Section 3.7 Fourier series
Below we discuss briefly Fourier series. The reason why this is importance to us is that it turned out that the trigonometric polynomials, namely polynomials in the functions
\begin{equation*}
1,\cos(x),\sin(x),\cos(2x),\sin(2x),\dots,
\end{equation*}
are a Schauder basis for the \(L^p\) spaces. This is arguably the most well known and most studied Schauder basis. This is a good moment to point out a fact that is elementary but often leads to confusion. When we say that, in a vector space, \(f_n\to f\) under some norm \(\|\cdot\|\text{,}\) all we mean is that \(\|f_n-f\|\to0\text{.}\) In case of functional spaces, "vectors" are functions and so are also characterized by their values on their domain. So we are sometimes led to think that the fact that, under a norm \(\|\cdot\|\text{,}\)
\begin{equation*}
f = \sum_{n=1}^\infty c_n\cdot e_n,
\end{equation*}
where the \(e_n\) are functions and the \(c_n\) are scalars, means that
\begin{equation*}
f(x) = \sum_{n=1}^\infty c_n\cdot e_n(x)
\end{equation*}
for all \(x\) over which the functions are defined. The fact that it is necessarily so is illustrated very well by the history of the study of Fourier series. In his 1822 treatise on heat, J. Fourier exposed the wonderful idea that any function on an interval, say [0, 1], could be represented as superposition of functions \(\sin(2\pi nx)\) and \(\cos(2\pi nx)\text{,}\) \(n\in\bN\text{.}\) Since these functions are eigenfunctions for the operator \(d/dx\text{,}\) such expressions facilitated the solution of differential equations. The issue of convergence, which in those days could only have meant pointwise convergence, was recognized. The first publication proving pointwise convergence was by P. Dirichlet in 1829, although the device in the proof had appeared in an earlier manuscript of Fourier whose publication was delayed. B. Riemann’s 1854 Habilitationschrift concerned the representability of functions by trigonometric series. In 1915 N. Luzin conjectured that Fourier series of functions in \(L^2(\bS^1)\) converge almost everywhere pointwise. Decades later, in 1966, L. Carlson proved Luzin’s conjecture. In 1968, R. Hunt generalized this to \(L^p(\bS^1)\) functions for \(p > 1\text{.}\) On the other side, in 1876 P. du Bois-Reymond found a continuous function whose Fourier series diverges at a single point. Via the uniform boundedness theorem, one can show that there are continuous functions whose Fourier series diverges at any given countable collection of points. A. Kolmogorov (1923/26) gave an example of an \(L^1(\bS^1)\) function whose Fourier series diverges pointwise everywhere. The density of trigonometric polynomials (finite Fourier series) in the space of continuous function \(C^0(\bS^1)\) can be made to follow from Weierstrass’ aproximation theorem or using the Fejer kernel. In 1904, L. Fejer gave an even more direct proof of the density of trigonometric polynomials in \(L^2(\bS^1)\text{,}\) in effect using an approximate identity made directly in terms of trigonometric polynomials. See A. Zygmund’s, Trigonometric Series, I, II, for much more bibliographic and historical information. The original Fourier's idea can be extended to periodic functions in \(n\) variables, namely functions on the \(n\)-torus \(\bT^n\text{,}\) the Cartesian product of \(n\) copies of the unit circle \(\bS^1\text{.}\) We will not include proofs because they tend to be long and technical and we would not use them anywhere else in the textbook. Moreover, in order to keep the discussion simple, we will just refer to the case \(n=1\text{.}\) The basis
\begin{equation*}
\{e_1=1,e_2=\cos(x),e_3=\sin(x),e_4=\cos(2x),e_5=\sin(2x),\dots\}.
\end{equation*}
This is arguably the most important and studied Schuder basis. The expansion of a function in that basis is called Fourier series and the coefficients are called Fourier coefficients. The regularity of a function can be read from the asymptotics of the Fourier coefficients. Theorem 3.7.1. Fourier series converge in \(L^p, 1<p<\infty\).
The \(\{e_i\}\) is a Schauder basis for every \(L^p(\bS^1)\) for every \(1<p<\infty\) and the Fourier series of each function is the same for every \(p\text{.}\) Namely, if \(f\in L^2(\bS^1)\cap L^p(\bS^1)\text{,}\) then the Fourier series of \(f\) converges in both \(L^2\) and \(L^p\text{.}\) This basis, though, is unconditional only for \(p=2\text{.}\)Proof.
Theorem 3.7.2. Fourier coefficients determine a function's regularity.
- \(f\in C^\omega(\bS^1)\) if and only if there are \(C>0\) and \(\alpha>0\) such that \(|f_n|\leq Ce^{-\alpha|n|}\) (exponential decay);
- \(f\in C^\infty(\bS^1)\) if and only for every \(k\geq0\) there is a \(C_k>0\) such that \(|f_n|\leq \frac{C_k}{n^k}\) (decay faster than any polynomial);
- if \(f\in C^k(\bS^1), k\geq1,\) then \(|f_n|=O(1/|n|^k)\text{;}\)
- if \(|f_n|=O(1/|n|^{k+1}), k\geq1,\) then \(f\in C^k(\bS^1)\text{;}\)
- \(f\in H^s(\bS^1), s\geq0,\) if and only if \(\sum_{i=1}^\infty (1+n^2)^s|f_n|^2<\infty\text{;}\)
- if \(|f_n|=O(1/|n|)\text{,}\) then \(f\in L^2(\bS^1)\text{;}\)
- if \(|f_n|=O(1/|n|^{1+\eps})\text{,}\) then \(f\in C^0(\bS^1)\text{;}\)
- \(f\) is a distribution (see Chapter 5) if and only if there are \(C>0\) and \(k\geq0\) such that \(|f_n|\leq C n^k\text{.}\)