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Cesàro summability and Fejér’s theorem

March 27, 2012

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.

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