Abstract
SATTES (1980) has considered best uniform approximation by smooth functions in the space C([−1,1]) of continuous real valued functions on an interval and has characterised such best approximations. GLASHOFF (1980) and PINKUS (1980) have considered a family of similar problems. We state a general problem which contains those of Sattes, Glashoff and Pinkus as special cases. A general theorem which characterises best approximations is proved in Section 2 and in Section 4 it is used to obtain an alternative proof of Sattes’ result. The link between the general theorem and the proof of Sattes’ result is provided by a theorem, concerning the zeros of certain functions, which is proved in Section 3. Section 3 is a small contribution to the theory of total positivity.
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References
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© 1984 Springer Basel AG
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Brown, A.L. (1984). Best Approximation by Smooth Functions and Related Problems. In: Brosowski, B., Deutsch, F. (eds) Parametric Optimization and Approximation. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 72. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6253-0_5
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DOI: https://doi.org/10.1007/978-3-0348-6253-0_5
Publisher Name: Birkhäuser, Basel
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