From: "G. A. Edgar" Subject: Re: Iterated Composite Functions Date: Tue, 30 Nov 1999 13:11:19 -0500 Newsgroups: sci.math.research Keywords: Hilbert's 13th problem (multivariable functions from univariate ones) [[ This message was both posted and mailed: see the "To," "Cc," and "Newsgroups" headers for details. ]] In article , Daz wrote: > A student of mine, John Mahoney, asks the following question: > > Consider some nice class of functions of 2 variables such as > arbitrary (arb), continuous (C^0), smooth (C^oo), or real analytic (C^w). > > [Let CLASS denote one of arb, C^0, C^oo, or C^w in what follows.] > > -------------------------------------------------------------------------- > Let Iter(n, 2; CLASS) denote the set of all functions f: R^n --> R that > are finitely-iterated compositions of CLASS functions of 2 variables > (per function). > > For example, > > f(x,y,z,w) = g(p(y,z), h(k(w,z), m(z,x))) > > where g, p, h, k, m : R^2 --> R are all of class CLASS. > > [N.B. Here we used 5 functions (of 2 variables); there is no limit on how > large the finite number of them may be.] > > GENERAL QUESTION: How extensive is Iter(n, 2; CLASS)? > > As a start, here's a CONCRETE QUESTION: > ************************************************************************** > Is there a C^0 function R^3 --> R which is NOT in Iter(3, 2; C^0) ??? > ************************************************************************** > > If convenient, please cc any response to John Mahoney at > > johnm@mail.csuchico.edu > > ....thanks! > > --Dan Asimov Looks like Hilbert's 13th problem. Kolmogorov (1957) showed Iter(n, 2; C^0) = C^0... indeed, to get any continuous function R^n -> R it is enough to use some continuous functions of a single variable and only one function of two variables (addition). In the other direction, Iter(3, 2; C^1) does not include all C^1 functions of 3 variables. A survey of results was given by G. G. Lorenz in his chapter of the book Mathematical Developments Arising from Hilbert Problems edited by F E Browder -- Gerald A. Edgar edgar@math.ohio-state.edu