Generalized problem of moments and the Padé approximation (Q788148)

The generalized moment problem is to find a measure \(\mu\) (x) and two sequences \(\{a_ j(x)\}_ 0^{\infty}\) and \(\{b_ k(x)\}_ 0^{\infty}\) in \(L^ 2(X,d\mu(x))\) so that \[ s_{j+k}=\int_{X}a_ j(x)b_ k(x)d\mu(x)\quad for\quad some\quad set\quad X\subset {\mathbb{R}}. \] Assuming the existence of a solution the author constructs a system of biorthogonal polynomials from \(\{a_ j(x)\}\) and \(\{b_ k(x)\}\) and relates them to the Padé approximants of the series \(f(z)=\sum^{\infty}_{0}s_ jz^ j\) assumed analytic in \(\{z:\quad | z|<1\}\) and such that the Hankel determinants of the \(\{s_ j\}_ 0^{\infty}\) are different from zero.