asymptotic behaviour of functions..

tpm

let be f(x) and g(x) 2 arithmetical functions related by:

[tex] g(x)= \sum_{n=0}^{\infty} f(x/n)h(n) [/tex]


where h(n) is a known function, my question is, if we know that

[tex] g(x) \sim x^{a} [/tex] a>0 and real.

What could we say about [tex] f(x) \sim ? [/tex] knowing the value of h(n).

i have tried approximating the sum by an integral and we get:

[tex] g(x)= \int_{0}^{\infty} f(x/u)h(u)du [/tex]