5

I would like to determine if the following sequence of linear functionals on $C[0,1]$ converges weakly (i.e. pointwise) and strongly (with respect to the norm) $$ \phi_n(f)=\int\limits_{0}^{1}(nt^2-[nt^2])f(t)dt,$$ where $[x]$ means greatest integer that is less than or equal to $x$. I do not know how to investigate this question. Thanks in advance for your help.

Mars Plastic
  • 4,239
Timur B.
  • 492
  • 1
    Hint: Draw a graph of $nt^2 -[nt^2]$ for large $n$. – s.harp May 19 '21 at 20:55
  • 1
    @OliverDiaz what does $\implies$ mean? – copper.hat May 19 '21 at 21:53
  • 1
    @OliverDiaz I'm grinding through a simple computation at the moment, I am not convinced of the ${ nx^2}$ case yet. I would expect some 'area multiplier' at minimum. – copper.hat May 19 '21 at 22:20
  • 2
    @OliverDiaz I should do some paying work too :-). – copper.hat May 19 '21 at 22:49
  • 1
    @OliverDiaz Looks good, I was expecting some constant (the ${ 1\over 2}$ below), I am still (slowly) trying to compute it for $f=1$. – copper.hat May 19 '21 at 23:24
  • 2
    @OliverDiaz Not surprisingly (given your answer) $\int\limits_{0}^{1} {nt^2} dt \to {1 \over 2}$, but off the bat it was not what I thought. – copper.hat May 19 '21 at 23:30
  • 1
    Pointwise convergence is weak-star convergence, which matters here. – daw May 20 '21 at 06:34
  • 1
    @TimurB. I ignore if whether there is simpler solution, but when the measures in $[0,1]$ or any other finite interval $[0,T]$ for that matter, you have measures of the form $\mu_n(dx)=f(nx+b_n),dx$ where $f$ defined on $[0,T]$, one may try to extend $f$ the whole real line as periodic functions. The result by Féjer can then applied to suitable test functions ($\mathcal{C}[0,1]$ in your case). – Mittens May 20 '21 at 20:26
  • 2
    This is a perfectly good question with an insightful answer. I really do not get this rush to close & delete. there are plenty of rubbish, truly low quality questions, sure they should be dealt with. – copper.hat May 22 '21 at 04:45

1 Answers1

4

The function $h(t)=\{t\}=t-[t]$ is $1$-periodic. The change of variable $u=t^2$ gives $$\int^1_0f(t)\{nt^2\}\,dt=\frac{1}{2}\int^1_0 \frac{f(\sqrt{u})}{\sqrt{u}}\,h(nu)\,du$$

Notice that $g(u)=\mathbb{1}_{[0,1]}(u)\frac{f(\sqrt{u})}{\sqrt{u}}$ is integrable for any $f\in\mathcal{C}([0,1])$. By Féjer's formula, $$\begin{align} \frac12\int^1_0 \frac{f(\sqrt{u})}{\sqrt{u}}h(nu)\,du&= \frac12\int_\mathbb{R}g(u)\,h(n\,u)\,du\xrightarrow{n\rightarrow\infty}\frac12\Big(\int^1_0 h(u)\,du\Big)\int_\mathbb{R}g(u)\,du\\ &=\frac14\int^1_0 \frac{f(\sqrt{u})}{\sqrt{u}}\,du=\frac{1}{2}\int^1_0f \end{align}$$ Therefore, the sequence of measures $\phi_n(dt)=\{nt^2\}\,dt$ converge to $\frac{1}{2}dx$ in $(\mathcal{C}[0,1])^*$ with the weak-* topology.

Mittens
  • 39,145
  • Never seen this formula caller Fejer's. Do you have a reference? (No critic, just curious) – daw May 20 '21 at 06:31
  • 2
    @daw: I learnt of that result in a book of problems by Claude, George , Exercices et Problèms d'integration, BORDAS, Paris, 1980 who attributed to Féjer. Other versions (without attribution) and some restriction have been discussed in MSE, but the version I have applies for any bounded measurable $T$-peridoc function $h$ and any integrable function $g$. It is a beautiful result, in fact Riemann-Lebesgue's theorem, at least in $L_2(\mathbb{T}$, can be deduced from it. – Mittens May 20 '21 at 13:30