Optimal discretization and expansion order of arbitrary data

[laxsu19]

Hi all,

I am trying to figure out 1) What to call my problem so I can better research the literature, and 2) see if anyone here knows of a solution.

Essentially, I have a large set of f(x) vs x points (~20,000) which I need to split into subdomains in x, and within each subdomain calculate a functional expansion of f(x). I want to do this in an optimal manner such that 1) the number of subdomains is minimized - or at least manageable, and 2) the number of expansion orders (probably Legendre) within each subdomain is also minimized.

Does anyone have any idea what 'field' of math this could be considered, and where to begin searching around? Unfortunately, this is just a minor step in what I have to do so I don't want to expend much effort here.

Thanks for your help!

 

[Stephen Tashi]

Does your data contain "noise" or is the data simply known values of some precisely defined function? If your data is known values of a precisely defined function, then the general topic to research is "function approximation". For many functions, the simplest approximations (for a given mean square error) are done by using ratios of polynomials. That topic is "approximation by rational functions".

 

[AlephZero]

If you can fit each subdomain by a low order polynomial, some buzzwords are automatic knot placement for spline curve fitting. (The "knots" are the points at the end of each subdomain, i.e. the end of each spline segment).