Pick interpolation in several variables
Author:
Ryan Hamilton
Journal:
Proc. Amer. Math. Soc. 141 (2013), 20972103
MSC (2010):
Primary 47A57; Secondary 30E05, 46E22
Published electronically:
January 30, 2013
Fulltext PDF
Abstract 
References 
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Additional Information
Abstract: We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in .
 1.
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H. Bercovici and D. Westwood, The factorization of functions in the polydisc, Houston J. Math. 18 (1992), No. 1, 16. MR 1159434 (93b:32006)
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B.J. Cole, K. Lewis, and J. Wermer, Pick conditions on a uniform algebra and von Neumann inequalities, J. Funct. Anal. 107 (1992), 235254. MR 1172022 (93e:46059)
 8.
K.R. Davidson and R. Hamilton, NevanlinnaPick interpolation and factorization of linear functionals, Integral Equations Operator Theory 70 (2011), No. 1, 125149. MR 2786738 (2012e:47045)
 9.
R.G. Douglas and J. Sarkar, Some remarks on the Toeplitz corona problem, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, 51, 8189, Amer. Math. Soc., Providence, RI, 2010. MR 2648868 (2011i:46051)
 10.
S.G. Krantz, Function theory of several complex variables, The Wadsworth & Brooks/Cole Mathematics Series, Second Edition, Pacific Grove, CA, 1992. MR 1162310 (93c:32001)
 11.
S. McCullough, NevanlinnaPick type interpolation in a dual algebra, J. Funct. Anal. 135 (1996), No. 1, 93131. MR 1367626 (96j:47013)
 12.
B. Prunaru, A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces, Canadian Math. Bull., to appear.
 1.
 M.B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), 195203. MR 532320 (80j:30052)
 2.
 J. Agler and J.E. McCarthy, NevanlinnaPick interpolation on the bidisk, J. Reine Angew. Math. 506 (1999), 191204. MR 1665697 (2000a:47034)
 3.
 J. Agler and J.E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44, Amer. Math. Soc., Providence, RI, 2002. MR 1882259 (2003b:47001)
 4.
 H. Bercovici, The algebra of multiplication operators on Bergman spaces, Arch. Math. (Basel) 48 (1987), 165175. MR 878428 (88e:47057)
 5.
 H. Bercovici, C. Foiaş, and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, 56, Amer. Math. Soc., Providence, RI, 1985. MR 787041 (87g:47091)
 6.
 H. Bercovici and D. Westwood, The factorization of functions in the polydisc, Houston J. Math. 18 (1992), No. 1, 16. MR 1159434 (93b:32006)
 7.
 B.J. Cole, K. Lewis, and J. Wermer, Pick conditions on a uniform algebra and von Neumann inequalities, J. Funct. Anal. 107 (1992), 235254. MR 1172022 (93e:46059)
 8.
 K.R. Davidson and R. Hamilton, NevanlinnaPick interpolation and factorization of linear functionals, Integral Equations Operator Theory 70 (2011), No. 1, 125149. MR 2786738 (2012e:47045)
 9.
 R.G. Douglas and J. Sarkar, Some remarks on the Toeplitz corona problem, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, 51, 8189, Amer. Math. Soc., Providence, RI, 2010. MR 2648868 (2011i:46051)
 10.
 S.G. Krantz, Function theory of several complex variables, The Wadsworth & Brooks/Cole Mathematics Series, Second Edition, Pacific Grove, CA, 1992. MR 1162310 (93c:32001)
 11.
 S. McCullough, NevanlinnaPick type interpolation in a dual algebra, J. Funct. Anal. 135 (1996), No. 1, 93131. MR 1367626 (96j:47013)
 12.
 B. Prunaru, A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces, Canadian Math. Bull., to appear.
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Additional Information
Ryan Hamilton
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L–3G1, Canada
Email:
rhamilto@uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S000299392013115716
PII:
S 00029939(2013)115716
Keywords:
Pick interpolation,
reproducing kernel,
dual algebras
Received by editor(s):
October 5, 2011
Published electronically:
January 30, 2013
Additional Notes:
This work was partially supported by an NSERC graduate scholarship
Communicated by:
Richard Rochberg
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
