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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An extremal problem for analytic functions with prescribed zeros and $ r$th derivative in $ H\sp \infty$

Authors: A. Horwitz and D. J. Newman
Journal: Trans. Amer. Math. Soc. 295 (1986), 699-713
MSC: Primary 30C75; Secondary 30D50
MathSciNet review: 833704
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Abstract: Let $ ({\alpha _1}, \ldots ,{\alpha _n})$ be $ n$ points in the unit disc $ U$. Suppose $ g$ is analytic in $ U$, $ g({\alpha _1}) = \cdots = g({\alpha _n}) = 0$ (multiplicities included), and $ \Vert g\prime\Vert _{\infty } \leq 1$. Then we prove that $ \vert g(z)\vert \leq \vert\phi (z)\vert$ for all $ z \in U$, where $ \phi ({\alpha _1}) = \cdots = \phi ({\alpha _n}) = 0$ and $ \phi\prime(z)$ is a Blaschke product of order $ n - 1$. We extend this result in a natural way to convex domains $ D$ with analytic boundary. For $ D$ not convex we show that there is no extremal function $ \phi $.

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  • [1] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • [2] S. D. Fisher and Charles A. Micchelli, The 𝑛-width of sets of analytic functions, Duke Math. J. 47 (1980), no. 4, 789–801. MR 596114
  • [3] A. Horwitz, Restricted interpolation and optimal recovery of certain classes of functions, Doctoral Thesis, Temple University, Philadelphia, Pa., 1984.
  • [4] Charles A. Micchelli and Theodore J. Rivlin (eds.), Optimal estimation in approximation theory, Plenum Press, New York-London, 1977. The IBM Research Symposia Series. MR 0445154
  • [5] D. J. Newman, Polynomials and rational functions, Approximation theory and applications (Proc. Workshop, Technion—Israel Inst. Tech., Haifa, 1980) Academic Press, New York-London, 1981, pp. 265–282. MR 615416
  • [6] H. L. Royden, The boundary values of analytic and harmonic functions, Math. Z. 78 (1962), 1–24. MR 0138747,
  • [7] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR 0218588

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Article copyright: © Copyright 1986 American Mathematical Society

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