Number of clusters, deconvolution and classical problem of moments

In the paper there is given a connection between one special case of cluster analysis, deconvolution problem, and classical moment problem. Namely, the methods used there are applied to solve deconvolution problem for the case of one known distribution and another one concentrated in unknown finite number of points. These results can be applied to estimate a number of clusters for the case of scale or location mixture of identical distributions. keywords: number of clusters, deconvolution problem, classical moment problem, scale and location mixtures.