We prove convergence of diagonal multipoint Pad\'e approximants of Stieltjes-type functions when a certain moment problem is determinate. This is used for the study of the convergence of Fourier--Pad\'e and nonlinear Fourier--Pad\'e approximants for such type of functions.

We introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: $T((a_n)_n)=1/(a_0+\dots +a_n)$. We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers $(1+1/2+\dots +1/(n+1))^{-1}$.

For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures $\mu$ in $V_{2n}$ (this is the set of solutions of the problem of degree $2n$) for which the polynomials up to degree $n$ are dense in the corresponding space ${\cal L}^2(\mu)$. These matrices of measures are exactly the extremal measures of the set $V_n$.