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)
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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