Complex Orthogonal Polynomials and Numerical Quadrature via Hyponormality

The operator approach to the truncated multidimensional moment problem

“…Notice that the truncated two-dimensional moment problem with rectangular data appeared in [8], [29]. The general case of the set K d ,d ,...,dn was proposed to the author by Vasilescu (private communication).…”

“…Theorem 2. Let the moment problem (1) with K = K + K, for an admissible nite set K and some S = (s k ) k∈K , be given and conditions (8), (17) hold. Fix an arbitrary l ∈ Z ,n .…”

“…Theorem 5. Let the moment problem (1) with K = K + K, for an admissible nite set K and some S = (s k ) k∈K , be given and conditions (8), (17) hold. For every j ∈ Ω , denote by α t (j) (t ∈ Ω ) an arbitrary complex solution of the following linear algebraic system:…”

“…Let the moment problem (1) with K = K + K, for a set K as in Relation (5) and some S = (s k ) k∈K , be given. Suppose that conditions (8), (17) hold and that S is dimensionally stable. Then S is completely self-adjoint and the moment problem (1) has a solution.…”

“…The validity of the inverse implication in (44) is also of interest. Suppose that for an admissible nite set K ⊂ Z n + the moment problem (1), with K = K + K and some S = (s k ) k∈K , is given and conditions (8), (17) hold. De ne the associated Hilbert space H and its subspace H .…”