Assume that $h:=h_n>0$ is a sequence satisfying $h\to 0, nh\to\infty$ as $n\to\infty$. Let $K$ be a real function which is nonnegative, symmetric, bounded, Lipschitz continuous and with compact support. Define $K_h(u)=1/hK(u/h)$.
I want to know if for any $x\in[0,1]$
$$\bigg\lvert \frac{1}{n}\sum_{i=1}^n K_h(i/n-x)-\int_0^1 K_h(y-x)dy \bigg\rvert\leq \frac{C}{n}?$$ for some constant $C>0$. I saw results that suggest that this is the case.
Reference: Wand and Jones p. 121 (They actually assumed less things than me about the function $K$, except that $K'$ is bounded. But Lipschitz continuity implies $K'$ bounded. This might be crucial)
I will print the important parts:
(only the first equality above is of interest here) with
When $l=0$, his result is equivalent to what I'm asking for. See Big oh Notation.