Fourier series need not converge at points of continuity
Exercise 2.8 Verify that $\frac{1}{2i}\sum_{n\neq 0} \frac{e^{inx}}{n}$ is the Fourier series of the $2\pi$-periodic sawtooth function, defined by $f(0)=0$, ...
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L2 Recovery of Integrable Functions and Consequences
Corollary 1.2 (Parseval’s Identity) Let $f$ be integrable function, and $a_n= \hat{f}(n)$. Then $\lim_{N\to\infty}\sum_{n=-N}^N\lvert a_n\rvert^2$ converges ...
Fourier Series in the Language of Infinite Dimensional Vector Space
Preliminaries Let $\{a_n: n \in\mathbb{Z}\}$ denote a sequence of complex numbers. We define $\ell_2$ norm of $\{a_n\}$ by
Dirichlet Problem on the Unit Disc
Preliminaries Suppose one has an infinite plate $(\mathbb{R}^2)$ with an initial heat distribution. Let $u(x,y)$ denote the temperature of the place at posit...
Cesaro and Abel Summability
Definition 1.1 Given a sequence $\{c_n\}$, let $s_n:=\sum_{k=0}^nc_k$ be the sequence of partial sums. We define $N$-th Cesaro mean $\sigma_N$ of the sequenc...