Subspace Approximation vs. the Riesz-Fischer Theorem

Lecture 8

The previous theorem on page took $f\in L^2(a,b)$ as given and then constructed a sequence $\{ c_1,c_2, \cdots\}$ , which by Bessel's inequality

$$\sum_{k=1}^N \vert c_k \vert^2 \le \Vert f \Vert^2 \quad \textrm{for all}~N$$

is square-summable, i.e.

$$\{c_i \} \in l^2\quad .$$

By contrast, the Riesz-Fischer theorem on page  allows us to turn this theorem around: Given a square summable sequence,

$$\{c_i \} \in l^2\quad ,$$

the R-F theorem considers the concomitant Cauchy sequence $\sum_{k=1}^N c_k u_k,~N=1,2,\cdots$ in $\mathcal{H}=L^2(a,b)$ , and then guarantees the existence of a square-intergrable function

$$f\in L^2(a,b)\quad .$$

This function is related to $\{c_i \} \in l^2$ by virtue of the property that

$$\langle u_k,f \rangle=c_k \quad \quad k=1,2,\cdots$$

with the result that

$$\lim_{N\rightarrow \infty}\Vert f- \sum_{k=1}^N c_k u_k\Vert ^2=0\quad .$$ (115)

Often this is written as

$$f\doteq\sum^\infty_{k=1} c_k u_k$$

and one says that ``$f$ equals $\sum\limits^\infty_{k=1} c_k u_k$ in the least square sense or in the mean''.

This is to be compared with pointwise equality, which is expressed by the statement that

$$f(x)=\sum_{k=1}^\infty c_k u_k(x) \quad .$$ (116)

The difference between ``$\doteq$ '' and ``$=$ '' manifests itself only when the square-summable sequence $\{ c_k\}$ yields a function which has one or more discontinuities, then one does not have pointwise equality, Eq.(1.16). Instead, one has the weaker condition, Eq.(1.15). This condition does not specify the value of $f$ at the point(s) of discontinuity. Instead, it specifies an equivalence class of functions, all having the same graph everywhere except at the point(s) of discontinuity.