Cesàro summability and Fejér’s theorem
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
giving us property 3.
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.
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.
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.
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.
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.