Approximaths

Stieltjes functions II

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In the previous post, we studied the direct problem of constructing a Stieltjes function from a given weight function and explored the associated Stieltjes series and Padé approximants. Now, let’s consider the opposite problem and first define the weight function. In this case, we choose:

\displaystyle \rho(t)=\begin{cases}1 & \text{if } 0 < t < 1 \\ 0 & \text{otherwise}\end{cases}

this implies that

\displaystyle a_n = \int_{0}^{\infty} \rho(t) t^n \, dt
\displaystyle = \int_{0}^{1} t^n \, dt = \frac{1}{n+1}

This implies that the corresponding Stieltjes series is (see previous post):

\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n+1}

(which has radius of convergence 1 and coincides with f(x) for |x|<1) and the Stieltjes function is:

\displaystyle f(x) = \frac{\log(1+x)}{x}

(where \log denotes the principal branch, with cut along (-\infty,-1]). Thus f is analytic in \mathbb{C}\setminus(-\infty,-1].

Properties of Stieltjes functions:

  • f(x) is analytic in the cut complex plane \mathbb{C}\setminus(-\infty,0] (in the present example the natural cut is actually shorter: \mathbb{C}\setminus(-\infty,-1]).

  • f(x)\ge 0 for all x>0.

  • f(x) \to 0 as |x| \to \infty in the cut complex plane, under appropriate assumptions on \rho(t).

  • f(x) \sim \sum_{n=0}^{\infty} (-1)^n a_n x^n as |x| \to 0 in the cut complex plane.

  • -f(x) is Herglotz (i.e., \mathrm{Im}(-f(x)) has the same sign as \mathrm{Im}(x)).

Under appropriate additional hypotheses, a converse characterization also holds: a function satisfying the corresponding analyticity, asymptotic, positivity, and Herglotz properties is a Stieltjes function. It can be proved rigorously that appropriate Padé approximants of a Stieltjes series converge. This is a particularly useful result (which we have already illustrated with examples), as it implies that the Padé approximants constructed from the Stieltjes series converge to the Stieltjes function for x in the cut complex plane, under appropriate hypotheses.

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