Separation of Variables
Series representation of solutions
Consider the homogeneous problem \begin{align} \label{e:H1}\tag{H1} \partial_t u - \lap u &= f &&\text{in } U_T \,,\\ \label{e:H2}\tag{H2} u &= 0 && \text{on } \partial U \,,\\ \label{e:H3}\tag{H3} u_0 &= g && \text{on } U \,. \end{align}
Suppose $\set{\varphi_k}$ are a family of eigenfunctions with corresponding eigenvalues $\set{\lambda_k}$ such that \begin{align} -\lap \varphi_k &= \lambda_k \varphi_k &&\text{in } U\,, \\ \varphi_k &= 0 &&\text{in } \partial U\,. \end{align} normalized so that $\ip{\varphi_i, \varphi_j} = \delta_{i, j}$. Here $\ip{f, g} = \int_U f g \, dx$ denotes the $L^2$ inner-product. If $$\label{e:F}\tag{F} f = \sum \ip{f, \varphi_k} \varphi_k \,, $$ then $$ u_t = \sum_k e^{-\lambda_k t} \ip{f, \varphi_k} \varphi_k \,, $$ solves \eqref{e:H1}–\eqref{e:H3}, provided the series converges fast enough.
To use this decomposition we need to justify \eqref{e:F} holds. This can be done for a large class of domains.
Example 1. If $U = [0, L]$ then $\varphi_k = (\frac{2}{L})^{1/2} \sin( \frac{k \pi}{L})$, $k \in \set{1, 2, \dots}$ are the desired eigenfunctions, with corresponding eigenvalues $k^2 \pi^2 / L^2$.
Example 2 (Periodic boundary conditions). If $U = \R^d / \Z^d \cong \T^d$ is the $d$-dimensional torus, we look for solutions to \eqref{e:H1} which are $1$-periodic in $x$. In this case it’s convenient to use complex eigenfunctions and write $$ e_k(x) = e^{2 \pi i k \cdot x} \quad \text{for } k \in \Z^d, x \in \R \,. $$ These functions are orthonormal when we use the complex $L^2$ inner-product defined by $\ip{f,g} = \int_U f \bar g \, dx$.
Example 3. Let $U = B(0, R) \subseteq \R^2$. For $m \in \Z$, the $m$-th Bessel function of the first kind is the solution of $$ x^2 J_m\dprime (x) + x J_m’(x) + (x^2 - m^2) J_m(x) = 0 \quad\text{such that } \lim_{x \to 0^+} \frac{J_m(x)}{x^{\abs{m}}} = \frac{(-1)^{(\abs{m}-m)/2}}{2^{\abs{m}}\abs{m}!} \,. $$ For $n \in \set{1, 2, \dots}$, define $$ \varphi_{m,n}(r,\theta) = \frac{1}{R\sqrt{\pi}\abs{J_{m+1}(j_{m,n})}} J_m\paren[\Big]{\frac{j_{m,n}r}{R}} e^{i m \theta} \,, \qquad \lambda_{m,n} = \frac{j_{m,n}^2}{R^2} \,, $$ where $j_{m,n}$ is the $n$-th positive zero of $J_m$. Then $\set{\varphi_{m,n}}$ is a orthonormal family of eigenfunctions of $-\lap$ with homogeneous Dirichlet boundary conditions, with corresponding eigenvalues $\lambda_{m,n}$.
Complex Fourier Series
Let’s first restrict to $\T^1 = S^1$.
Definition 4. If $f \in L^1(\T^1)$, $n \in \Z$, define the $n^\text{th}$ Fourier coefficient of $f$ by $\hat f(n) = \ip{f, e_n}$.
Definition 5. For $N \in \N$, let $S_N f = \sum_{-N}^N \hat f(n) e_n$, be the $N$-th partial sum of the Fourier Series of $f$.
Our goal is to get basic convergence results showing $S_N f \to f$.
Proposition 6. $S_N f = D_N * f$, where $$ D_N = \displaystyle \frac{ \sin ( 2 \pi (N + \frac{1}{2}) x ) }{\sin(\pi x)} \,. $$ The functions $D_N$ are called the Dirichlet Kernels.
Proposition 7. Define the Cesàro sum by $\sigma_N f = \frac{1}{N} \sum_{0}^{N-1} S_n f$. Then $\sigma_N f = F_N*f$, where $F_N$ is the Fejér kernel defined by $$ F_N = \displaystyle \frac{1}{N} \paren[\Big]{ \frac{\sin(N \pi x)}{\sin (\pi x)} }^2 \,. $$
Proposition 8. The Fejér kernels are an approximate identity, but the Dirichlet kernels are not.
Corollary 9. If $p \in [1, \infty)$ and $f \in L^p$, then $\sigma_N f \to f$ in $L^p$.
Corollary 10. If $f \in L^2$ then $S_N f \to f$ in $L^2$.
Proof sketch. For any $g \in \Span \set{e_{-N}, \dots, e_N}$, $\ip{f - S_Nf, g} = 0$. Hence $\norm{f - S_N f}_2 \leq \norm{f - g}_2$ for every such $g$. Choose $g = \sigma_N f$.
Theorem 11 (Parseval’s equality). If $f \in L^2([0, 1])$ then $\norm{\hat f}_{\ell^2} = \norm{f}_{L^2}$.
Theorem 12. If $p \in (1, \infty)$, $f \in L^p$ then $S_N f \to f$ in $L^p$.
Proof. The proof requires boundedness of the Hilbert transform and is beyond the scope of this course.
Theorem 13. If $f \in L^\infty$ and is Hölder continuous at $x$ with any exponent $\alpha > 0$, then $S_n f(x) \to x$.