There are multiple series and functional bases (e.g. Taylor, Fourier) that allow approximating a function.
In some sense, such approximations associate a function with an infinite sequence of real or complex numbers (e.g. coefficients). Also, there are often theorems that bound the approximation errors for a truncated series.
One problem I have with these approaches is that they use real numbers which contain infinite amount of information.
I'm interested in an approximation technique where the function is associated with an infinite series of bits. Ideally there should be a theorem that bounds the approximation errors for a truncated series of bits.
Are there any information-based approximation methods?