Suppose I have some function $f(x)$.
I know that this function can be represented in several different bases. For example, we can express this using Legendre polynomials, Hermite polynomials, Fourier series, Taylor series, etc.
In general,
$$f(x)=\sum_i c_i P_i(x)$$
Here $P_i(x)$ represents some orthonormal basis, like the Legendre polynomials for example.
Now suppose I want to compare which of the expansions converges into the actual function fastest. For example, if I only consider the first 100 terms, can I say that expanding $f(x)$ using Legendre polynomials is better than the Hermite ones? For a given function, is it possible to compare the expansion in different bases, and then find out which of the basis expansions converge to our function quickest?
Is it possible to do it analytically, and not by using the brute force method, where we calculate the terms of the series by hand and then see which series is more accurate if truncated at the $n$th term. Can I just examine the basis itself and draw conclusions on which series would be more accurate for a given $f(x)$ ? Ofcourse, all the series would give me a perfect approximation, if I consider all infinite terms. However, can I find out which series converges quickest for some finite number of terms ?
Different bases will yield faster convergence for different functions. If the function is itself a Hermite polynomial, then clearly the Hermite expansion will be the most accurate....