Inverse of infinite Hankel moment matrices. (English) Zbl 1412.42064
For a given indeterminate Hamburger moment sequence \((s_n)_{n\ge0}\), let \(\mathcal{H}\) denote the matrix \(\{s_{m+n}\}\), which is a positive definite Hankel matrix. The authors address the question of the existence of an infinite symmetric matrix \(\mathcal{A}=\{a_{j,k}\}\) such that \(\mathcal{AH=I}\), where \(\mathcal{I}\) is the identity matrix, and the product \(\mathcal{AH}\) is defined by absolutely convergent series. In this case, the authors say that the moment problem has the property (aci). Considering mainly symmetric indeterminate moment problems, the authors exhibit several sufficient conditions implying (aci), in terms of the recurrence coefficients of the associated orthonormal polynomials.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 44A60 | Moment problems |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
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