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This is an applied math problem.

I need to approximate a function which (for physical reasons) must be symmetrical and monotonic. A few other things which are known is that the function is continuous (and probably has continuous derivatives of any order), and smooth in some sense.

The exact function is not known, but there is a (very computationally expensive) way of testing how close any particular function is to the exact function.

To come up with a good approximation, I need a way to parametrize all possible symmetrical, monotonic, continuous functions, using something like a basis set of functions, so that I can use that parametrization to search for a good approximation to the exact function. A good basis set would be ordered such that the smoother (for some definition of smooth) members are first, similar to the lower-frequency sine functions in Fourier transform, so that (based on the smoothness properties) only the first few elements of the basis would have to be considered.

The question: what is a good basis set to use in this case?

P.S. In one restricted version of this problem the function is only defined on inputs in [0,1], and it is known that f(all zeros) = 0, f(all ones) = 1. An example of this kind of function is $f(x) = \prod x_i$

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    Depending on what you mean, [Symmetric Polynomials](https://en.wikipedia.org/wiki/Symmetric_polynomial) might be useful. – 2017-01-17
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    @Mark Interesting... I did come up with that when I did a quick search but it wasn't obvious to me how to determine how smooth any particular polynomial is. How does one form a basis set of symmetric polynomials? How to arrange it by smoothness? (... and how to make sure they're all monotonic?) – 2017-01-17
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    What do you mean by symmetrical ($f(-x)=f(x)$ ?) and monotonic (for example increasing, I think on $(0,\infty)$, not on $(-\infty,\infty)$ which is impossible for a symmetrical function) ? – 2017-01-17
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    @JeanMarie Symmetrical in the sense of $f(x, y) = f(y, x)$, also for any number of variables $f(x_0, ... x_n) = f(permutation(x_0, ... x_n))$ – 2017-01-17
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    Polynomials have derivatives of all order (although after a while they become zero constantly), so will be as smooth as you want. The basis set of symmetric polynomials in $k$ variables is defined already. The harder part will be finding a way to express it in the basis (I haven't done this before, but I'd imagine you'd need to do something ugly polynomial long division many times). – 2017-01-17
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    [This paper](http://dept.math.lsa.umich.edu/~fomin/Papers/sparse.ps) seems like it might be useful for you (it talks about quickly computing a symmetric polynomial that interpolates some symmetric function). – 2017-01-17

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