Summability Methods
1. Cesàro Sums and Fejér Kernels
Recall that we may express partial sums of the Fourier series in terms of convolutions with Dirichlet kernels, i.e.,
\[ S_N f(x) := \sum_{n=-N}^N \widehat{f}(n) e^{2 \pi i nx} = f * D_N(x) \]
where
\[ D_N(x) = \sum_{n=-N}^N e^{2 \pi i n x} = \frac{\sin (2N+1) \pi x}{\sin \pi x}. \]
(From this point forward, we restrict attention to the case of \(1\)-periodicity. All other cases may be derived from this one using a change of variables.)
Understanding the convergence properties of Fourier series turns out to be surprisingly delicate, particularly in comparison to the much simpler behavior of Taylor series. There is a very real sense in which this complexity is caused by the fact that partial sums are in some sense a crude way of approximating infinite series. Here, the crude feature is that some terms (namely, those with index between \(-N\) and \(N\)) are included at “full strength” in the \(N\)-th symmetric partial sum, while nearby terms are entirely absent from the same partial sum. It turns out that the convergence properties can be greatly enhanced by smoothly transitioning between inclusion and exclusion so that adjacent terms are always handled in similar ways.
The simplest way to accomplish this is to take what are known as Cesàro partial sums. Roughly speaking, we simply average the first \(N\) symmetric partial sums:
\[ T_N f(x) := \frac{1}{N} \sum_{k=0}^{N-1} S_N f(x). \]
Using what we know about the partial sums, it's got to be the case that the Cesàro means are also given by convolution with \(f\) against kernels, this time known as Fejér kernels:
\[ T_N f(x) = f * F_N(x) \]
where
\[ F_N(x) := \frac{1}{N} \sum_{k=0}^{N-1} D_k(x). \]
Proposition
The following formulas hold for the Fejér kernels:
\[ F_N (x) = \sum_{n=-N+1}^{N-1} \left( 1 - \frac{|n|}{N}\right) e^{2 \pi i n x}, \]
\[ F_N (x) = \frac{1}{N} \left( \frac{\sin \pi N x}{\sin \pi x} \right)^2. \]
(In the latter case, the expression is understood to equal \(N\) when \(x=0\).) From the first formula, we can see that the Fourier coefficients of \(F_N\) transition linearly from \(\widehat{F_N}(0) = 1\) to \(\widehat{F_N}(\pm N) = 0\) (and \(\widehat{F_N}(k) = 0\) for \(|k| > N\) also.)
Proof
Meta (Main Idea)
For the first identity, we use the fact that
\[{}\frac{1}{N} \sum_{k=0}^{N-1} \sum_{j=-k}^{k} e^{2 \pi i j x}{}\]
\[{}= \frac{1}{N} \sum_{k=0}^{N-1} \sum_{j = -N+1}^{N-1} \chi_{|j| \leq k} e^{2 \pi i j x}{}\]
\[{}= \frac{1}{N} \sum_{j = -N+1}^{N-1} \sum_{k=0}^{N-1} \chi_{|j| \leq k} e^{2 \pi i j x}{}\]
\[{}= \frac{1}{N} \sum_{j = -N+1}^{N-1} \left( \sum_{k=0}^{N-1} \chi_{|j| \leq k} \right) e^{2 \pi i j x}{}\]
\[{}= \frac{1}{N} \sum_{j = -N+1}^{N-1} \left( N-|j| \right) e^{2 \pi i j x}{}\]
\[{}= \sum_{j = -N+1}^{N-1} \left( 1 - \frac{|j|}{N} \right) e^{2 \pi i j x}.{}\]
For the second identity, we sum the trigonometric expressions for the Dirichlet kernels themselves and use the identity
\[{}\sin \pi x + \cdots + \sin (2N-1) \pi x{}\]
\[{}= \operatorname{Im} \left( \sum_{k=0}^{N-1} e^{(2k+1)\pi i x} \right){}\]
\[{}= \operatorname{Im} \left( e^{\pi i x} \frac{e^{2 \pi i N x}-1}{e^{2 \pi i x}-1} \right){}\]
\[{}= \operatorname{Im} \left( \frac{e^{2 \pi i N x}-1}{e^{\pi i x} - e^{- \pi i x}} \right){}\]
\[{}= \operatorname{Im} \left( \frac{e^{2 \pi i N x}-1}{2 i \sin \pi x} \right){}\]
\[{}= - \operatorname{Re} \left(\frac{e^{2 \pi i N x}-1}{2 \sin \pi x} \right){}\]
\[{}= \frac{1 - \cos 2 \pi N x}{2 \sin \pi x}{}\]
\[{}= \frac{1 - \cos^2 \pi N x + \sin^2 \pi N x}{2 \sin \pi x}{}\]
\[{}= \frac{\sin^2 \pi N x}{\sin \pi x}.{}\]
Below is a plot of \(F_4, F_8\), and \(F_{16}\) on the interval \([-\frac{3}{2},\frac{3}{2}]\).
Notice how the kernels become more and more concentrated around the integers. The decay away from integer points is much more rapid than was the case for Dirichlet kernels. This turns out to be what makes these kernels nicer from the standpoint of convergence.
2. Uniform Approximation of Continuous Functions
Proposition (Key Properties of Fejér Kernels)
- For each \(N\), \(F_N(x) \geq 0\) for all \(x\).
- For each \(N\), \(\int_{-1/2}^{1/2} F_N(x) dx = 1\).
- For each \(\delta > 0\), \(\int_{\delta < |x| < \frac{1}{2}} F_N(x) dx \rightarrow 0\) as \(N \rightarrow \infty\).
Proof
Theorem (Uniform Approximation via Convolution with Good Kernels)
If \(\{F_N\}_{N=1}^\infty\) is a family of good kernels which are \(1\)-periodic, then for any continuous, 1-periodic function \(f\), \(f * F_N\) converges uniformly to \(f\) as \(N \rightarrow \infty\). In particular, there is pointwise convergence \(f * F_N(x) \rightarrow f(x)\) at every point \(x\) as \(N \rightarrow \infty\).
The first step in the proof of this theorem is to write out \(f * F_N(x) - f(x)\) and simplify:
\[{}f * F_N (x) - f(x){}\]
\[{}= \int_{-\frac{1}{2}}^{\frac{1}{2}} F_N(y) f(x-y) dy - f(x){}\]
\[{}= \int_{-\frac{1}{2}}^{\frac{1}{2}} F_N(y) \left[ f(x-y) - f(x) \right] dy{}\]
There are two things to observe. The first is that we have used symmetries of convolution to write the integral (in \(y\)) as an integral over \([-\frac{1}{2},\frac{1}{2}]\). We have also chosen \(F_N\) to be a function of \(y\) only inside the integral (so that \(f\) is evaluated at \(x-y\)). The other important trick is that we combine the two terms into one on the last line. This can be done exactly because the integral of \(F_N(y)\) is one, so
\[ f(x) = f(x) \cdot 1 = f(x) \int_{-\frac{1}{2}}^{\frac{1}{2}} F_N(y) dy. \]
Now we use the triangle inequality and the fact that the \(F_N\)'s are nonnegative:
\[{}|f * F_N (x) - f(x)|{}\]
\[{}\leq \int_{-\frac{1}{2}}^{\frac{1}{2}} F_N(y) \left| f(x-y) - f(x) \right| dy.{}\]
Because \(f\) is periodic and continuous, it is uniformly continuous. Thus there is some \(\delta\) (without loss of generality less than \(1/2\)) such that \(|y| < \delta\) implies \(|f(x-y) - f(x)| < \epsilon/2\) for any fixed choice of \(\epsilon\). Because the integral of each \(F_N\) is at most \(1\), it follows that
\[{}|f * F_N (x) - f(x)|{}\]
\[{}\leq \int_{-\delta}^{\delta} F_N(y) \left| f(x-y) - f(x) \right| dy{}\]
\[{}+ \int_{\delta < |x| < \frac{1}{2}} \! \! F_N(y) \left| f(x-y) - f(x) \right| dy{}\]
\[{}\leq \frac{\epsilon}{2} \int_{-\delta}^{\delta} F_N(y) dy{}\]
\[{}+ \int_{\delta < |x| < \frac{1}{2}} \! \! F_N(y) \left| f(x-y) - f(x) \right| dy{}\]
\[{}\leq \frac{\epsilon}{2}{}\]
\[{}+ \int_{\delta < |x| < \frac{1}{2}} \! \! F_N(y) \left| f(x-y) - f(x) \right| dy.{}\]
For the remaining integral, since \(f\) is continuous and periodic, it is bounded in magnitude by some constant \(M\). Thus
\[{}|f * F_N (x) - f(x)|{}\]
\[{}\leq \frac{\epsilon}{2} + 2M \int_{\delta < |x| < \frac{1}{2}} F_N(y) dy.{}\]
By the third property of good kernels, we see that the second term on the right-hand side can be made less than \(\epsilon/2\) for all sufficiently large indices \(N\). This choice is independent of the particular value of \(x\), so the convergence is uniform.
Note
If the function \(f\) is merely continuous at some point, we can modify this argument slightly to show that \(f * F_N(x) \rightarrow f(x)\) at this particular value of \(x\) provided that everything makes sense (e.g., if \(f\) is Riemann integrable). The convergence would not be uniform anymore in such a case.
Exercise
Show that Fourier coefficients uniquely determine continuous functions. In other words, show that, if \(f\) and \(g\) are continuous, 1-periodic functions with the property that \(\widehat{f}(n) = \widehat{g}(n)\) for all \(n \in {\mathbb Z}\), then \(f = g\).