1.20 Error of the best approximation in an orthobasis

As an application of Parseval's Theorem, say ${\left\{v_k\right\}}_{k=1}^{\infty }$ is an orthobasis for an inner product space of $V$ .

Let $A$ be the subspace spanned by the first 10 elements of $\left\{v_k\right\}$ , i.e., $A=\mathrm{span}\left(\left\{v_1,,,.,.,.,,,v_{10}\right\}\right)$

1. Given $x\in v$ , what is the closest point in $A$ (call it $\widehat{x}$ ) to $x$ ? We have seen that it is $\widehat{x}={\sum }_{k=1}^{10}〈x,,,v_k〉v_k$
2. How good of an approximation is $\widehat{x}$ to $x$ ? Measured with ${\parallel ·\parallel }_V$ :
   $$\begin{array}{cc}\hfill \parallel x-\widehat{x}{\parallel }_V^2& ={∥\sum _{k>10},⟨x,v_k⟩,v_k∥}_V^2\hfill \\ & =\sum _{k>10}{\left|⟨x,v_k⟩\right|}^2\hfill \end{array}$$

Since we also have that ${\parallel x\parallel }_V^2={\sum }_{k=1}^{\infty }{\left|⟨x,v_k⟩\right|}^2$ , the approximation $\widehat{x}$ will be “good” if the first 10 transform coefficients contain “most” of the total energy.Constructing these types of approximations is exactly what is done in image compression.