1.18 Orthobasis expansions

Suppose that the ${\left\{v_j\right\}}_{j=1}^N$ are a finite-dimensional orthobasis. In this case we have

But what if $x\in \mathrm{span}\left(\left\{v_j\right\}\right)=V$ already? Then we simply have

for all $x\in V$ . This is often called the “reproducing formula”. In infinite dimensions, if $V$ has an orthobasis ${\left\{v_j\right\}}_{j=1}^{\infty }$ and $x\in V$ has

then we can write

In other words, $x$ is perfectly captured by the list of numbers $〈x,,,v_1〉,〈x,,,v_2〉,...$

Sound familiar?

The general lesson is that we can recreate a vector $x$ in an inner product space from the coefficients $\left\{〈x,,,v_k〉\right\}$ . We can think of $\left\{〈x,,,v_k〉\right\}$ as “transform coefficients.”