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I wanted to do this problem for practice but I believe there may be a typo. The problem is: If $F_1,F_2,\ldots,F\in NBV$ and $F_j\rightarrow F$ pointwise, then $T_F\leq \liminf T_{F_j}$

I believe the corrected version should be: If $F_1,F_2,\ldots,F_n\in NBV$ and $F_j\rightarrow F$ pointwise, then $T_F\leq \liminf T_{F_j}$

I just want to know if these two statements are the same or is there some typo there. Any suggestions or comments is greatly appreciated.

Attempt (not complete) - Let $F_1,F_2,\ldots, F\in NBV$ and $F_j\rightarrow F$ pointwise. Given $\epsilon > 0$ choose points $x_0 < \ldots < x_n = x$ where $x\in\mathbb{R}$ such that $$\sum_{1}^{n}|F(x_j) - F(x_{j-1}| \geq T_F(x) - \epsilon$$ Then $T_F(x_0) \leq \epsilon$. Thus we have \begin{align*} T_F(x) - \epsilon &= \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} - \epsilon\\ &\leq \sum_{1}^{n}|F(x_j) - F(x_{j-1}|\\ &\leq \inf\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} \end{align*}

I am not sure if what I have done is on the right track or not.

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Either I'm missing something or everyone else is. It seems clear to me that if $F_j\to F$ pointwise than $T_F\le\liminf T_{F_j}$ without any assumption on any of the functions having bounded variation. And there's no epsilons needed either.

If $x_0<\dots<x_n$ then $$\begin{aligned}\sum_k|F(x_k)-F(x_{k-1})| &=\lim_j \sum_k|F_j(x_k)-F_j(x_{k-1})| \\&=\liminf_j \sum_k|F_j(x_k)-F_j(x_{k-1})| \\&\le\liminf_jT_{F_j}.\end{aligned}$$Now taking the sup over all choices of $x_0<\dots<x_n$ shows that $T_F \le\liminf T_{F_j}$.

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Wolfy, the two statements are correct, but your statement is stronger since it applies also to the case $T_F = \infty $. If $T_F$ is finite, then you can proceed as @Qiyu indicates. Qiyu is assuming implicitly that $T_F$ is finite at the statemnt: "Chose some partion $P = \{x_0,\dots,x_n\}$ such that $T-T(P) < \epsilon$ ...", which cannot be done if $T_F = + \infty $.

Your attempt is in the right track, but there is at lest one typo and it needs some cleaning.

Qiyu's proof can be modified to avoid assuming $T_F < \infty $. Here is how. Choose a partition $P = \{x_0,\dots,x_n\}$ of the interval $[a,b]$ with $a=x_0 < ... < x_n=b$ and choose $\epsilon > 0$.

By the assumption of pointwise convergence, there is $k_0$, that depends on $\epsilon$ and the partition, such that for all $k\ge k_0$ we have $|F_k(x_i) - F(x_i)| < \frac{\epsilon}{2n}$ for $i=0,...,n$.

Then for any $k\ge k_0$, and for $i=1,..,n$ $$|F(x_i) - F(x_{i-1})| \le |F(x_i) - F_k(x_{i})| + |F_k(x_i) - F_k(x_{i-1})| + |F(x_{i-1}) - F_k(x_{i-1})| $$ $$ < \frac{\epsilon}{n} + |F_k(x_i) - F_k(x_{i-1})| .$$

Therefore
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| < \epsilon + \sum_{1}^{n}|F_k(x_i) - F_k(x_{i-1})| \le \epsilon + T_{F_k}.$$

Now we can take lim inf over $k \ge k_0$ to get:
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| \le \epsilon + \liminf_{j\to \infty}T_j$$
and whether the lim-inf is finite or not, since this holds for any $\epsilon > 0$ we get:
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| \le \liminf_{j\to \infty}T_j .$$

Since the last inequality holds for any partition $P$, taking supremum over all partitions we get the result.$\square$

$$ \dots \dots \dots $$

In particular, this shows that point-wise convergence to $F$ + finite lim-inf implies the limit $F$ is of bounded variation.

Let's work a couple of examples. Let's take one of the standard examples of unbounded variations $$ f(x) = \begin{cases} \frac{1}{x}\cos(\frac{\pi}{x}) & \text{ for $ x \in (0,1]$},\\ 0 & \text{for $x=0$}. \end{cases} $$ The graph of $f$ oscilates between the lines $y=x$ and $y=-x$ as $x \rightarrow 0^+.$ Take the partition $P=\{ 0, \frac{1}{n},\frac{1}{n-1},.., \frac{1}{2}, 1 \}$. Since $f(\frac{1}{k}) = \frac{(-1)^k}{k}$ for $k=1,2,..$ we get
$$|f(\frac{1}{k}) - f(\frac{1}{k-1}) | = \frac{1}{k} + \frac{1}{k-1}$$
and
$$\sum_{1}^{n}|f(\frac{1}{k}) - f(\frac{1}{k-1})| = 2*ln(n)*(1+ o(1)) ,$$ therefore $T_f = \infty .$

Let $$F_u(x) = \begin{cases} f(x) & \text{ for $x \in (\frac{2}{2u+1},1]$},\\ 0 & \text{otherwise}. \end{cases} $$ Since $| F_u(x)-f(x)| \le \frac{2}{2u+1}$ for all $x \in [0,1]$, it follows that $F_j\rightarrow f$ uniformly, and pointwise.
$F_u$ is differentiable, except at $x=\frac{2}{2u+1}$, and the derivative is bounded, hence $F_u \in BV([0,1])$. Consider the partition $P=\{ 0, \frac{2}{2u+1}, \frac{1}{u},\frac{1}{u-1},.., \frac{1}{2}, 1 \}$. Because $F_u(0)=F_u(\frac{2}{2u+1}) = 0$, the variation of $F_u$ over this partition is the same sum as above, i.e.: $$\sum_{1}^{u} \frac{1}{k} +\frac{1}{k-1} = 2*ln(u)*(1+ o(1)) ,$$ so $\liminf_{u\to \infty}T_{F_u} \ge \liminf_{u\to \infty}2ln(u) = + \infty$. We have an example where both, $\liminf_{u\to \infty}T_{F_u}$ and $T_f$ are infinite. Your re-statement of the exercise in Folland applies.

$$ \dots \dots \dots $$

Let's next take
$$ f_u(x)=\begin{cases} f(s)& \text{for $x \in (\frac{2}{2u^2+1},\frac{2}{2u+1})$},\\ 0 & \text{otherwise}. \end{cases} $$

Since $|f_u(x)| \le \frac{2}{2u+1}$, we have $f_u\rightarrow 0$ uniformly, and in hence pointwise as well. Clearly $T_0 = 0$, so the limit function is of bounded variation. Consider the partition with partion points $$ 0, \frac{2}{2u^2+1}, \text{$\frac{1}{k},$ (for $k=u+1,..,u^2 )$} , \frac{2}{2u+1}, 1 .$$

The variation of $f_u$ over this partition is
$$\sum_{u+1}^{u^2} \frac{1}{k} +\frac{1}{k-1} = 2*ln(u)*(1+ o(1)) ,$$
therefore $\liminf_{u\to \infty}T_{f_u} \ge \liminf_{u\to \infty}2ln(u) = + \infty$. This is an example where $\liminf_{u\to \infty}T_{f_u} = + \infty $ and $T_f$ is finite.

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The problem in the book is well posed. For simplicity, I will denote the sums over partition $P$ by $T_j(P)$ and $T(P)$, and the total variations by $T_j$ and $T$.

Given $\epsilon > 0$, choose some partition $P = \{x_0,\dots,x_n\}$ such that $T-T(P) < \epsilon$. Choose $N$ such that $ |F_j(x_k) - F(x_k)| < \epsilon/2n$ for all $j>N$ and $k = 0,1,\dots,n$. Then $$ T(P) < T_j(P)+\epsilon \leq T_j + \epsilon $$ for all $j>N$, hence $$T < T(P) + \epsilon \leq \liminf_{j\to \infty}T_j+2\epsilon.$$ Since $\epsilon$ is arbitrary, $T \leq \liminf_{j\to \infty}T_j$.

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  • $\begingroup$ By stating it is well-posed. Are you implying that my suggested corrected statement is incorrect? $\endgroup$ – Wolfy Feb 17 '16 at 20:21
  • $\begingroup$ @MorganWeiss Well, I do not immediately see what the consequence would be if your $F_{n+1}, \dots, F$ aren't assured to be of bounded variation. $\endgroup$ – Qiyu Wen Feb 17 '16 at 22:49
  • $\begingroup$ Is your partition $P$ the same as the points $x_0 < x_1 < \ldots < x_n = x$ for $x\in\mathbb{R}$? $\endgroup$ – Wolfy Jul 29 '16 at 1:41
  • $\begingroup$ I think my proof is clear enough on its own, and I don't understand your attempt, if that's what you are trying to relate my proof to. $\endgroup$ – Qiyu Wen Jul 30 '16 at 23:03
  • $\begingroup$ Is my attempt incorrect? As in did I make any mistakes? $\endgroup$ – Wolfy Jul 30 '16 at 23:12

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