(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an aspect of the question I am asking with a bunch of example $\alpha$'s. Please let me know if the question is not clear.)
I'm wanting to get a grasp on the degree to which polynomials, complex exponentials, and Bessel functions can serve as bases for the same function spaces.
Let $n\in \mathbb{Z}$. In a given finite interval, let $T$ represent the set of all functions f(x) that are equal to their Taylor series, $L$ be the set of all functions that are equal to their Laurent series, $F$ be the set of all functions that are equal to their Fourier series, $J_{n}$ the same for 1st-kind Bessel series, and $Y_{n}$ similarly.
My guess (and this is only a guess) regarding the relation between these sets is $$T\subset L = F = J_{n} \neq Y_{n}$$ with $J_{n} \cup Y_{n}=\emptyset$ (for any given $n$, implying that I also have e.g., $J_{1}=J_{-15}$).
What is the correct string of equalities etc. that should be in place of mine? Furthermore, letting $\alpha \in \mathbb{C}$, how would $J_{\alpha}$ and $Y_{\alpha}$ be compared with these sets. E.g., all functions in $J_{\alpha \notin \mathbb{R^{-}}-\mathbb{Z^{-}}}$ must be singular at the origin, and all functions in $Y_{\alpha}$ must be singular at the origin regardless to $\alpha$. Are these sets the same for certain regimes of $\alpha$, e.g., would any of the following equalities be true: $Y_{0}=Y_{1}=Y_{-3}=Y_{1.5}=Y_{-2.5}=Y_{-2i}=Y_{5i}=J_{-1.5}=J_{2i}=J_{-2i}=$?
Thanks very much.