
more precisely, check that if
MNJ(B)
=
f(x)e
21rilm,
then
(MNf)f'·(n)
=
](n- N).
Finally check that
i(MNHM_N)f'(n)
=
sgn(n-
N)](n).
Similarly check that:
i(M-NHMN )f'(n)
=
sgn(n
+
N)](n).
The exercise implies that
SN
=
~(MNfiM_N-
M_NfiMN)·
EXERCISE
1.5. Show that the
SN's
are uniformly (inN) bounded in
LP,
for
each 1 p oo. Deduce, from the Uniform Boundedness Principle, that
Therefore the convergence in
LP
of the partial Fourier sums is a consequence
of the boundedness of the Hilbert transform in those spaces.
1.1.3.
Connection to stationary processes.
In the 50's Wiener and Massani
studied stationary Gaussian processes, see
[MW].
A
discrete stationary process
is a sequence {
~n
}nEZI:
of random variables in the probability space (D, P) such
that
E(~n)
=
0 and
E(~~)
oo
3
;
and
E(~n~k)
=
'T(k-
n)
(stationary condition).
This last condition implies that the correlation matrix has a Toeplitz structure.
Notice that the sequence
{'l(k)}kEZI:
is a positive definite
sequence4.
They were interested in the geometry of such a sequence of random variables
in L
2
(D,
P).
The inner product is naturally given by
(~,
ry)
=
E(~TJ),
and let
1{
be
the closure of the linear span of the sequence {
~n
}nEZI:
in the norm induced by this
inner product. The problem they had was to predict
6
knowing the predecessors
~k,
k
:S:
0. The best predictor in the Hilbert space context is the orthogonal projection
of
6
onto the span of {
~k
:
k
:S:
0}.
The celebrated Herglotz-Bochner-Schwartz theorem allows us to move into
more familiar ground: given a positive definite sequence
{'l(k)}kEZI:
there exists
a unique positive measure
p,
2:
0 on the unit circle
TI'
(parameterized by
z
=
3
as usual the expectation is given by E(.;) =
j
0
.;dP
4i.e.
L:k,n "!(k- n)xkXn
= El
L:k Xk.;kl
2 2:
0 for all Xk·
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