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 $\sum_{n \in \textbf{N}} c_n$ of complex numbers, recall that the $N^{th}$ partial sum $s_N$ of the series is defined as $s_N = \sum_{n=0}^{N}c_k.$ Define the $N^{th}$ Cesàro mean $\sigma_N$ as

$\sigma_N = \displaystyle\frac{s_0 + s_1 + \cdots + s_{N-1}}{N}.$

It is the average of the first $N$ partial sums. We say that a series $\sum c_n$ is Cesàro summable to $\sigma$ if it’s sequence of Cesàro means $\{\sigma_N\}$ converges to $\sigma$. For example, take  the series $1 - 1 + 1 - 1 + \cdots =\sum_{n \in \textbf{N}}(-1)^n.$ The partial sums of this series form the sequence $\{1, 0, 1, 0, \cdots \}$ 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 $1/2$ in a weaker sense than usual. Well, taking the Cesàro means $\sigma_N$, we see that $\sigma_N \rightarrow 1/2$ as $N \rightarrow \infty,$ so $\sum_{n \in \textbf{N}}(-1)^n$ is in fact Cesàro summable to $1/2$. 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 $\sum c_n$ converges to $s$, then $\sum c_n$ is Cesàro summable to $s$.

So why is Cesàro summability important to us? Recall that at the end of my last post, I mentioned that the Dirichlet kernels $D_N$ isn’t a family of good kernels. However, their averages are! Consider the $N^{th}$ Cesàro mean of the Fourier series,

$\sigma_N(f)(x) = \displaystyle\frac{S_0(f)(x) + \cdots + S_{N-1}(f)(x)}{N}.$

Recall from my last post that the convolution is distributive on additon. Since $S_n(f) = f * D_n,$ we thus get $\sigma_N(f)(x) = (f*F_N)(x)$, where $F_N(x)$ is the $N^{th}$ Fejér kernel given by

$F_N(x) := \displaystyle\frac{D_0(x) + \cdots D_{N-1}(x)}{N}.$

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

$F_N(x) = \displaystyle\frac{1}{N} \frac{\sin^2(Nx/2)}{\sin^2(x/2)}.$

(The proof uses Lagrange’s trigonometric identities, which will be derived in exercice 3). From this expression, we see that $\int_{-\pi}^{\pi}|F_n(x)|dx = \int_{-\pi}^{\pi}F_n(x)dx$ since $F_N(x)$ is positive. As remarked in my last post, we have $\frac{1}{2\pi}\int_{-\pi}^{\pi} D_N(x)dx = 1,$ so the same is true of $F_N$ (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 $\delta \leq |x| \leq \pi$, then it is easy to see that $\sin^2(x/2) \geq c_{\delta}$ for some $c_{\delta} > 0.$ But then $F_N(x) \leq \frac{1}{Nc_{\delta}},$ from which it follows that

$\displaystyle\int_{\delta \leq |x| \leq \pi}|F_N(x)|dx \rightarrow 0$ as $N \rightarrow \infty,$

giving us property 3.

QED

The theorem about good kernels proven in my last post now gives

Theorem (Fejér): If $f$ is an integrable function on the circle, then the Fourier series of $f$ is Cesàro summable to $f$ at every point of continuity of $f.$ Moreover, if $f$ is continuous on the circle, then the Fourier series of $f$ is uniformly Cesàro summable to $f.$

Let’s now look at some important corollaries of Fejér’s theorem:

Corollary 1: If $f$ is integrable on the circle and $\hat{f}(n) = 0$ for all $n$, then $f = 0$ at all points of continuity of $f.$

Proof: Since the partial sums are given by $S_N(f)(x) = \sum_{n=-N}^{N} \hat{f}(n)e^{inx}$, they are all 0, hence all the Cesàro means are 0. We therefore get $\sigma_N(f) \rightarrow 0$ as $N \rightarrow \infty$. But the theorem says that $\sigma_N(f) \rightarrow f$ at all point of continuity of $f$, so by the uniqueness of the limit, $f = 0$ at those points.

QED

Note that this implies that for functions $f$ and $g$ on the circle, $S(f) = S(g) \Rightarrow f = g$ except maybe at points of discontinuity of $f$ or $g.$

Corollary 2: If the Fourier series of a function $f$ converges, it converges to $f$ at all points of continuity of $f.$

Proof: If the Fourier series of a function $f$ converges to some function $g,$ we know from exercice 2 that it is Cesàro summable to $g$ everywhere. But the theorem says that it is Cesàro summable to $f$ at all points of continuity. Hence $f = g$ 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 $f$ on the circle are implicitly identified with continuous functions on $[-\pi, \pi]$ with $f(-\pi) = f(\pi).$ Thus we get a proof of the following:

Corollary 4 (Weierstrass): Let $f$ be a continuous function on the closed and bounded interval $[a,b] \in \textbf{R}.$ Then, for any $\epsilon > 0,$ there exists a polynomial $P$ such that

$\displaystyle\sup_{x \in [a,b]} |f(x)-P(x)| < \epsilon.$

Proof: Without loss of generality, we can assume that $[a,b] = [0, \pi].$ Indeed, we may consider the function $\tilde{f}(x) = f(x(\frac{b-a}{\pi}) + a)$ defined on $[0, \pi],$ which is just $f$ in disguise. Then we extend it to a periodic function on $[-\pi, \pi]$ by declaring $\tilde{f}(x) = \tilde{f}(-x)$ for $x < 0.$ Now corollary 3 applies so there exists a trigonometric polynomial $p(x) = \sum c_n e^{inx}$ approximating $\tilde{f}$ uniformly. But we know that $e^{inx} = \sum_{k=0}^{\infty} \frac{(inx)^k}{k!}$ 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.