Preface


It turns out that many interesting problems in Fourier analysis do not depend on the pointwise convergence of Fourier series. Actually, Fourier series is based on another type of convergence, called 𝔏² convergence. Of course pointwise convergence is important for Fourier series; however, it is more convenient ti use Cesàro summation, which is a topic of next section.

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Introduction to Linear Algebra with Mathematica

Convergence of Fourier Series


One of the main issues of Fourier series applications is a possiblity of restoring the function from its Fourier series. This ill-posed problem is related to the convergence of Fourier series of a periodic function. This topic is a field known as classical harmonic analysis, a branch of pure mathematics. Historically, Fourier series was the first example of expansion of a function with respect to eigenfunctions corresponding to some Sturm--Liouville problem. Previously, Taylor series were mostly in use and their expansions require a usage of pointwise convergence, uniform convergence, and absolute convergence. Taylor's series coefficients are defined by derivatives at one point---the center of convergence, so only infenisimal information is used for their determination. In opposite, eigenfunction expansions are based on the integration over whole interval. This caused development of another kinds of convergence: 𝔏² spaces, summability, and the the Cesàro mean.

The Fourier series is an example of an eigenfunction expansion:

\begin{equation} \label{EqConverge.1} f(x) \sim \frac{a_0}{2} + \sum_{k\ge 0} \left[ a_k \cos \frac{k\pi x}{\ell} + \sin \frac{k\pi x}{\ell} \right] = \sum_{n=-\infty}^{\infty} \alpha_n e^{n{\bf j} \pi x/\ell} , \end{equation}
where Fourier coefficients αn and 𝑎n, bn are determined through Euler--Fourier formulas
\begin{equation} \label{EqFourier.2} \alpha_k = \alpha_k (f) = \frac{1}{2\ell} \int_{-\ell}^{\ell} f(x)\, e^{-k{\bf j} \pi x/\ell} \,{\text d} x = \frac{1}{T} \int_{0}^{T} f(x)\, e^{-2k{\bf j} \pi x/T} \,{\text d} x , \qquad k=0, \pm 1, \pm 2, \ldots ; \end{equation}
and
\begin{equation} \label{EqFourier.5T} \begin{split} a_k &= \frac{2}{T} \int_{0}^{T} f(x)\, \cos \frac{2k \pi x}{T} \,{\text d} x , \qquad k= 0, 1, 2, 3, \ldots ; \\ b_k &= \frac{2}{T} \int_{0}^{T} f(x)\, \sin \frac{2k \pi x}{T} \,{\text d} x , \qquad k= 1, 2, 3, \ldots . \end{split} \end{equation}

 

Convergence in mean square sense, 𝔏²


 

Pointwise convergence


The first convergence theorem was proved by the German mathematician Peter Gustav Lejeune Dirichlet (1805--1859) in 1829.
Theorem (Dirichlet): Let f(x) be a periodic function with period T = 2ℓ and satisfy the following conditions (that are usually referred to as Dirichlet's conditions): Then for all x, the Fourier series
\[ \frac{a_0}{2} + \sum_{k\ge 1} \left[ a_k \cos \frac{k\pi x}{\ell} + b_k \sin \frac{k\pi x}{\ell} \right] = \sum_{n=-\infty}^{\infty} \alpha_n e^{n\pi{\bf j}x/\ell} = \frac{1}{2} \left[ f(x+0) + f(x-0) \right] \]
converges to the mean values of the function f at that point. Here
\[ f(x+0) = f(x^{+}) = \lim_{\varepsilon \downarrow 0} f(x+\varepsilon ) , \qquad f(x-0) = f(x^{-}) = \lim_{\varepsilon \downarrow 0} f(x-\varepsilon ) , \qquad \varepsilon > 0, \]
denotes the right/left limits of f(x).     ⧫
The absolutely integrability condition is a sufficient condition to guarantee existence of Fourier coefficients. The last two conditions of Dichlet's theorem are imposed to apply the second mean value theorem:
Second mean value Theorem: Let g(x) be a bounded, increasing real valued function defined on the interval [𝑎,b], that is continuous at 𝑎 and b; and f(x) a Riemann integrable function on [𝑎,b]. Then there exists a real number c∈[𝑎,b] such that
\[ \int_a^b g(x)\,f(x)\,{\text d} x = g(a+0) \int_a^c f(x)\,{\text d} x + g(b-0) \int_c^b f(x)\,{\text d} x . \]

In section on Fourier series, a class of functions, called piecewise smooth, was introduced. Such functions satisfy Dirichlet's conditions.

Lemma 1: If f is integrable on the interval [-ℓ,ℓ], then
\[ \sum_{k=-n}^n \alpha_k (f)\, e^{{\bf j}kx\pi /\ell} = \frac{1}{2\ell} \,\int_{-\ell}^{\ell} f(x-t)\, D_n (t)\,{\text d} t , \]
where
\[ D_n (t) = \csc \left( \frac{\pi t}{2\ell} \right) \sin \left( \frac{(2n+1)\pi t}{2\ell} \right) , \qquad t\ne 0, \]
is called the Dirichlet kernel.     ⧫
Note that for the complex Fourier series, the Dirichlet kernel is the same:
\[ D_n (t) = \sum_{n=-N}^N e^{{\bf j} nx} = \frac{\sin \left( \left( N + \frac{1}{2} \right) x \right)}{\sin \left( x/2 \right)} . \]

Lemma 2: Let f be a continuous periodic function. If its Fourier series converges absolutely, then the Fourier series converges to f uniformly.    ; ⧫

Theorem 3: For piecewise smooth f(x), the Fourier series of f(x) is continuous and converges to f(x) for −ℓ ≤ x ≤ ℓ if and only if f(x) is continuous and f(−ℓ) = f(ℓ).     ⧫

Theorem 4: A Fourier series for a 2ℓ-periodic function f(x) that is continuous can be differentiated term by term if its derivative f'(x) satisfies Dirichlet's conditions.     ⧫

 

Order of Growth


Theorem 4 (Riemann lemma): Let function f∈𝔏¹ be absolutely integrable on an interval of length T = 2ℓ and 𝑎n, bn be the Fourier coefficients of f. Then
\[ \lim_{n\to +\infty} a_n = \lim_{n\to +\infty} b_n = 0, \]
which means that
\[ \lim_{n\to +\infty} \int_0^T f(x)\,\cos \left( \frac{2\pi nx}{T} \right) {\text d}x = \lim_{n\to +\infty} \int_0^T f(x)\,\sin \left( \frac{2\pi nx}{T} \right) {\text d}x = 0, \]
From the Bessel inequality, it follows that
\begin{equation*} \frac{1}{2} \left\vert a_0 \right\vert^2 + \sum_{n\ge 1} \left( \left\vert a_n \right\vert^2 + \left\vert b_n \right\vert^2 \right) = 2 \sum_{n= -\infty}^{\infty} \left\vert \alpha_n \right\vert^2 \le \frac{2}{T} \int_0^T \left\vert f(x) \right\vert^2 {\text d} x . \end{equation*}

Therefore,
\[ \lim_{n\to +\infty} a_n = \lim_{n\to +\infty} b_n = 0 \]
because a convergent series must have its entries to approach zero. ﹡ ⁎ ✱ ✲ ✳ ✺ ✻ ✼ ✽ ❋

 

Cesàro summation


L² convergence

 

  1. Convergence of Fourier series, Wikipedia.
  2. Hewitt, E. and Hewitt, R.E., The Gibbs--Wilbraham phenomenon: An episode in Fourier analysis, Archive for History of Exact Science, 1979, Vol. 21, No. 2, pp. 129--160.

 

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