My notes when reading Fourier Analysis An Introduction by Stein and Shakarchi. This is for chapter 2 Basic Properties of Fourier Series, section 1 Examples and formulation of the problem.

1 Examples and formulation of the problem

Riemann Integrable

A real-valued function $$f$$ defined on [0, L] is Riemann integrable if it isbounded, and if for every $$\epsilon >0$$, there is a subdivision $$0=x_0

$$ \mathcal{U}=\sum_{j=1}^N[\sup_{x_{j-1}\leq x\leq x_j} f(x)](x_j-x_{j-1}) $$

and

$$ \mathcal{L}=\sum_{j=1}^N[\inf_{x_{j-1}\leq x\leq x_j} f(x)](x_j-x_{j-1}) $$

then we have $$\mathcal{U}-\mathcal{L}<\epsilon$$.

1.1 Main definitions and some examples

Fourier Coefficients

If $$f$$ is an integrable function given on an interval $$[a,b]$$ of length $$L$$ (that is, $$b-a=L$$), then the n-th Fourier coeffcient of $$f$$ is defined as

$$ \hat{f}(n)=\frac{1}{L}\int_a^bf(x)e^{-2\pi inx/L}dx\mathrm{,}\quad n\in\mathbb{Z} $$

The Fourier series of $$f$$ is given formally by

$$ \sum_{n=-\infty}^{\infty}\hat{f}(n)e^{2\pi inx/L}\mathrm{.} $$

Dirichlet kernel

The trigonometric polynomial defined for $$x\in [-\pi,\pi]$$ by

$$ D_N(x)=\sum_{n=-N}^N e^{inx} $$

is called the N-th Dirichlet kernel. Its Fourier coefficients $$a_n$$ have the property that $$a_n=1$$ if $$\lvert n\rvert\leq N$$ and $$a_n=0$$ otherwise.

Poisson kernel

The function $$P_r(\theta)$$, called the Poisson kernel, is defined for $$\theta\in[-\pi,\pi]$$ and $$0\leq r<1$$ by the absolutely and uniformly convergent series

$$ P_r(\theta)=\sum_{n=-\infty}^\infty r^{\lvert n\rvert}e^{in\theta}\mathrm{.} $$