Wisniewool

Answered

2022-07-04

The purpose of this problem is that I want to prove that for any $\lambda $ integrable function f on a bounded closed interval [a,b] holds

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{[a,b]}f(x)\mathrm{sin}(nx)d\lambda =0.$

I have submitted a proof below.

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{[a,b]}f(x)\mathrm{sin}(nx)d\lambda =0.$

I have submitted a proof below.

Answer & Explanation

treccinair

Expert

2022-07-05Added 18 answers

The smooth functions with compact suppport ${\mathcal{C}}_{0}^{\mathrm{\infty}}([a,b])$ functions are dense over the Lebesgue integral functions ${L}^{1}[a,b]$. Then for every f there is a sequence $({f}_{k}{)}_{k}\subset {\mathcal{C}}_{0}^{\mathrm{\infty}}([a,b])$ such that $\Vert f-{f}_{k}{\Vert}_{1}\stackrel{k\to \mathrm{\infty}}{\to}0$. We observe that

$\begin{array}{r}|{\int}_{a}^{b}{f}_{k}(x)\mathrm{sin}(nx)dx|=|-\frac{1}{n}{\int}_{a}^{b}{f}_{k}^{\prime}(x)\mathrm{cos}(nx)dx|=\frac{1}{{n}^{2}}|{\int}_{a}^{b}{f}_{k}^{\u2033}(x)\mathrm{sin}(nx)dx|\le \frac{{M}_{k}}{{n}^{2}}\stackrel{n\to \mathrm{\infty}}{\to}0\end{array}$

Where ${M}_{k}=\underset{x\in [a,b]}{sup}|{f}_{k}^{\u2033}(x)|(b-a)<\mathrm{\infty}$. Now

$\begin{array}{rl}|{\int}_{a}^{b}f(x)\mathrm{sin}(nx)dx|& \le |{\int}_{a}^{b}{f}_{k}(x)\mathrm{sin}(nx)dx|+|{\int}_{a}^{b}(f(x)-{f}_{k}(x))\mathrm{sin}(nx)dx|\\ & \le \frac{{M}_{k}}{{n}^{2}}+\Vert {f}_{k}-f{\Vert}_{1}\stackrel{n\to \mathrm{\infty}}{\to}\Vert {f}_{k}-f{\Vert}_{1}\stackrel{k\to \mathrm{\infty}}{\to}0.\end{array}$

$\begin{array}{r}|{\int}_{a}^{b}{f}_{k}(x)\mathrm{sin}(nx)dx|=|-\frac{1}{n}{\int}_{a}^{b}{f}_{k}^{\prime}(x)\mathrm{cos}(nx)dx|=\frac{1}{{n}^{2}}|{\int}_{a}^{b}{f}_{k}^{\u2033}(x)\mathrm{sin}(nx)dx|\le \frac{{M}_{k}}{{n}^{2}}\stackrel{n\to \mathrm{\infty}}{\to}0\end{array}$

Where ${M}_{k}=\underset{x\in [a,b]}{sup}|{f}_{k}^{\u2033}(x)|(b-a)<\mathrm{\infty}$. Now

$\begin{array}{rl}|{\int}_{a}^{b}f(x)\mathrm{sin}(nx)dx|& \le |{\int}_{a}^{b}{f}_{k}(x)\mathrm{sin}(nx)dx|+|{\int}_{a}^{b}(f(x)-{f}_{k}(x))\mathrm{sin}(nx)dx|\\ & \le \frac{{M}_{k}}{{n}^{2}}+\Vert {f}_{k}-f{\Vert}_{1}\stackrel{n\to \mathrm{\infty}}{\to}\Vert {f}_{k}-f{\Vert}_{1}\stackrel{k\to \mathrm{\infty}}{\to}0.\end{array}$

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