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·