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Sampling and interpolation of entire functions and exponential systems in convex domains. (English) Zbl 0819.30021
For an appropriate convex function \(h\) on the unit circle, the authors consider the Bargmann-Fock spaces of entire functions \(f\) under the norm \[ \int \int_ C \bigl | f(z) \bigr |^ 2 \exp \bigl( - 2h (\arg z) | z |^ 2 \bigr) dm(z) \] or \(\sup | f(z) | \exp ('')\). For such spaces of entire functions the authors study associated sequences of complex numbers called sets of sampling and sets of interpolation and characterise such sets through NASC in terms of lower and upper angular densities. Cfr. H. J. Landau, Acta Math. 117, 37- 52 (1967; Zbl 0154.153); K. Seip, J. Reine Angew. Math. 429, 91- 106 (1992; Zbl 0745.46034); K. Seip and R. Wallstén, ibid. 107-113 (1992; Zbl 0745.46033)]. The sampling and interpolation results are applied to study expansions in series of exponentials in the Smirnov space \(E^ 2 (G)\), \(G\) being a bounded convex set in \(C\), that consists of the closure of all polynomials in \(z\) with respect to the norm \(\int_{\partial G} | f(z) |^ 2 | dz |\). There is an appendix devoted to the construction of certain entire functions of regular growth, called “analogues of sine type functions”.

MSC:
30E05 Moment problems and interpolation problems in the complex plane
30D10 Representations of entire functions of one complex variable by series and integrals
30D15 Special classes of entire functions of one complex variable and growth estimates
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