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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On dual sets generated by lacunary polynomials


Authors: Hans Dobbertin and Volker Kasten
Journal: Proc. Amer. Math. Soc. 111 (1991), 323-330
MSC: Primary 30C10; Secondary 30C15, 30C50
MathSciNet review: 935105
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Abstract: The notion of dual sets of analytic functions has been developed by Ruscheweyh. In terms of this theory a well-known convolution theorem of Szegö states that the set of all polynomials $ 1 + {a_1}z + \cdots + {a_n}{z^n}$ nonvanishing in the unit disc $ {\mathbf{D}}$ is the dual hull of $ {(1 - z)^n}$. More general for $ T = \{ {m_1}, \ldots ,{m_n}\} $ let $ {\hat P_T}$ denote the set of all lacunary polynomials $ 1 + {a_{{m_1}}}{z^{{m_1}}} + \cdots + {a_{{m_n}}}{z^{{m_n}}}$ nonvanishing in $ {\mathbf{D}}$. In this paper we investigate whether the sets $ {\hat P_T}$ are generated in a similar way. Some necessary conditions are given, and the case $ \vert T\vert \leq 3$ is completely solved.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-0935105-4
PII: S 0002-9939(1991)0935105-4
Keywords: Convolution, dual sets, lacunary polynomials, Szegö's Theorem
Article copyright: © Copyright 1991 American Mathematical Society