Entropy-convergence in Stieltjes and Hamburger moment problem
Let \(\{\mu_n\}^\infty_0\) be a sequence of moments for the Hamburger moment problem and let \(f(x)\) be the (unknown) density function: \(\mu_n= \int_\mathbb{R} x^n f(x)dx\), \(n=0,1, \dots\) The maximal entropy principle is a formal procedure for obtaining approximants \(f_M\) of \(f\) with first \(M+1\) moments \(\{\mu_j \}^M_0\) equal to the given ones. It is shown that if the moment sequence uniquely determines the density function, then \(\lim_{M \to \infty} H[f_M] =H[f]\), where \(H[f]=-\int_\mathbb{R} f(x)\ln f(x) dx\) is the Boltzmann-Shannon entropy. The same result is obtained for the Stieltjes moment problem. In a previous publication the authors considered the case of finite moment sequences [J. Math. Phys. 35, No. 12, 6748-6756 (1994; Zbl 0830.41020)].
scientific article; zbMATH DE number 1183162