Math related to representing functions as compositions of others?

1. Dec 12, 2011

Stephen Tashi

I'd like to hear people's thoughts on the general topic of representing (or approximating) given functions as compositions of others.

Of course, one task is define precisely what this means. What are some interesting problems (theoretical or statistical) that require it?

Perhaps a Google virtuoso can do better than my attempts. I've used terms like "function decomposition" and I get lots of hits, but they are mostly about organizing businesses or computer programs. Abstractly, such problems do have some relevance. For example, a typical computer program needs to compute some function F(x,y,z,w...) and it's useful to do this in steps that compute simpler functions like g(x,y), h(z,w) and thus F becomes F(g(x,y),h(z,w)).

Several years ago, I saw a paper where some people were analying a way to evaluate the creditworthiness of loan applicants in a baltic country and they proposed a method of decomposing a function F(x,y,z,...) specified by a numerical table into simpler functions, also given by tables. I don't recall the specifics.

Writing a function as a multivariate Taylor series is a decomposition. I'm curious if there is interesting math besides that approach and besides the more general approach of representing a function as a summation over a set of orthogonal functions. Or maybe someone has proven that all approaches amount to these summation approaches?

2. Dec 12, 2011

Staff: Mentor

there are several approaches depending on the type of function:

f(x,y,z) = X(x) * Y(y) * Z(z) is used in solving some types of partial differential eqns

other types of product sequences can be used to approximate a function as in the product sequence for the Reimmann functions.

Anyway Wikipedia has a brief article on it:

http://en.wikipedia.org/wiki/Function_decomposition

which might point you in other directions. Mathworld search lists a bunch of methods specific to it but no general duscussion.