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Orthogonal polynomials, \(L^ 2\)-spaces and entire functions. (English) Zbl 0879.42015

We consider determinate measures \(\mu\) having moments of every order and finite index of determinacy. The latter means that there exists a polynomial \(p\) such that the measure \(|p|^2\mu\) is indeterminate. If the smallest degree of a polynomial \(p\) with this property is \(k+1\), we call \(k\) the index of determinacy, denoted \(\text{ind}(p)= k\). In a previous paper [Trans. Am. Math. Soc. 347, No. 8, 2795-2811 (1995; Zbl 0863.42019)], we have characterized these measures as those derived from \(N\)-extremal indeterminate measures by removal of \(k+1\) point masses. In particular, they are discrete measures.
We prove that the space \(L^2(\mu)\) can be realized as a Hilbert space of entire functions in the sense of de Branges, and that the entire functions occurring are of minimal exponential type in the Cartwright class.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
44A60 Moment problems
30D15 Special classes of entire functions of one complex variable and growth estimates
30H05 Spaces of bounded analytic functions of one complex variable

Citations:

Zbl 0863.42019
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