Does this "show" that the fourier basis is more "powerful" than the polynomial basis?

Question

I'm sure the answer is negative, they are both called "basis" after all, but there is a "paradox" that I can't wrap my head around.

We can write the dirac delta function in the fourier basis as $$ \delta(x) = \frac{1}{2\pi} + \frac{1}{\pi}\sum_{n=1}^\infty \cos nx $$ yet since the delta function is not a "function" per se, it seems that it can't have a taylor series. Does this mean that it can't be expressed in the polynomial basis?

I guess there are 2 possible answers to this question:

1. It can still be expressed in the polynomial basis, just by coefficients that are different from those given by a taylor series
2. Since the dirac delta is not a function, "basis" theory loses its meaning, and so it does make sense that it can only be expressed by sins and coses. In this sense, is the fourier basis "more powerful/expressive" than the polynomial basis (ie it can also express things such as distributions)

Answer 1

It's not a Taylor series, but you can also write the Dirac delta as a sum (convergent in the sense of distributions on the interval $(-1,1)$) of Chebyshev polynomials $T_n(x)$:

$$ \delta(x) = \frac{1}{\pi} + \frac{2}{\pi} \sum_{k=1}^\infty (-1)^k T_{2k}(x) $$

Answer 2

To write a function, or a "generalized function" (distribution? hyperfunction?) as an infinite sum of simpler functions (rigorously, as opposed to heuristically... even though many physicists have made tremendous use of physically intuitive heuristics! E.g., Dirac!) is probably best done when/if one has a suitable (!) topology on a relevant space of functions (or its dual, or its extension, or both simultaneously, most usefully, as in the case of distributions, tempered distributions, etc.)

Somewhat ironically, the vector space of polynomials on a closed interval is a bit clumsy, insofar as polynomials are not eigenfunctions for any convenient self-adjoint (bounded or not) operator. That is, despite their intuitive appeal, they are not as well adapted to subtler situations as, for example, Fourier series of various sorts (where exponentials or sines and cosines are indeed eigenfunctions for Laplacians...)

A related example where things turn out somewhat better is the case of the "quantum harmonic oscillator" $-\Delta+x^2$ on $\mathbb R$, which does have an orthonormal basis of eigenfunctions in $L^2(\mathbb R)$, and whose subtler theory does allow expression of tempered distributions as infinite sums of terms which are constant multiples of $n$th Hermite polynomial times suitable Gaussian.