Cesàro summability and Fejér’s theorem
This is the follow-up of my last post, Convolution and good kernels. I am still following Stein and Shakarchi’s Fourier Analysis.
Given a series of complex numbers, recall that the
partial sum
of the series is defined as
Define the
Cesàro mean
as
It is the average of the first partial sums. We say that a series
is Cesàro summable to
if it’s sequence of Cesàro means
converges to
. For example, take the series
The partial sums of this series form the sequence
which obviously does not have a limit so it is a divergent series in the usual sense. However, one could argue that this is not a wildly divergent series, that it could be convergent to
in a weaker sense than usual. Well, taking the Cesàro means
, we see that
as
so
is in fact Cesàro summable to
. Moreover, we’ll see in exercice 2 that Cesàro summability “fits in” nicely with the usual theory of convergent series in the sense that if
converges to
, then
is Cesàro summable to
.
So why is Cesàro summability important to us? Recall that at the end of my last post, I mentioned that the Dirichlet kernels isn’t a family of good kernels. However, their averages are! Consider the
Cesàro mean of the Fourier series,
Recall from my last post that the convolution is distributive on additon. Since we thus get
, where
is the
Fejér kernel given by
Lemma (Stein’s book p. 53): The Fejér kernels form a family of good kernels.
Proof: First, we will see in exercice 4 that a closed form formula for the Fejér kernels is given by
(The proof uses Lagrange’s trigonometric identities, which will be derived in exercice 3). From this expression, we see that since
is positive. As remarked in my last post, we have
so the same is true of
(the Fejér kernel being the average value of Dirichlet kernels). This gives us property 1 and 2 of the good kernels. For the last property, note that if
, then it is easy to see that
for some
But then
from which it follows that
as
giving us property 3.
QED
The theorem about good kernels proven in my last post now gives
Theorem (Fejér): If
is an integrable function on the circle, then the Fourier series of
is Cesàro summable to
at every point of continuity of
Moreover, if
is continuous on the circle, then the Fourier series of
is uniformly Cesàro summable to
Let’s now look at some important corollaries of Fejér’s theorem:
Corollary 1: If
is integrable on the circle and
for all
, then
at all points of continuity of
Proof: Since the partial sums are given by , they are all 0, hence all the Cesàro means are 0. We therefore get
as
. But the theorem says that
at all point of continuity of
, so by the uniqueness of the limit,
at those points.
QED
Note that this implies that for functions and
on the circle,
except maybe at points of discontinuity of
or
Corollary 2: If the Fourier series of a function
converges, it converges to
at all points of continuity of
Proof: If the Fourier series of a function converges to some function
we know from exercice 2 that it is Cesàro summable to
everywhere. But the theorem says that it is Cesàro summable to
at all points of continuity. Hence
at all points of continuity.
QED
Note that I didn’t get this last corollary from a book so maybe I’m missing something.
Corollary 3: Continuous functions on the circle can be uniformly approximated by trigonometric polynomials.
Proof: This is immediate since the partials sums, hence the Cesàro means, are trigonometric polynomials.
QED
Note that continuous functions on the circle are implicitly identified with continuous functions on
with
Thus we get a proof of the following:
Corollary 4 (Weierstrass): Let
be a continuous function on the closed and bounded interval
Then, for any
there exists a polynomial
such that
Proof: Without loss of generality, we can assume that Indeed, we may consider the function
defined on
which is just
in disguise. Then we extend it to a periodic function on
by declaring
for
Now corollary 3 applies so there exists a trigonometric polynomial
approximating
uniformly. But we know that
with uniform convergence on every compact.
QED
I will now write the solutions to the exercises mentioned in the last two posts and resume my study of the Dirichlet problem in the unit disk in the next one.
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