Contrary to Fourier, Laplace, Cauchy, or Poisson transforms, the power moments of a positive measure, fast decaying at infinity on the real line, do not determine the original measure. The non-uniqueness phenomenon was analyzed in depth by Stieltjes, via continued fraction expansions of the formal generating series of moments. I will sketch the main ideas of Stieltjes celebrated memoir, to continue with an account of a not less foundational contribution put by Carleman in book format. Then we will touch on Marcel Riesz’s pioneering work on extensions of positive linear functionals, to return from another perspective to the Christoffel-Darboux kernel. All in 1D.

Some unfinished parallel studies in $n$D, marred by pitfalls and open problems will be discussed.

The video of this talk is available on the IISc Math Department channel.