Interpolating sequences in polydisks
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- by Eric P. Kronstadt PDF
- Trans. Amer. Math. Soc. 199 (1974), 369-398 Request permission
Abstract:
Let ${D^n}$ be the unit polydisk in ${{\mathbf {C}}^n}$, $A$ be a uniform algebra, ${H^\infty }({D^n},A)$, the space of bounded analytic $A$ valued functions on ${D^n}$, ${l^\infty }A$, the space of bounded sequences of elements in $A$. A sequence, $S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}$ will be called an interpolating sequence with respect to $A$ if the map $T:{H^\infty }({D^n},A) \to {l^\infty }A$ given by $T(f) = \{ f({a_i})\} _{i = 1}^\infty$ is surjective. In 1958, L. Carleson showed that for $n = 1,S$ is interpolating w.r.t. ${\mathbf {C}}$ iff $S$ satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where $n > 1$ and $A \ne {\mathbf {C}}$. In particular, we show that if $S \subset {D^n}$ is uniformly separated and $S \subset {W_1} \times {W_2} \times \cdots \times {W_n}$ (where each ${W_j}$ is a region in $D$ lying between two circular arcs which intersect twice on the boundary of $D$) then $S$ is an interpolating sequence w.r.t. any uniform algebra. If $S \subset {D^n}$ is uniformly separated and $S \subset D \times {W_2} \times \cdots \times {W_n}$ then $S$ is interpolating w.r.t. ${\mathbf {C}}$. Other examples and generalizations of interpolating sequences are discussed.References
- William G. Bade and Philip C. Curtis Jr., Embedding theorems for commutative Banach algebras, Pacific J. Math. 18 (1966), 391–409. MR 202001, DOI 10.2140/pjm.1966.18.391
- Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
- Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
- Alain Dufresnoy, Sur les compacts d’interpolation du spectre d’une algèbre uniforme et la propriété d’extension linéaire bornée, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A568–A571 (French). MR 300094
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Irving Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415–435. MR 173957, DOI 10.1090/S0002-9947-1962-0173957-5
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
- Eric P. Kronstadt, General interpolating sequences in disks and polydisks, Bull. Amer. Math. Soc. 80 (1974), 132–137. MR 350427, DOI 10.1090/S0002-9904-1974-13388-4
- A. G. Naftalevič, On interpolation by functions of bounded characteristic, Vilniaus Valst. Univ. Moksl Darbai. Mat. Fiz. Chem. Moksl Ser. 5 (1956), 5–27 (Russian). MR 0120387
- D. J. Newman, Interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 92 (1959), 501–507. MR 117350, DOI 10.1090/S0002-9947-1959-0117350-X
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892
- Kam-fook Tse, Nontangential interpolating sequences and interpolation by normal functions, Proc. Amer. Math. Soc. 29 (1971), 351–354. MR 274777, DOI 10.1090/S0002-9939-1971-0274777-4
- Nicholas Th. Varopoulos, Ensembles pics et ensembles d’interpolation pour les algèbres uniformes, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A866–A867 (French). MR 279590
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 199 (1974), 369-398
- MSC: Primary 32E25; Secondary 46J10
- DOI: https://doi.org/10.1090/S0002-9947-1974-0417451-0
- MathSciNet review: 0417451