The Fourier series can approximate any $\mathbb{R} \rightarrow \mathbb{R}$ periodic function. One way to write this is
$$\sum_{j=-N}^N \mathcal{Re} \left[ c_j e^{i 2 \pi j x / P} \right]$$
The Fourier transform looks like
$$\int_{-\infty}^{\infty} f(x) e^{-i 2\pi \xi x} dx; \xi \in \mathbb{R}$$
The Laplace transform looks very similar to the Fourier transform, except there is an additional real term $a$ that causes the sinusoids to decay (and the $2\pi$ constant is no longer present)
$$\int_0^\infty f(t) e^{- (a + i \omega)t} dt$$
By setting the real term $a=0$, we can make the Laplace transform look nearly identical to the Fourier transform
$$\int_0^\infty f(t) e^{-i \omega t} dt$$
My question is: what if we were were to take the negative exponential decay term from the Laplace transform, and put it into a Fourier series like so
$$\sum_{j=-N}^N \mathcal{Re} \left[ c_j e^{(i 2 \pi j - a) x / P} \right]$$
Is there a name for this "Laplace Series"? Can anything be said about the approximation capabilities of this series? It must at least be able to represent anything a Fourier series can when $a=0$. But what about when $a \neq 0$?