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Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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dense subalgebra in measurable functions set

This is a partial answer for the case that your algebra is inverse closed, i.e., if $f,g$ are in the algebra and $g$ is invertible, then so is $\frac f g$.Consider the intersection of your algebra wit …
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