📂Hilbert Space

Generalized Fourier Coefficients and Fourier Series in Hilbert Spaces

Definition[^1]

Let $H$ be a Hilbert space, and let $\left\{ u_{\alpha} \right\}_{\alpha\in A}$ be a normal orthogonal system in $H$. Then, for a fixed $x\in H$, let’s define the complex function $\hat{x} :A\to \mathbb{C}$ as follows.

$$\hat{x}(\alpha)=\left\langle x,u_{\alpha} \right\rangle$$

The values above are referred to as the Fourier coefficients of $x$ with respect to $\left\{ u_{\alpha} \right\}$.