# Approximating a Riemann integrable function with a smooth function

Let $$F: [0,1]^n \to \mathbb{R}$$ be a Riemann integrable function. Let $$J(H) =\int_0^1.. \int_0^1 (\int_{[0,1]} H(t_1, t_2, ..., t_n) dt_1)^2 dt_2 ... dt_n.$$ Given $$\epsilon > 0$$ I want to find $$G$$ a smooth function such that $$|J(F) - J(G)| < \varepsilon.$$ I was wondering how I could start with this. Any comments are appreciated! Thank you!

Edit: My apologies for the confusion! I had mistakenly oversimplified the part of the article I am trying to understand.

Let $$J^{(m)}(H) =\int_0^1.. \int_0^1 (\int_{[0,1]} H(t_1, t_2, ..., t_n) dt_m)^2 dt_1 ... dt_{m-1} dt_{m+1}...dt_n.$$ $$I(H) = \int_0^1.. \int_0^1 H^2(t_1, t_2, ..., t_n) dt_1 ...dt_n.$$

Let $$M = \sup_{F \in S} \frac{\sum_{m=1}^nJ^{(m)} (F)} {I(F)},$$ where $$S$$ is the set of all Riemann integrable functions on $$[0,1]^n$$. Let $$\delta > 0$$ small. Then there exists $$F_0 \in S$$ such that $$\sum_{m=1}^nJ^{(m)} (F_0) > (M- \delta) I(F_0) > 0.$$ This I understand. But then they claim that since $$F_0$$ is Riemann integrable, there exists $$F_1$$ a smooth function such that $$\sum_{m=1}^nJ^{(m)} (F_1) > (M- 2\delta) I(F_1) > 0.$$ I was wondering how this follows. I had mistakenly thought that if I could understand what I had asked above then I could deduce this, but I don't think this is the case seeing the comments and answers. Any explanation for this would be appreciated! Thank you!

• I believe you mean $|J(F-G)| < \varepsilon$, (otherwise you can just take $G$ to be a constant function of value $J(F)$, which is only a very rough approximation of $F$). Aug 4 '19 at 15:49
• Thank you for the comments/answers! I have fixed the question. Aug 5 '19 at 11:52

As written, is utterly trivial. You can take a constant (what constant?) $$G$$.
If the limits can vary, you can't. Take e.g. the function that is 1 up to $$1/2$$, 2 afterwards. No smooth function can have such a derivative.