The strong truncated Hamburger moment problem with and without gaps

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