Test functions in constrained interpolation
Authors:
Michael A. Dritschel and James Pickering
Journal:
Trans. Amer. Math. Soc. 364 (2012), 55895604
MSC (2010):
Primary 47A57; Secondary 32C15, 46E20, 46E22, 47B32
Published electronically:
June 8, 2012
MathSciNet review:
2946923
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Abstract 
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Additional Information
Abstract: We give a set of test functions for , the algebra of bounded holomorphic functions on the disk with first derivative equal to 0, whose interpolation problem was studied by Davidson, Paulsen, Raghupathi and Singh (2009). We show that this set of test functions is minimal by relating these ideas to realization and interpolation problems.
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 [Abr79]
 Marine B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), no. 2, 195203. MR 532320 (80j:30052)
 [AHR08]
 Jim Agler, John Harland, and Benjamin J. Raphael, Classical function theory, operator dilation theory, and machine computation on multiplyconnected domains, Memoirs of the American Mathematical Society 191 (2008), no. 892, viii+159. MR 2375060 (2009d:47011)
 [AM02]
 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, American Mathematical Society, 2002. MR 1882259 (2003b:47001)
 [BBT08]
 Joseph A. Ball, Vladimir Bolotnikov, and Sanne Ter Horst, A constrained NevanlinnaPick interpolation problem for matrixvalued functions, ArXiv: 0809.2345, September 2008.
 [BC96]
 Joseph A. Ball and Kevin F. Clancey, Reproducing kernels for Hardy spaces on multiply connected domains, Integral Equations and Operator Theory 25 (1996), 3557. MR 1386327 (97f:46042)
 [DM05]
 Michael A. Dritschel and Scott McCullough, The failure of rational dilation on a triply connected domain, Journal of The American Mathematical Society 18 (2005), no. 4, 873918. MR 2163865 (2008i:47024)
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 , Test functions, kernels, realizations and interpolation, Operator Theory, Structured Matrices and Dilations: Tiberiu Constantinescu Memorial Volume, pp. 153179, Theta Foundation, Bucharest, 2007. MR 2389623 (2008k:47033)
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 Scott McCullough and Vern Paulsen, envelopes and interpolation theory, Indiana University Mathematics Journal 51 (2002), no. 2, 479505. MR 1909298 (2003e:46086)
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 James Pickering, Counterexamples to rational dilation on symmetric multiply connected domains, Complex Analysis and Operator Theory 4 (2010), no. 1, 5595. MR 2643788
 [Rag08]
 Mrinal Raghupathi, NevanlinnaPick interpolation for , Integral Equations and Operator Theory 63 (2009), 103125. MR 2480640 (2010f:47033)
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Additional Information
Michael A. Dritschel
Affiliation:
Department of Mathematics, University of NewcastleuponTyne, NewcastleuponTyne, NE1 7RU, United Kingdom
Email:
m.a.dritschel@ncl.ac.uk
James Pickering
Affiliation:
Department of Mathematics, University of NewcastleuponTyne, NewcastleuponTyne, NE1 7RU, United Kingdom
Email:
james.pickering@ncl.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299472012055152
PII:
S 00029947(2012)055152
Keywords:
Interpolation,
realizations,
NevanlinnaPick,
test functions
Received by editor(s):
April 21, 2009
Published electronically:
June 8, 2012
Additional Notes:
This paper is based on work contributing to the second author’s Ph.D. thesis, at the University of NewcastleuponTyne, under the supervision of the first author. The work was funded in part by the Engineering and Physical Sciences Research Council.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
