Geometric condition for universal interpolation in $\hat {\mathcal {E}}’$
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- by William A. Squires PDF
- Trans. Amer. Math. Soc. 280 (1983), 401-413 Request permission
Abstract:
It is known that if $h$ is an entire function of exponential type and $Z(h) = {\{ {z_k}\} _{k = 1}}$ with $|h’({z_k})| \geqslant \varepsilon \exp (- c|{z_k}|)$ for constants $\epsilon$, $C$ independent of $k$, then $\{ {z_k}\} _{k = 1}^\infty$ is a universal interpolation sequence. That is, given any sequence of complex numbers $\{ {a_k}\} _{k = 1}^\infty$ such that $|{a_k}| \leqslant A \exp (B|{z_k}|)$ for constants $A,B$ independent of $K$ then there exists $g$ of exponential type such that $g({z_k}) = {a_k}$. This note is concerned with finding geometric conditions which make $\{ {z_k}\} _{k = 1}^\infty$ a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for $\{ {z_k}\} _{k = 1}^\infty$ to be a universal interpolation sequence is that $\int _0^{|{z_k}|} {n({z_k},t) dt/t \leqslant C|{z_k}| + D,k = 1} , 2,\ldots$, where $n({z_k},t)$ is the number of points of $\{ {z_k}\} _{k = 1}^\infty$ in the disc of radius $t$ about ${z_k}$, excluding ${z_k}$, and $C,D$ are constants independent of $k$. Results for the space $\hat {\mathcal {E}}^\prime = \{ f\;{\text {entire}}||f(z)| \leqslant A\;\exp [B|\operatorname {Im} z| + B\log (1 + |z|^{2})]\}$ are given but the theory is not as complete as for the above example.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 280 (1983), 401-413
- MSC: Primary 30E05; Secondary 30D15, 42A38
- DOI: https://doi.org/10.1090/S0002-9947-1983-0712268-7
- MathSciNet review: 712268