Doctor of Philosophy
We show that any smooth closed surface has polynomial density 3 and that any connected compact smooth surface with boundary has polynomial density 2.
Summary for Lay Audience
We assume any surfaces we consider are all in one piece. The distance between a pair of functions is defined to be the maximum over all points of the distance between their values. (When choosing functions of a particular form to approximate a given function, this distance can be thought of as an error tolerance.) We say that a specific function f can be approximated by a set of functions F if we can always find a function from F within the distance specified, no matter how small we chose the distance to be.
We say a surface has polynomial density n if there exist smooth functions g_1, g_2, ..., g_n such that the set of polynomials in them can approximate any continuous function on the surface. We confirm that for a surface without a boundary component, a closed surface, the polynomial density is three. On the other hand, for a surface with a boundary component, we show that the polynomial density must be two.
Broemeling, Luke P., "Polynomial Density Of Compact Smooth Surfaces" (2023). Electronic Thesis and Dissertation Repository. 9387.
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