# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Beurling type density theorems in the unit disk. (English) Zbl 0789.30025
We consider two equivalent density concepts for the unit disk that provide a complete description of sampling and interpolation in ${A}^{-n}$ (the Banach space of functions $f$ analytic in the unit disk with ${\left(1-|z|}^{2}{\right)}^{n}|f\left(z\right)|$ bounded). This study reveals a ‘Nyquist density’: A sequence of points is (roughly speaking) a set of sampling if and only if its density in every part of the disk is strictly larger than $n$, and it is a set of interpolation if and only if its density in every part of the disk is strictly smaller than $n$. Similar density theorems are also obtained for weighted Bergman spaces.

##### MSC:
 3e+06 Moment problems, interpolation problems
##### Keywords:
uniform densities of sequences; sampling; Bergman spaces
##### References:
 [1] Amar, E.: Suites d’interpolation pour les classes de Bergman de la boule et du polydisque deC n . Can. J. Math.30, 711-737 (1978) · Zbl 0385.32014 · doi:10.4153/CJM-1978-062-6 [2] Beurling, A.: In: Carleson, L., Malliavin, P., Neuberger, J., Wermer, J. The collected Works of Arne Beurling, vol. 2, Harmonic Analysis, pp. 341-365. Boston: Birkhäuser 1989 [3] Bruna, J., Pascuas, D.: Interpolation inA ??. J. Lond. Math. Soc.40 452-466, (1989) · Zbl 0652.30026 · doi:10.1112/jlms/s2-40.3.452 [4] Carleson, L. An interpolation problem for bounded analytic functions. Am. J. Math.80, 921-930 (1958) · Zbl 0085.06504 · doi:10.2307/2372840 [5] Coifman, R.R., Rochberg, R.: Representation theorems for holomorphic and harmonic functions inL p . Astérisque77, 11-66 (1980) [6] Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory36, 961-1005 (1990) · Zbl 0738.94004 · doi:10.1109/18.57199 [7] Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc.72, 341-366 (1952) · doi:10.1090/S0002-9947-1952-0047179-6 [8] Hoffman, K.: Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice-Hall: 1962 [9] Korenblum, B.: An extension of the Nevanlinna theory. Acta Math.135, 187-219 (1975) · Zbl 0323.30030 · doi:10.1007/BF02392019 [10] Landau, H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math.117, 37-52 (1967) · Zbl 0154.15301 · doi:10.1007/BF02395039 [11] Luecking, D.: Closed range restriction operators on weighted Bergman spaces. Pac. J. Math.110, 145-160 (1984) [12] Rochberg, R.: Interpolation by functions in Bergman spaces. Mich. Math. J.28, 229-237, (1982) [13] Seip, K.: Regular sets of sampling and interpolation for weighted Bergman spaces. Proc. Am. Math. Soc.117, 213-220 (1993) · doi:10.1090/S0002-9939-1993-1111222-5 [14] Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J. Reine Angew. Math.429, 91-106 (1992) · Zbl 0745.46034 · doi:10.1515/crll.1992.429.91 [15] Seip, K., Wallstén, R.: Density theorems for sampling and interpolation in the Bargmann-Fock space. II. J. Reine Angew. Math.429, 107-113 (1992) [16] Shields, A.L., Williams, D.L., Bounded projections, duality, and multipliers in spaces of analytic functions. Trans. Am. Math. Soc.162, 287-302 (1971) [17] Young, R.M.: An Introduction to the Theory of nonharmonic Fourier Series. New York: Academic Press 1980 [18] Zhu, K.: Operator Theory in Function Spaces. New York: Marcel Dekker 1990