Propagation of microcontinuity for nonstandard polynomials |
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Authors: | Chris Impens |
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Institution: | (1) Rijksuniversiteit gent, Seminarie voor Wiskundige Analyse, Galglaan 2, B-9000 Gent, Belgium |
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Abstract: | This paper explores the differences between real and complex microcontinuity for hyperreal polynomials, with hypernatural
degree and nonstandard coefficients. On the real line, complex microcontinuity differs from real microcontinuity in replacing
the coefficients with their absolute values. Apart from this feature, not much analogy is found between (absolute) convergence
of series and (absolute) microcontinuity of infinite polynomials, even if these are infinite partial sums of a standard series.
Real microcontinuity may be confined to isolated monalds, whereas complex microcontinuity always propagates a noninfinitesimal
distance. An infinite partial sum of a power series can be microcontinuous outside the circle of convergence. |
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