Measures with Finite Index of Determinacyor a Mathematical Modelfor

In this note measures with nite index of determinacy (i.e. determinate measures for which there exists a polynomial p such that jpj 2 is indeterminate), are characterizated in terms of the operator associated to its Jacobi matrix. Using this characterization, we show that such determinate measures with nite index of de-terminacy (Jekyll) turn out to be indeterminate (Hyde) when considered as matrices of measures. 1. Results. By M we denote the set of positive measures on R having moments of every order and innnite support. A measure 2 M is determinate if no other measure has the same moments as those of , otherwise is indeterminate. With 2 M we can associate the sequence (p n) n of orthonormal polynomials. We always assume that p n is of degree n with positive leading coeecient, and this condition together with orthonormality determines (p n) n uniquely from. The sequence of polynomials (p n) n satisses the so-called three-term recurrence formula tp n (t) = a n+1 p n+1 (t) + b n p n (t) + a n p n?1 (t); where a n > 0, b n 2 R and p ?1 (t) = 0. The well-known theorem of Favard establishes that this algebraic relation characterizes the orthonormality of (p n) n with respect to a positive measure. The so-called Jacobi matrix is deened from the three-term recurrence relation as follows: It is the matrix representation of the operator of multiplication by t in the space P of complex polynomials with respect to the orthonormal basis (p n).