Linear discrete operators on the disk algebra

Authors:
Ivan V. Ivanov and Boris Shekhtman

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1987-1993

MSC (2000):
Primary 46-XX

DOI:
https://doi.org/10.1090/S0002-9939-00-05774-9

Published electronically:
December 13, 2000

MathSciNet review:
1825906

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that

and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.

**[Bo]**S. Bochkariev (1985):*Construction of interpolating basis in the space of continuous functions based on the Fejer kernel*, MIAN, SSR 172, 29-59. MR**87a:41004****[C]**L. Carleson (1958):*Research on interpolation problems*, Air Force Technical Report, AF 61(052)-238, AD 278-144.**[CPS]**P. Casazza, R. Pengra and C. Sundberg (1980):*Complemented ideals in the disk Algebra*, Israel J. Math., 37: 76-83. MR**82d:30029****[D]**P. Duren (1970):*Theory of**spaces*, Academic Press. MR**42:3552****[ISh]**I. Ivanov and B. Shekhtman (1998):*Linear Discrete Operators and Recovery of Functions*, Proceedings of Approximation Theory IX, Vol 1: 159-166. CMP**2000:11****[LR]**J. Lindenstrauss and H. Rosenthal (1969):*The**spaces*, Israel J. Math., 7:325-349. MR**42:5012****[Ni]**N. Nikol'skii (1986):*Treatise on the Shift Operator*, Springer- Verlag, Berlin Heidelberg. MR**87i:47042****[P]**J. Partington (1997):*Interpolation, Identification and Sampling*, Clarendon Press, Oxford. MR**99m:41001****[Sh]**B. Shekhtman (1992):*Discrete approximating operators on function algebras*, Constructive Approximation 8: 371-377. MR**93h:46071****[So]**G. Somorjai (1980):*On discrete operators in the function space*, Constructive Approximation Theory 77, Sofia, 489-496.**[To]**V. Totik (1984):*Recovery of**- Functions*, Proceedings of the American Mathematical Society, Vol. 90, 531-537. MR**85j:30073**

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Additional Information

**Ivan V. Ivanov**

Affiliation:
Department of Mathematics, Texas A & M University, College Station, Texas 77843

Email:
ivanov@math.tamu.edu

**Boris Shekhtman**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620

Email:
boris@2chuma.cas.usf.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05774-9

Received by editor(s):
November 30, 1998

Received by editor(s) in revised form:
October 15, 1999

Published electronically:
December 13, 2000

Additional Notes:
The results in this paper are part of the Ph.D. thesis of the first author.

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2000
American Mathematical Society