## 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

Preliminaries Let $\{a_n: n \in\mathbb{Z}\}$ denote a sequence of complex numbers. We define $\ell_2$ norm of $\{a_n\}$ by

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...

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...

Definition 1.1 Let $f,g:\mathbb{R}\to\mathbb{C}$ be $2\pi$-periodic functions. The convolution $f*g$ of $f$ and $g$ is the function defined by $[-\pi, \pi]$ ...

Basic Knowledge We focus on a class of functions