In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution.
Let F denote a real or complex-valued function defined on the positive real axis such that $F(x) = 0$ for $0<x<1$.
$\alpha$ is any arithmetical function.
Generalized convolution is then defined by $(\alpha \circ F)(x)= \sum_{n \leq x} \alpha(n)F(\frac{x}{n})$
Then the text says:
" If F(x) = 0 for all nonintegral values of x, the restriction of F to integers is an arithmetical function and we find that $(\alpha \circ F)(m) =(\alpha \ast F)(m)$"
Here $(\alpha \ast \beta)(n) = \sum_{d | n} \alpha(n)\beta(\frac{n}{d})$.
My question is why does F(x) have to be 0 for all nonintegral values of x for this to work?
Edit: Is the point that we want $F(\frac{x}{n})$ to be 0 when $\frac{x}{n}$ is not an integer?
Yes. For example, to isolate $F$, take $\alpha$ to be the unit function $u$. Then the generalized convolution for integer $m \geq 1$ is
$(u \circ F)(m) = \sum_{n \leq m} F(\frac{m}{n})$
If $F(\frac{m}{n}) = 0$ whenever $\frac{m}{n}$ is not an integer, the above sum reduces to
$\sum_{n | m} F(\frac{m}{n}) = (u \ast F)(m)$
I think Apostol could have simply said "If $F(x) = 0$ for all nonintegral values of x, we find that $(\alpha \circ F)(m) = (\alpha \ast F)(m)$ for all integers $m \geq 1$..."