Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function.
I know one can bound sums of the form
$$
\sum\_{x \leq X} e(f(x))
$$
via for example Van der Corputs's result, provided we make assumptions 
on the range of the derivatives. I was wondering if there were results of this type
for weighted sums. Can we bound a sum of the form 
$$
\sum\_{x \leq X} g(x) e(f(x)) ?
$$
If so I was wondering what kind of assumption would be needed for g(z).
(In particular, would it possible for $g(z) = 2^z$, for example? ) Thanks!



Bound on exponential sum with weights
You can use van der Corputs's method for weighted sums as well, see Ch. III in



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> Karatsuba A. A., Voronin S. M. The Riemann zeta-function Walter de Gruyter & Co., 1992.
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