There exists many such bases, you could also take all the monomes and approximate a function by its Taylor development. However, it does not mean that such bases can be efficiently approximated in a reasonable dimension, nor that conversion from/to textual representation is easy. A deep learning network would achieve those points.
I searched the article, but could not find basis or uncountable. Could you point where exactly I should look?
equation 1 (in formal definition). the basis is e^(-st). if you don't know how that's a basis you need to read a little bit about functional analysis but just look at the integral as a continuous sum and f(t) as the basis coefficients and e^(-st) starts to look like a vector space basis (hilbert space) basis.
I am afraid it is a bit over my head. Would you be able to point me to the source?
Not the most explanatory sources but function spaces essentially are vector spaces: