Lecture 3

Motivation: Convolution and approximation¶

Last time we worked on convolution and showed how to approximate an integrable function by continuous function by convolution. The following code shows the approximation process in problem 4 of problem set 1. It plots $\Pi*\Lambda_{1/n}$ for $n = 5,10,20$.

x = PolynomialRing(QQ,'x').gen()
T = piecewise([[(-1,0),1+x],[(0,1),1-x]])
f = piecewise([[(-1/2,1/2),1]])
P = plot(f)
Q = plot(T)
#N=1
#TN=piecewise([[(-1/N,0),N*(1+x*N)],[(0,1/N),N*(1-x*N)]])
#TNF = f.convolution(TN)
#P=plot(TN)
for N in [5,10,20]:
    TN=piecewise([[(-1/N,0),N*(1+x*N)],[(0,1/N),N*(1-x*N)]])
    Q = Q + plot(TN,color='red')
    TNF=f.convolution(TN)
    P=P+plot(TNF)
P.show(title='convolution')
Q.show(title='lambda n')

The idea to do pointwise approximation is, suppose we want to approximate $f(x)$ by a sequence $f*g_n$. We will have to control $|f(x)-f*g_n(x)| = |\int_{0}^1f(x-y)g(y)dy - f(x)|$. With assumption $\int_{0}^1g_n(y)dy = 1$ for every $n$, we can write the RHS as $|\int_{0}^1(f(x-y) - f(x))g_n(y)dy| \leq \int_{0}^1|f(x-y)-f(x)||g_n(y)|dy$.

When in particular $g_n$ has very small support $(-\delta,\delta)$, the integral is taken on $y\in(-\delta,\delta)$, so that $f(x-y)$ is a small perturbation of $f$ near $x$. For example, when $f$ is continuous at $x$, the term $|f(x-y)-f(x)|$ will be small.

Good kernels¶

Let $(K_n)_{n=1}^{\infty}$ be a family of kernels. It is called good provided that it satisfies the following properties:

(1) $\int_{0}^1 K_n(x)dx = 1$ for every $n$.

(2) For every $\delta > 0$, $\int_{(0,1)\setminus(-\delta,\delta)}|K_n(x)|dx \to 0$.

(3) Either $K_n\geq 0$, or $\|K_n\|_{L^1}<M$ for some $M>0$ and all $n$.

A good family of kernels is also called an “approximation to the identity”.

The $\Lambda_n$ satisfies (1) has support shrinking to $\{0\}$, so it is automatically a good kernel. But the Fejer kernel and Dirichlet kernel has no “shrinking support”, as can be seen from appendix of lecture 1.

The notion of “good kernel” is designed so that the following holds:

Theorem. Let $(K_n)_{n=1}^{\infty}$ be a good family of kernels. Let $f\in L^1(\mathbb{T})$, then

• If $f$ is continuous at $x$, then $\lim_{n\to \infty}(f*K_n)(x) = f(x)$.
• If $f$ is continuous everywhere on $\mathbb{T}$, the limit is uniform.

Proof. The first properties implies

For $\epsilon > 0$ choose δ such that when $y\in (-\delta,\delta)$, $|f(x-y) - f(x)|<\epsilon$, so $\int_{-\delta}^{\delta}|f(x-y)-f(y)||K_n(y)|dy\leq \epsilon \int_{-\delta}^\delta |K_n(y)|dy \leq \epsilon M \cdot 2 \delta$.

The other part of integration is

$\int_{(0,1)\setminus(-\delta,\delta)})|f(x-y)-f(x)||K_n(y)|dy\leq 2\sup_{x\in [0,1]}|f(x)| \cdot \int_{(0,1)\setminus(-\delta,\delta)}|K_n(y)|dy \to 0$ when $n\to \infty$ by assumption (2) of good kernel.

We conclude that when $n$ is large enough (depend on $x$), $|f*K_n(x)-f(x)|\leq \int_{0}^1|f(x-y)-f(x)||K_n(y)|dy < C\epsilon$ for some constatn $C>0$.

If $f$ is everywhere continuous on $\mathbb{T}$, then it is uniformly continuous on $\mathbb{T}$, so the δ in the proof can be chosen independent of $x$, so the “when $n$ large enough part” no longer depend on $x$, this implies uniform convergence. $\square$

Cesaro means and Fejer kernel¶

Definition. (Cesaro sum) Let $f\in L^1(\mathbb{T})$. Let $S_n f$ be its $n$-th Fourier partial sum. The $N$-th Cesaro sum of $f$ is

Let $D_n = \sum_{k=-n}^n e^{2\pi i k x}$ be the Dirichlet kernels. Let

Then $\sigma_n(f)(x) = f*F_N(x)$. The $F_N$ is called Fejer kernel.

Similar to the Dirichlet kernel, the Fejer kernel has also a simpler form.

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

Proof. In problem set 2.

Lemma. The Fejer kernel is a good kernel.

Proof. Note that since $F_N\geq 0$, it sufficies to verify (1) and (2) in the definition of good kernel.

• To verify $(1)$, $\int_{-\frac{1}{2}}^{\frac{1}{2}} F_N(x)dx = \int_{0}^1 \frac{D_0(x)+\cdots+D_{N-1(x)}}{N}dx = \frac{N}{N} = 1$.
• To verify (2), let $A_\delta$ denote the set $[-\frac{1}{2},\frac{1}{2}]\setminus (-\delta,\delta)$. Observe that for $\delta > 0$, $F_N\to 0$ uniformly on $A_\delta$. Indeed, choose $c_\delta$ such that $\sin^2(\pi x)\geq c_\delta$, so $F_N(x)\leq \frac{1}{N}\cdot \frac{1}{c_\delta}$ on $A_\delta$, which goes to 0 when $N\to \infty$. By uniform convergence $\lim_{N\to \infty}\int_{A_\delta}F_N = \int_{A_\delta}\lim_{N\to \infty}F_N = 0.$

As a consequence of goodness of $F_N$, we obtain the Fejer’s theorem

Theorem. $f\in L^1(\mathbb{T})$. Then

• $\sigma_N(f)(x)\to f(x)$ when $f$ is continuous at $x$.
• If $f$ is continuous on $\mathbb{T}$, then the convergence is uniform.

In particular, we have showed that every continuous function on $\mathbb{T}$ can be uniformly approximated by trignomic polynomials, where the trignomic polynomials can be taken as its Cesaro sums. Later, we shall use this result to prove the $L^2$ convergence of Fourier series.

Further remarks on “approximated identity”¶

A family of good kernels are also called “an approximated identity”, because it is really an “approximation” to the “identity”. An “identity” element usually means something similar to 1, multiplying with the identity element will return itself. In our case, the identity is the “identity” of convolution. Imagine a “function” $I$ such that $f*I = f$, what shall the $I$ be? By the convolution theorem, $\hat{f}(n) = \widehat{f*I}(n) = \hat{f}(n)\hat{I}(n)$, so one has $\hat{I}(n) = 1$ for every $n$. Then the Fourier series of $I$ should look like $\sum_{n = -\infty}^{+\infty}e^{2\pi i n x}$, so $I$ should be something like “$\lim_{N \to \infty}D_N$”. With the notion of distributions and measures we can make this discussion precise and the “identity” will be the δ-measure, and an approximate identity is an approximation to the δ-measure in weak sense.

The “goodness” of good kernels is not restricted to continuous category. One can show for example that if $f\in L^p(\mathbb{T})$, then $\sigma_n f \to f$ in $L^p$. See Hoffman’s book chapter 2 for more general results in this direction.