notas.itmens
From Classical to Abstract Harmonic Analysis
Fourier series#
The study of harmonic analysis began with the study of Fourier series, which originates in the solutions of the classical partial differential equations of mathematical physics. As an illustration consider the one dimensional wave equation
\[\partial^2_x u = \frac{1}{a^2}\partial^2_t u.\]derived first by d'Alembert as the law governing the motion of a vibrating string in 1747. The general solution is of the form
\[u(x,t) = f_1(x + at) + f_2(x-at)\]where \(f_1, f_2\) are twice differentiable. Imposing Dirichlet boundary conditions \(u(0,t) = u(L,t) = 0\) and initial conditions, it can be seen that the solution actually takes the form
\[u(x,t) = f(x+at) - f(at - x)\]where
\[f(x) = a_0 + \sum_1^\infty\left( a_n \cos(n\pi x/L) + b_n \sin(n\pi x/L)\right).\]This is the superposition of \(n\)th fundamental modes or \(n\)th harmonics \(\sin(n\pi x/L) \cos(n\pi at /L)\) and \(\sin(n\pi x/L)\sin(n\pi at/L)\) (note that \(\sin(x+y) = \sin x\cos y+ \cos x\sin y\) and \(\cos(x + y) = \cos x \cos y - \sin x \sin y\)). Fourier was the first to study these series. Nowadays the series is written as
\[f(x) = \sum_{-\infty}^\infty c_n e^{in x}\]where \(c_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) e^{-inx}dx\) putting \(L=2\pi\); \(x\) now parametrizes the unit circle \(\mathbb{T}\). The convergence of the series is most appropriate when the notion is that of \(L^2\)-convergence. Hence on \(L^2(\mathbb{T})\) there is a good theory of Fourier series: the functions \(e_n(x) = e^{inx}\) form a complete orthonormal system of the Hilbert space \(L^2(\mathbb{T})\), and the Fourier series is simply the expansion of \(f\in L^(\mathbb{T})\) in terms of this orthonormal basis (with each coefficients \(c_n = \langle f, e_n\rangle\)). The convergence means \(L^2\)-convergence
\[\lim_{N\to \infty}||f - \sum_{n=-N}^N c_n e_n|| = 0.\]And we have the Parseval identity
\[||f||^2 = \sum_{n=-\infty}^\infty | c_n|^2.\]Classical Harmonic Analysis#
There are, broadly speaking, two interpretations of the Fourier series, or rather, the basis functions \(e_n(x)\).
- The functions \(e_n\) are eigenfunctions of the Laplacian operator \(\Delta\) on \(\mathbb{T}\).
- The functions \(e_n\) are all the continuous homomorphisms from the circle \(\mathbb{T}\) into the multiplicative group of nonzero complex numbers \(\mathbb{C}^\ast \cong GL(\mathbb{C})\) which are the 1-dimensional irreducible representations of \(\mathbb{T}\).
Eigenfunctions of the Laplacian operator#
Let \(\Delta = (-iD)^2\) and call it Laplacian, which is a self-adjoint differential operator on \(\mathbb{T}\), we have \(\Delta e_n = n^2 e_n\) for all \(n\). Hence the Fourier series of \(f\) is the expansion of \(f\) in therms of eigenfunctions \(e_n\) of the Laplacian.
For a differential equation on \(\mathbb{T}\) of the form
\[\Delta u = f + \hat{f}_0,\]up to a constant the solution \(u\) is uniquely determined by the equations
\[\hat{u}_n = \frac{1}{n^2} \hat{f}_n\]for \(n\neq 0\) where \(\hat{g}_n\) is the \(n\)th Fourier coefficient; this can be seen from two integration by parts: \(\widehat{\Delta u}_n = n^2 \hat{u}_n\). Suppose that we know \(\hat{f}_n\), then what is \(u\)? It can be summed from \(\hat{u}_n\), but a better way is to consider the following. We want to find a function \(f_1\ast f_2\), for which the Fourier coefficients are \(\widehat{f_1 \ast f_2}_n = \hat{f_1}_n \hat{f_2}_n\), it is easy to verify that it is given by the convolution
\[(f_1 \ast f_2)(x) = \frac{1}{2\pi} f_1(y) f_2(x-y) dy\]of \(f_1\) and \(f_2\). Now let \(g\) be the function \(g(x) = \frac{1}{2} x^2 - \pi x + \frac{1}{3} x^2\) for \(x\in [0,2\pi]\), then \(\hat{g}_n = \frac{1}{n^2}\) for \(n\neq 0\) and \(g_0 = 0\), and we have \(u = g\ast f + C\) where \(C\) is an arbitrary constant.
Group representation#
The functions \(e_n\) are all the continuous homomorphisms from the circle \(\mathbb{T}\) into the multiplicative group of nonzero complex numbers \(\mathbb{C}^\ast \cong GL(\mathbb{C})\) which are the 1-dimensional irreducible representations of \(\mathbb{T}\). Moreover, under pointwise multiplication, these homomorphisms themselves form a group that is isomorphic to the additive group \(\mathbb{Z}\) of integers, which is called the dual group of \(\mathbb{T}\).
Recall that according to representation theory of finite groups given a finite group \(G\) and the irreducible represesntations \(V\), we have the following decomposition of the function space \(C(G)\):
\[C(G) \cong \bigoplus\text{dim}(V) V.\]This is an isomorphism of algebras taking convolution as the multiplication on \(C(G)\); \(C(G)\) is called the group algebra of \(G\).
There's an analogy, heuristically, the Fourier series then can be seen as the expansion of a function \(f\in C(\mathbb{T})\) in the direct sums of the irreducible representations \(e_n\) of the group \(\mathbb{T}\), with representation spaces being all \(\mathbb{C}\).