Brunei Math Club

Our main question has been: What type of functions can be represented as a Fourier series? We have seen two types of convergence of $S[f]$ to $f$.  Uniform convergence: If $f\in C_{2\pi }^1$ (continuously differentiable), then $S[f]$ converges uniformly to $f$. $L^2$ convergence: If $f\in {\mathcal{R}}_{2\pi }^2$ (square-integrable), then $S[f]$ converges to $f$ in the $L^2$ norm. These are ``global'' convergence properties, so to speak, in the sense that they characterize the function $f$ for the whole period $[-\pi ,\pi ]$. Now, we consider ``local'' convergence properties such as the problem of pointwise convergence.  To begin with, we have the following result: Theorem (du Bois-Reymond) Let $f$ be a periodic continuous function. Then, the Fourier series of $f$ may not converge pointwise. But we don't really need the continuity of $f$ for its Fourier series to converge to $f(x_0)$ at $x_0$. Thus, let's define the following. Defi...

Let us approximate a function $f(x)$ by a linear combination of $\{e^{ikx}|k=-n,\cdots ,0,\cdots ,n\}$. We show that the partial Fourier sum is the ``best'' of such approximators. Then, we prove that the Fourier series of the function $f$ converges to $f$ in the $L^2$ norm. Lemma (``best'' approximator) Let any $n\in \mathbb{N}$, consider the set of $2n+1$ functions of the form $${\phi }_k(x)=e^{ikx}(|k|\le n).$$ Let $v_n$ be an arbitrary linear combination of these functions: $$v_n(x)=\sum _{|k|\le n}{\gamma }_k{\phi }_k(x)$$ where ${\gamma }_k\in \mathbb{C}$ are constants. Suppose $f\in C_{2\pi }^0$. Then, $$\Vert f-S_n\Vert \le \Vert f-v_n\Vert$$ where $S_n=S_n[f]$ is the partial sum of the Fourier series $S[f]$. In other words, $S_n[f]$ is the best approximator of $f$ among all the linear combinations of $\{{\phi }_k{\}}_{|k|\le n}$ with respect to the $L^2$ norm. Proof .  \[\begin{eqnarray*} \|f - v_n\|^2 & = & \|(f ...

Here, we present a different way of looking at the Fourier series using  kernels . We see that the partial Fourier sum of a function $f(x)$ can be expressed as the convolution between $f$ and the Dirichlet kernel. We have already seen that the partial Fourier sum converges uniformly to $f$ if $f(x)$ is continuously differentiable. Similarly, we show that the convolution of the function $f$ and the Fejér kernel converges uniformly to $f$, but this time, if $f(x)$ is continuous. These kernels are used to prove the $L^2$ convergence of the Fourier series . Definition (Kernel function) Let $X$ be a set, and $K:X\times X\to \mathbb{R}$ be a two-variable function. $K(x,y)$ is said to be a  kernel function  (or  integral kernel  or  nucleus ) if it satisfies the following conditions: (symmetry) For any $x,y\in X$, $$K(x,y)=K(y,x).$$  (positive semi-definiteness) For any $n\in \mathbb{N}$, $\{x_1,x_2,\cdots ,x_n\}\subset X$...

We study the collection of functions ${\mathcal{R}}_{2\pi }^2$ (square-integrable functions with period $2\pi$) as a vector space. We define the $L^2$ norm and $L^2$ inner product on this vector space so that we can investigate the ``geometric'' structure of the space of functions.  Let us show that ${\mathcal{R}}_{2\pi }^2$, the set of square-integrable functions with period $2\pi$, is a vector space over $\mathbb{C}$. First, we need to define addition and scalar multiplication. Let $f,g\in {\mathcal{R}}_{2\pi }^2$. We define $f+g\in {\mathcal{R}}_{2\pi }^2$ by $$\begin{array}{}\text{(eq:add)}& (f+g)(x)=f(x)+g(x),x\in \mathbb{R}.\end{array}$$ Note that the ``$+$'' on the left-hand side is defined between the two functions $f$ and $g$, whereas the ``$+$'' on the right-hand side is the addition between two complex numbers $f(x)$ and $g(x)$. Next, we define scalar multiplication. Let $\alpha \in \mathbb{C}$ and $f\in {\mathcal{R}}_{2\pi }^2$. We define \(\alpha ...

Our goal in this post is to prove the following: If $f$ is ``piece-wise smooth,'' then the (complex) Fourier series of $f$ converges uniformly to $f$. Note that the condition in this theorem does not involve the Fourier coefficients of $f$; hence, it is more direct. That is, we can tell if the Fourier series of a function converges uniformly from the property of the function alone. Let $C_{2\pi }^m$ be the collection of all functions of class $C^m$ such that all derivatives up to the $m$-th order have period $2\pi$ .  We say a function $f$ is piece-wise smooth if $f\in C_{2\pi }^0$ (i.e., continuous) and $f^{\prime }\in {\mathcal{R}}_{2\pi }^2$ (i.e., the derivative is square-integrable, not necessarily continuous). In the following, we will consider the complex Fourier series: $$f\sim \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{e}^{ikx}$$ where the Fourier coefficients are $$c_k=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)e^{-ikx}dx.$$ Lemma (Bessel's inequality) If \(f\in\mathca...

The Fourier series we have studied so far is a special case of the complex Fourier series. In many cases, the complex Fourier series is more convenient than the (real) Fourier series. The complex Fourier series is defined as a series of the form $$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{e}^{ikx}$$ where $i=\sqrt{-1}$ is the imaginary unit and $c_k\in \mathbb{C}$. Note that the variable $x$ is still real, $x\in \mathbb{R}$.  In the following, the collections ${\mathcal{R}}_{2\pi }^1$ and ${\mathcal{R}}_{2\pi }^2$ also include complex-valued functions. If $f\in {\mathcal{R}}_{2\pi }^1$, the complex Fourier coefficients $c_k$ of $f$ are defined by $$\begin{array}{}\text{(eq:ck)}& c_k=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)e^{-ikx}dx.\end{array}$$ Here, the right-hand side is the definite integral of a complex-valued function. See also : Calculus of complex-valued functions For $m,n\in \mathbb{Z}$, if $m\ne n$, then \[\begin{eqnarray*} \int_{-\pi}^{\pi}e^{imx}e^{-inx}dx &=& \int_{-\pi}^{\pi}e^...

When a function is an odd function, its Fourier series contains only the sine functions. Such a Fourier series is called a Fourier sine series . When a function is an even function, its Fourier series contains only the cosine functions (and possibly a constant). Such a Fourier series is called a Fourier cosine series . Example . Consider the function $f\in {\mathcal{P}}_{2\pi }$ defined by $$f(x)=x(-\pi <x<\pi ).$$ Note that $f$ is an odd function (i.e., $f(-x)=-f(x)$). The Fourier coefficients are obtained as (exercise!) $$\begin{array}{rcl}{a}_n& =& \frac{1}{\pi }{\int }_0^{2\pi }f(x)\cos (nx)dx=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (nx)dx\\ & =& 0\\ b_n& =& \frac{1}{\pi }{\int }_0^{2\pi }f(x)\sin (nx)dx=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx.\\ & =& (-1{)}^{n+1}\frac{2}{n}.\end{array}$$ Thus, we have $$x\sim 2[\frac{\sin x}{1}-\frac{\sin (2x)}{2}+\frac{\sin (3x)}{3}-\cdots +(-1{)}^{n+1}\frac{\sin (nx)}{n}+\cdots ]$$ f...

To study the Fourier series, it is convenient to define collections of functions with a period of $2\pi$. Definition (Collection of periodic functions) We denote by ${\mathcal{P}}_{2\pi }$ the collection (set) of all functions on $\mathbb{R}$ that have the period of $2\pi$ and that have at most finitely many discontinuous points in each period. For each $f\in {\mathcal{P}}_{2\pi }$, its value may not be defined at its discontinuous points. Definition (Collections of integrable functions) We denote by ${\mathcal{R}}_{2\pi }^1$ the collection of all $f\in {\mathcal{P}}_{2\pi }$ such that $f$ absolutely integrable on the period, that is, $${\int }_0^{2\pi }|f(x)|dx<+\mathrm{\infty }.$$ We denote by ${\mathcal{R}}_{2\pi }^2$ the collection of all $f\in {\mathcal{P}}_{2\pi }$ such that $f$ is square-integrable, that is, $${\int }_0^{2\pi }|f(x){|}^{2}dx<+\mathrm{\infty }.$$ In these definitions, the integral may be improper. Remark . The $\mathcal{R}$ in ${\mathcal{R}}_{2\pi }^1$ and \(\mathcal{R}_{2\...

Joseph Fourier introduced the Fourier series to solve the heat equation in the 1810s. In this post, we show how the Fourier transform arises naturally in a simplified version of the heat equation. Suppose we have the unit circle $S$ made of a metal wire. Pick an arbitrary point $A$ on the circle. Any point $P$ on the circle is identified by the distance $x$ from $A$ to $P$ along the circle in the counter-clockwise direction (i.e., $x$ is the angle of the section between $A$ and $P$ in radian). Let $u(t,x)$ represent the temperature at position $x$ and time $t$. The temperature distribution at $t=0$ is given by $u(0,x)=f(x)$. Assuming no radiation of heat out of the metal wire, $u(t,x)$ for $t>0$ and $0\le x\le 2\pi$ is determined by the following partial differential equation  (PDE) called the  heat equation : $$\gamma \frac{\partial u}{\partial t}=\kappa \frac{{\partial }^{2}u}{\partial x^2}$$ and the initial condition \[u(0,x) = f(x...

We study some of the basic properties of the Fourier series. In particular, we show that if the series of the Fourier coefficients of $f$ converges absolutely, then Fourier series $S[f]$ converges uniformly to $f$ at the continuous points of $f$. Lemma (Condition for the uniform convergence of Fourier series) Let $f\in {\mathcal{R}}_{2\pi }^1$ with its Fourier coefficients satisfying the conditions $$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}|a_n|& <& +\mathrm{\infty },\\ \sum _{n=1}^{\mathrm{\infty }}|b_n|& <& +\mathrm{\infty }.\end{array}$$ Then, the Fourier series $S[f]$ converges uniformly with respect to $x\in \mathbb{R}$. In particular, $S[f]$ is continuous. Proof . Consider the following positive (non-negative) term series: $$\frac{1}{2}{a}_0+\sum _{n=1}^{\mathrm{\infty }}(|a_n|+|b_n|).$$ By assumption, this series converges. Furthermore, this is a dominating series of $S[f]$. Thus, $S[f]$ converges uniformly. All terms of $S[f]$ are continuous (they are jus...

The theory of the  Fourier series is based on a wild assumption: any "well-behaved" periodic function can be represented as a linear combination of sine and cosine functions, and that expression is unique for the given function. This theory (eventually) provided much of the foundations of modern mathematics. But it is also of tremendous practical importance. Trigonometric functions are periodic. But what is a periodic function in general? Definition (Periodic function) A function $f(x)$ is said to be a periodic function if there exists a real number $T>0$ such that $$f(x+T)=f(x).$$ In this case, $T$ is called a period of the function. Note that if $T$ is a period of $f(x)$, then $2T$, $3T$, $\cdots ,nT$ ($n\in \mathbb{N}$) is are also periods of $f(x)$. In fact,  $$f(x+nT)=f(x+(n-1)T+T)=f(x+(n-1)T)=\cdots =f(x).$$ The smallest (non-zero) period is called the fundamental period . Remark . If we say simply a period, we usually mean ...

Here we list two useful theorems without proofs. If you learn the theory of measure and Lebesgue integral, you will encounter more general versions of these theorems. However, the following theorems do not require the Lebesgue integral but hold for the Riemann integral. Theorem (Arzelà's convergence theorem) Suppose that the sequence of continuous functions $$f_n=f_n(x)(n=1,2,\cdots )$$ is uniformly bounded on the closed bounded interval $I=[a,b]$. That is, there exists an $M>0$ such that $$|f_n(x)|\le M(x\in I,n=1,2,\cdots ).$$ In addition, suppose that $\{f_n\}$ converges to a continuous function $f_0=f_0(x)$ at each $x\in I$. That is, $$\underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=f_0(x)(x\in I).$$ Then, we have $$\underset{n\to \mathrm{\infty }}{lim}{\int }_a^b{f}_n(x)dx={\int }_a^b{f}_0(x)dx.$$ In other words, the limit as $n\to \mathrm{\infty }$ and the integral are ``commutative'' (i.e., the order of limit and integration can be swapped). Perhaps it is clearer if we write \[\...

The general solution of an inhomogeneous linear differential equation can be obtained as the sum of a special solution and the general solution of the corresponding homogeneous differential equation. We study the method of variation of parameters in particular. Consider the linear differential equation of the form $$F(D)y=q(x)$$ where $F(t)$ is a polynomial and $q(x)$ is a function. When $q(x)=0$, this is a homogeneous linear differential equation. Here, we assume $q(x)\ne 0$. Suppose we can factorize $F(t)$ as $F(t)=G(t)H(t)$. If $y=y(x)$ is a solution of $F(D)y=q$, then $z=H(D)y$ is a solution of $G(D)z=q$. Thus, the given differential equation $F(D)y=q$ is decomposed into two parts: $G(D)z=q$ (a linear differential equation of $z$, given $q$), $H(D)y=z$ (a linear differential equation of $y$, given $z$), and we can process one after the other. Thus, by factorizing the polynomial $F(D)$, we only need to consider the case w...

Consider the homogeneous linear differential equation $$\begin{array}{}\text{(Eq:code)}& y^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=0\end{array}$$ where $a_0,a_1,\cdots ,a_{n-1}\in \mathbb{R}$ are constants.  Using the differential operator $D=\frac{d}{dx}$ and its polynomial $$\begin{array}{}\text{(Eq:epoly)}& E=D^n+a_{n-1}{D}^{n-1}+\cdots +a_{1}D+a_0,\end{array}$$ (Eq:code) can be expressed as $$Ey=0.$$ Now, consider the polynomial of variable $t$ $$F(t)=t^n+a_{n-1}{t}^{n-1}+\cdots +a_{1}t+a_0.$$ Then, we have $$E=F(D)$$ and (Eq:code) is expressed as $$F(D)y=0.$$ We can use the properties of polynomials to solve this type of differential equation. We need some results from algebra. Definition (Relatively prime, coprime) Polynomials $F_1(t)$ and $F_2(t)$ are said to be relatively prime  or coprime if the equations $F_1(t)=0$ and $F_2(t)=0$ do not have common solutions within $\mathbb{C}$. Lemma The polynomials $F_1(t)$ and $F_2(t)$ in $t$ are relatively prime if and only if th...