Abstract
In this paper we prove Carleman's approximation type theorems in the framework of slice regular functions of a quaternionic variable. Specifically, we show that any continuous function defined on $\mathbb{R}$ and quaternion valued, can be approximated by an entire slice regular function, uniformly on $\mathbb{R}$, with an arbitrary continuous "error" function. As a byproduct, one immediately obtains result on uniform approximation by polynomials on compact subintervals of $\mathbb{R}$. We also prove an approximation result for both a quaternion valued function and its derivative and, finally, we show some applications.
Citation
Sorin G. Gal. Irene Sabadini. "Carleman Type Approximation Theorem in the Quaternionic Setting and Applications." Bull. Belg. Math. Soc. Simon Stevin 21 (2) 231 - 240, may 2014. https://doi.org/10.36045/bbms/1400592621
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