In this work, the maximum entropy method is used to solve the extension problem associated with a positive-definite function, or distribution, defined on an interval of the real line. Garbardo computes explicitly the entropy maximizers corresponding to various logarithmic integrals depending on a complex parameter and investigates the relation to the problem of uniqueness of the extension. These results are based on a generalization, in both the discrete and continuous cases, of Burg's maximum entropy theorem.

Readership

Research Mathematicians.

Table of Contents

• Facts and definitions
• The discrete case
• Positive-definite distributions on an interval \((-A,A)\)
• The non-degenerate case
• A closure problem in \(L^2_\mu (\hat {\mathbb R})\)
• Entropy maximizing measures in \(\scr M_A(Q)\)
• Uniqueness of the extension