Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: December 13, 2000
MathSciNet review: 1825906
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let ${\cal A}$ be the disk algebra. In this paper we address the following question: Under what conditions on the points $z_{k,n} \in \mathbf{ D}$ do there exist operators $L_n :{\cal A} \to {\cal A}$ such that

\begin{displaymath}L_n f = \sum^{m_n}_{k=1} f(z_{k,n})l_{k,n},\qquad f, l_{k,n} \in {\cal A}, \end{displaymath}

and $L_nf \to f$, $n \to \infty$, for every $f \in {\cal A}$? Here the convergence is understood in the sense of $sup$ norm in $\cal A$. Our first result shows that if $z_{k,n}$ satisfy Carleson condition, then there exists a function $f \in {\cal A}$ such that $L_nf \not\to f$, $n \to \infty$. 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 $L_n$ are required to be projections, then for any choice of $z_{k,n}$ the operators $L_n$ 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.

References [Enhancements On Off] (What's this?)

  • [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 $H^p$ 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 ${\cal L}_{p}$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 $\cal A$, Constructive Approximation Theory 77, Sofia, 489-496.
  • [To] V. Totik (1984): Recovery of $H^p$ - Functions, Proceedings of the American Mathematical Society, Vol. 90, 531-537. MR 85j:30073

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46-XX

Retrieve articles in all journals with MSC (2000): 46-XX

Additional Information

Ivan V. Ivanov
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843

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

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

American Mathematical Society