Exercise 7.22

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Exercise 22: 
    Assume $f\in\mathscr R(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim_{n\rightarrow\infty}\int_a^b|f-P_n|^2\,d\alpha=0.$$

analambanomenos · Accepted Answer

\def\eps{\varepsilon}

Recall the notation of Exercise 6.11: $||u||_2=\bigl(\int_a^b|u^2|\,d\alpha\bigr)^{1/2}$. We want to show that, if $\varepsilon>0$, there is a polynomial $P$ such that $||f-P||_2<\varepsilon$.
    By Exercise 6.12, there is a continuous function $g$ on $[a,b]$ such that $||f-g||_2<\varepsilon/2$. By Theorem 7.26, there is a polynomial $P$ such that $\sup\big|g(x)-P(x)\big|<
    \varepsilon/\bigl(2\sqrt{\alpha(b)-\alpha(a)}\bigr)$. Then $$||g-P||_2^2=\int_a^b\big|g(x)-P(x)\big|^2\,d\alpha<\frac{\varepsilon^2}{4\bigl(\alpha(b)-\alpha(a)\bigr)}\bigl(\alpha(b)-\alpha(a)\bigr)=
    \frac{\varepsilon^2}{4}.$$ Hence, by Exercise 6.11, $||f-P||_2\le||f-g||_2+||g-P||_2<\varepsilon$.

Exercise 7.22

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Exercise 7.22

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