Papers › The strong truncated Hamburger moment problem with and without gaps
The strong truncated Hamburger moment problem with and without gaps
Aljaž Zalar
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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∖{0}, such that βᵢ=∫xⁱdμ (-2k₁≤i≤2k₂). Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. 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. 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 β_(2k₂-1), which also gives the solution of the TMP with variety x²y=1 as a special case, first studied by Fialkow.
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