The strong truncated Hamburger moment problem with and without gaps

Abstract

The strong truncated Hamburger moment problem (STHMP) of degree (−2k1,2k2) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R, such that βi=∫xidμ(−2k1≤i≤2k2). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. Then, using the equivalence with the STHMP of degree (−2k,2k), we obtain the solution of the 2–dimensional truncated moment problem (TMP) of degree 2k with variety xy=1, first solved by Curto and Fialkow [22]. Our addition to their result is the fact previously known only for k=2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k1,2k2) with one missing moment in the sequence, i.e., β−2k1+1 or β2k2−1, which also gives the solution of the TMP with variety x2y=1 as a special case, first studied by Fialkow in [33].