integral equation

Question

Given the integral equation

$$\exp(x)-1=\int_0^{\infty} \frac{\mathrm dt}{t}\operatorname{frac}\left(\frac{ \sqrt x}{\sqrt t}\right) f(t)\;,$$

where $\operatorname{frac}$ denotes the fractional part of a number, $ \operatorname{frac}(x)= x-\lfloor x\rfloor$.

My questions are:

1. Can we deduce from this integral equation that $ f(x)= O(x^{1/4+\epsilon}) $ for some positive $\epsilon$?

2. Can we solve this integral by the Hilbert-Schmidt method?

Answer 1

You cannot conclude anything for the asymptotics of the integrand, because the function can have very high, very narrow peaks that contribute almost nothing to the integral.