Riemann sum for improper integral

Question

Suppose that $f:[0,\infty)\to[-1,1]$ is a continuous function at zero and at all but finite number of points such that $f(0)=1$ and $\int_0^\infty f^2(x)dx<\infty$. Let $\{a_n\}_{n\ge1}$ be a sequence of positive integers such that $a_n\to\infty$ as $n\to\infty$. The right Riemann sum $$ \frac1{a_n}\sum_{j=1}^{a_n}f^2(j/a_n) $$ converges to the integral $\int_0^1f^2(x)dx$ as $n\to\infty$. Also, for a positive integer $m$, we have that $$ \frac1{a_n}\sum_{j=1}^{ma_n}f^2(j/a_n)\to\int_0^mf^2(x)dx\quad\text{as}\quad n\to\infty. $$ Let $\{b_n\}_{n\ge1}$ be a sequence of positive integers such that $b_n/a_n\to\infty$ as $n\to\infty$. Intuitively, it seems that $$ \frac1{a_n}\sum_{j=1}^{b_n}f^2(j/a_n)\to\int_0^\infty f^2(x)dx\quad\text{as}\quad n\to\infty. $$ Is it true? How can I show this?

References are also welcome. Any help is much appreciated!

Answer 1

I'm not sure why you square $f.$ Why not just assume $f:[0,\infty)\to [0,1]$ continuously, and $\int_0^\infty f(x)\,dx < \infty?$

A counterexample given by Alex Francisco shows the answer to your question is no. I'm going to describe another answer that uses the same idea, but may be simpler.

Over each $n=2,3,\dots$ consider a thin triangle centered over $n,$ of base $1/n^2$ and height $1.$ Let $f$ be the tops of these triangles, with $f=0$ everywhere else. Then $f$ is continuous on $[0,\infty),$ and

$$\int_0^\infty f(x)\, dx = \sum_{n=2}^{\infty}\left (\frac{1}{2}\cdot\frac{1}{n^2}\cdot 1\right ) <\infty.$$

However, thinking of $a_n=n, b_n = n^3,$ we have

$$\frac{1}{n}\sum_{j=1}^{n^3}f(j/n) \ge \frac{1}{n}(f(2) + f(3) + \cdots +f(n^2)) = \frac{1}{n}(n^2-1) \to \infty.$$

Answer 2

The last proposition is false. Define$$ f(x) = \begin{cases} \sqrt{-2x + 1}; &\displaystyle 0 \leqslant x \leqslant \frac{1}{2}\\ \displaystyle \sqrt{\frac{1}{n} - \frac{2^n}{n} |x - n|}; &\displaystyle \exists n \in \mathbb{N}_+,\ |x - n| < \frac{1}{2^n}\\ 0; & \text{otherwise} \end{cases}. $$

The graph of $f^2$ is shown below.

On the one hand, $f(0) = 1$, $f$ is continuous on $[0, +\infty)$, $|f| \leqslant 1$, and$$ \int_0^\infty f^2(x) \,\mathrm{d}x = \frac{1}{4} + \sum_{n = 1}^\infty \frac{1}{n 2^n} < +\infty. $$

On the other hand, if $\displaystyle \frac{b_n}{a_n} \to +\infty \ (n \to \infty)$ and $b_n \geqslant a_n n^{a_n}$ holds for infinitely many $n \in \mathbb{N}_+$, then for these $n$,\begin{align*} \frac{1}{a_n} \sum_{j = 1}^{b_n} f^2\left(\frac{j}{a_n}\right) &\geqslant \frac{1}{a_n} \sum_{j = 1}^{a_n n^{a_n}} f^2\left(\frac{j}{a_n}\right) \geqslant \frac{1}{a_n} \sum_{k = 1}^{n^{a_n}} f^2(k) = \frac{1}{a_n} \sum_{k = 1}^{n^{a_n}} \frac{1}{k}\\ &> \frac{1}{a_n} \int_1^{n^{a_n}} \frac{\mathrm{d}x}{x} = \frac{\ln n^{a_n}}{a_n} = \ln n. \end{align*} Therefore,$$ \varlimsup_{n \to \infty} \frac{1}{a_n} \sum_{j = 1}^{b_n} f^2\left(\frac{j}{a_n}\right) = +\infty > \int_0^\infty f^2(x) \,\mathrm{d}x. $$