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The sequential stability index of a function space

  • J. M. Anderson (a1) and J. E. Jayne (a1)

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One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {zC: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each zD, Pn(Z) → f(z) as n → ∞,

where we have used the notation

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References

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1. Jayne, J. E., “Descriptive set theory in compact spaces”, Amer. Math. Soc. Notices, 17 (1970), 268.
2. Kuratowski, K., “Sur une généralisation de la notion d'homéomorphie”, Fund. Math., 22 (1934), 206220.
3. Kuratowski, K., Topology, Vol. 1 (Academic Press, New York, 1966).
4. Landau, E., Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie (Berlin, 1929).
5. Lebesgue, H., “Sur les fonctions représentables analytiquement”, Jour, de Math., 1 (1905), 139216.
6. Pelczήski, A. and Semadeni, Z., “Spaces of continuous functions III. (The space C(X) for X without perfect subsets.)”, Studia Math., 18 (1959), 211222.
7. Saks, S. and Zygmund, A., Analytic functions (Warszawa-Wroclaw, 1952).
8. Sarason, D., “On the order of a simply connected domain”, Michigan Math. J., 15 (1968), 129133.
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