## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1988), 100 - 115

### The convergence of entropic estimates for moment problems

#### Abstract

We show that if $x_n$ is optimal for the problem \[ sup\left\{\sum_{x_n}{1} log x(s)ds | \sum_0^1 (x(s)- \hat{x}(s))s^i ds = 0 , i = 0, \cdots,n , 0 \leq x \in L_1[0,1]\right\}, \] then $\frac{1}{x_n} \rightarrow \frac{1}{\hat{x}}$ weakly in $L_1$ (providing $\hat{x}$ is continuous and strictly positive). This result is a special case of a theorem for more general entropic objectives and underlying spaces.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416340107

**Mathematical Reviews number (MathSciNet)**

MR1009598

**Zentralblatt MATH identifier**

0673.41021

#### Citation

Lewis, A. S. The convergence of entropic estimates for moment problems. Workshop/Miniconference on Functional Analysis and Optimization, 100--115, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1988. https://projecteuclid.org/euclid.pcma/1416340107