Yes.

In an exercise, Stuart & Ord (*Kendall's Advanced Theory of Statistics*, 5th Ed., Ex. 3.12) quote a 1918 result of TJ Stieltjes (which apparently appears in his *[Oeuvres Completes][1],*):

> If $f$ is an odd function of period $\frac{1}{2}$, show that
> 
> $$\int_0^\infty x^r x^{-\log x} f(\log x) dx = 0$$
> 
> for all integral values of $r$.  Hence show that the distributions
> 
> $$dF = x^{-\log x}(1 - \lambda \sin(4\pi \log x))\ dx, \quad 0 \le x \lt \infty;\quad 0 \le |\lambda| \le 1,$$
> 
> have the same moments whatever the value of $\lambda$.

(In the original, $|\lambda|$ appears only as $\lambda$; the restriction on the size of $\lambda$ arises from the requirement to keep all values of the density function $dF$ non-negative.)  The exercise is easy to solve via the substitution $x = \exp(y)$ and completing the square.  The case $\lambda=0$ is the well-known [lognormal distribution](http://en.wikipedia.org/wiki/Log-normal_distribution).

![enter image description here][2]

*The blue curve corresponds to $\lambda=0$, a lognormal distribution.  For the red curve, $\lambda = -1/4$ and for the gold curve, $\lambda = 1/2$.*


  [1]: http://archive.org/details/oeuvrescompltesd02stie
  [2]: https://i.sstatic.net/2Qm8O.png