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 $\{p_n:n\ge1\}$ be a sequence of positive integers such that $p_n\to\infty$ as $n\to\infty$, but $p_n/n\to0$ as $n\to\infty$.
Is it true that $$ \frac1n\sum_{j=1}^{n-1}f^2\Bigl(\frac j{p_n}\Bigr)=O(p_n/n) $$ as $n\to\infty$?
If we replace the upper bound of the summation with $p_n$, then we would get the right Riemann sum and $$ \frac1{p_n}\sum_{j=1}^{p_n}f^2\Bigl(\frac j{p_n}\Bigr)\to\int_0^1f^2(x)dx $$ as $n\to\infty$, but since $p_n/n\to0$ as $n\to\infty$, we have more terms in the sum and I am not sure how to deal with them.
Any help is much appreciated!
It is not true. In fact for any such sequence $p_n$ there exists such an $f$ such that
$$\limsup_{n\to\infty}\frac1n\sum_{j=1}^{n-1}f^2\Bigl(\frac j{p_n}\Bigr)=1.$$
To see this, define $w_n=n/p_n$, and note that $w_n\to\infty$. Pick a sequence $n_k\to\infty$ such that $w_{n_k}/w_{n_{k+1}}\to 0$. Define $$S=\bigcup_{n\geq 1}(\tfrac{1}{p_{n_{k+1}}}\mathbb Z\cap[w_{n_k},w_{n_{k+1}}]).$$ $S$ is a discrete set, so we can pick an integrable function $f$ that is $1$ everywhere on $S$, for example around each point we can take a piecewise linear "hat" function that is zero outside the ball of radius $2^{-n}$ around the point, then taking the maximum value of all these hats. But $$\frac{1}{n_{k+1}}\sum_{j=1}^{n_{k+1}-1}f^2\Bigl(\frac j{p_{n_{k+1}}}\Bigr)\geq \frac{w_{n_{k+1}}-w_{n_k}-1}{w_{n_{k+1}}}\to 1.$$