Abstract
We generalize a classical Bernstein Theorem about approximation of functions on the real line by entire functions of the exponential type, i.e. for any function f continuous and bounded on an unbounded quasi-smooth (in the sense of Lavrentiev) curve L in the complex plane we construct entire functions of exponential type σ > 0 which in some sense converge optimally to f on L as σ → ∞.
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The work was supported in part by DFG-Research Grant (Germany) BL 272/10-1. The work of V.V.A. was also supported in part by NSF grant DMS-0554344 and was conducted while visiting the Katholische Universität Eichstätt-Ingolstadt, Germany.
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Andrievskii, V.V., Blatt, HP. On Approximation by Entire Functions on an Unbounded Quasi-Smooth Curve. Comput. Methods Funct. Theory 9, 525–550 (2009). https://doi.org/10.1007/BF03321743
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DOI: https://doi.org/10.1007/BF03321743
Keywords
- Entire functions of exponential type
- approximation
- continuous
- function
- quasiconformal mapping
- quasi-smooth curve