# Convolutions

The motivation of convolutions is below

Fact 2.3.1 Note that

$$S_N(f)(x)=(f*D_N)(x)$$

Definition 2.3.2 Suppose that $f,g\in \mathcal R(S^1)$, we define their convolutions $f*g$ by

$$(f*g)(x)=\dfrac1{2\pi}\int^\pi_{-\pi}f(t)g(x-t)\mathrm dt$$

Loosely speaking, convolutions correspond to weighted averages.

Property 2.3.3 Suppose that $f,g,h\in \mathcal R(S^1)$,then

1. $f*(g+h)=(f*g)+(f*h)$
2. $(cf)*g=c(f*g)=f*(cg),\forall c\in\mathbb C$
3. $f*g=g*f$
4. $(f*g)*h=f*(g*h)$
5. $f*g$ is continuous
6. $\widehat{f*g}(n)=\hat f(n)\hat g(n)$

From 5 we see that the $f*g$ is more regular than $f,g$

# Good kernels

The motivation of good kernels is below

Fact 2.4.1 We use the family of trigonometric polynomials below to proof Theorem 2.2.1

$$G_k=(\cos\theta +\epsilon)^k$$

as a result, we can isolate the behavior of $f$ at the origin.

Definition 2.4.2 A fmaily of kernels $\{K_n(x)\in \mathcal R(S^1)\}^\infty_{n=1}$ is said to be a family of good kernels if it satisfies the following properties

1. For all $n\geq 1$,

$$\dfrac 1{2\pi}\displaystyle\int^\pi_{-\pi}K_n(x)\mathrm dx=1$$

2. There exists $M>0$ such that for all $n\geq 1$

$$\int ^\pi_{-\pi}|K_n(x)|\mathrm dx\leq M$$

3. For every $\delta >0$,

$$\int_{\delta\leq |x|\leq \pi}|K_n(x)|\mathrm dx\to 0,\text{ as } n\to \infty$$

The importance of good kernels is highlighted by their use in connection with convolutions.

Theorem 2.4.3 Let $\{K_n\}^\infty_{n=1}$ be a family of good kernels ,and $f\in\mathcal R(S^1)$. Then

$$(f*K_n)(x)\to f(x),\text{ as }n\to \infty$$

whenver $f$ is continuous at $x$. Espeicially if $f\in C(S^1)$, then

$$(f*K_n)(x)\rightrightarrows f(x),\text{ as }n\to\infty$$

Because of the result, the family $\{K_n\}$ is referred to as an approximation to the identity

It is natural now for us to ask whether $D_N$ is a good kernel. Unfortunately, this is not the case, for

Fact 2.4.4 Note that

$$\int^\pi_{-\pi}|D_N(x)|\mathrm dx\geq c\ln N,\text{ as }n\to\infty$$

Considering that

$$S_N(f)(x)=(f*D_N)(x)$$

the conclusion suggests that the pointwise convergence of Fourier series is intricate, and may even fail at points of continuity.

# Cesàro and Abel summability

The motivation of Cesàro and Abel summability is upon. Since a Fourier series may fail to converge at individual points, we are led to try overcome this failure by interpreting the limit

$$\lim_{N\to\infty}S_N(f)=f$$

in a different sense.

# Cesàro means and summation

Definition 2.5.1 Let $\sigma_N$ be

$$\sigma_N=\dfrac{S_0+...+S_{N-1}}{N}$$

the quantity $\sigma_N$ is called the $N^{th}$ Cesàro mean of the sequence $\{S_k\}$ or the $N^{th}$ Cesàro sum of the series $\displaystyle \sum^{\infty }_{k=0}c_k$, and if

$$\sigma_N\to\sigma,\text{ as }N\to\infty$$

we say that the series $\displaystyle \sum c_k$ is Cesàro summable to $\sigma$.

And Cesàro summation is more inclusive than convergence.

Property 2.5.2 If a series is convergent to $s$, then it is Cesàro summable to $s$, but the converse doesn't work.

Though Dirichlet kernels fail to belong to the family of good kernels, their averages do form.

We form $N^{th}$ Cesàro mean of the Fourier series, which by definition is

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

# Fejér kernel

Definition 2.5.3 The $N^{th}$ Fejér kernel is given by

$$F_N(x)=\dfrac{D_0(x)+...+D_{N-1}(x)}{N}=\dfrac 1N\dfrac {\sin^2(Nx/2)}{\sin^2(x/2)}$$

The closed form is easy to calculate. And we have such observation that

Property 2.5.4 Note that

$$\sigma_N(f)(x)=(f*F_N)(x)$$

Property 2.5.5 The Fejér kernel is a good kernel.

Now applying Theorem 2.4.3 to Fejér kernel yields the following result

Theorem 2.5.6 If $f\in \mathcal R(S^1)$, then

$$\sigma_N(f)\to f,\text{ as } N\to\infty$$

at every point of continuity of $f$. Moreover, if $f\in C(S^1)$, then

$$\sigma_N(f)\rightrightarrows f,\text{ as }N\to\infty$$

Corollary 2.5.7 If $f\in C(S^1)$, then it can be uniformly approximated by trigonometric polynomials.

Corollary 2.5.7 is the periodic analogue of the Weierstrass approximation theorem for polynomials.

# Abel means and summation

Definition 2.5.8 A series of complex numbers $\displaystyle \sum^\infty_{k=0}c_k$ is said to be Abel summable to $s$ if the series

$$A(r)=\sum^\infty_{k=0}c_kc^k,\ \forall r\in [0,1)$$

converges, and

$$A(r)\to s,\ r\to 1^-$$

The quantities $A(r)$ are call the Able means of the series.

Property 2.5.9 If a series is convergent to $s$, then it is Abel summable to $s$, but the converse doesn't work. Moreover, when the series is Cesàro summable, it is always Abel summable to the same sum, but the converse doesn't work.

Loosely, we have the graph to show the connection between these kinds of summation

$$S\subset S(C)\subset S(A)$$

# Poisson kernel

Definition 2.5.10 If $f\in\mathcal R(S^1)$, we define the Able means of the function $f(\theta)$

$$A_r(f)(\theta)=\sum^\infty_{n=-\infty}r^{|n|}a_ne^{in\theta}$$

It is to adapt Abel summability to the context of Fourier series.

$$f(\theta)\sim \displaystyle \sum^\infty_{n=-\infty}a_ne^{in\theta}$$

And it's easy to tell that $A_r(f)$ converges absolutely and uniformly for each $0\leq r<1$.

Definition 2.5.11 For $0\leq r<1$, Poisson kernel is given by

$$P_r(\theta)=\sum^\infty_{n=-\infty}r^{|n|}e^{in\theta}$$

Property 2.5.11 Note that

$$A_r(f)(\theta)=(f*P_r)(\theta)$$

Lemma 2.5.12 If $0\leq r<1$, then

$$P_r(\theta)=\dfrac{1-r^2}{1-2r\cos \theta+r^2}$$

Property 2.5.13 Poisson kernel is a good kernel, as $r\to 1^-$.

Now applying Theorem 2.4.3 to Fejér kernel yields the following result

Theorem 2.5.14 If $f\in \mathcal R(S^1)$, then

$$A_r(f)\to f,\text{ as } r\to1^-$$

at every point of continuity of $f$. Moreover, if $f\in C(S^1)$, then

$$A_r(f)\rightrightarrows f,\text{ as }r\to1^-$$