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A Stone-Weierstrass theorem for increasing functions

20 Août 2012, 16:09pm

Publié par Fabien Besnard

I recently got interested into extensions of  the Stone-Weierstrass theorem for the set of continuous non-decreasing continuous functions on some compact poset M (if you want to know why, take a look at that). I thought that maybe some readers here might be interested by it.

 

Such a theorem already exists and is due to Nachbin (even if he did not formulate it that way) : a sufficient set of conditions for a bunch of continuous non-decreasing functions (call it S)  to be uniformly dense in the set of all these functions (call it I(M)) is :

 

1) to be a convex cone

2) to be a sublattice (stability under max and min)

3) to contain the constant functions

4) to generate the order

 

The final condition means that not only  x less than y => f(x) less or equal to f(y)  for every  f in S, but that the converse is true, that is, if f(x) is less or equal to f(y) for every f in S, then x has to be less or equal to y for the partial order on M. Basically, this is just a corollory of the lattice version of SW.

 

However, I wasn't satisfied with this theorem, mainly because I was interested in noncommutative generalizations, and though noncommutative max and min are possible to define (through the formula 1/2(x+y +/- |x-y|), which does not need  the commutativity of the product), they cease to satisfy lattice axioms, and are kind of cumbersome to calculate.

 

Now the classical SW theorem can be recast in a somewhat odd way : a non-empty subset S of the continuous real functions on some compact hausdorff set M is dense in the continuous functions if :

 

1) S is stable by sum,

2) the affine functions operate on S,

3) the function t -> t² operates on S.

 

2) and 3) mean that if f is in S, then af+b and f² also are. In fact, one can use any non-affine function instead of t -> t².

 

So I set out to prove a theorem of this sort for non-decreasing functions. Here is what I found : let S be a non-empty subset of the set of continuous non-decreasing functions on a compact poset M. Then S is uniformly dense in I(M) if :

 

1) S is stable by sum,

2) the continuous non-decreasing piecewise linear functions operate on S,

3) S generates the order on M.

 

If you want to look at the proof, here is the short note I have put on arxiv last week.

 

I'm happy with this theorem because I can change 2) by saying that any continuous non-decreasing function from R to R operates on S, which is really the weakest condition I could hope for for what I intend to do with that (I explain this at the end of this talk).

 

Now there are some obvious questions. The first that comes to mind is that maybe we don't really need the piecewise linear functions, but just the affine (non-decreasing function) plus a non-affine one (why not t -> t^3 ?). My first guess is that this won't work, but maybe some reader wants to check. My other concern is more vague : surely if S has to be dense in I(M), then the closure of S (that is I(M) itself) will be operated upon by all continous increasing functions from R to R, just because (posets+topology,continous non-decreasing maps) form a category. So maybe this hypothesis might be given a categorical meaning (I told you this was a vague question).

 

Ahem, there is just one more question : what the hell can you do with this theorem ? Even though I have something completely different in mind, I looked a bit for an application of this theorem in the realm of classical analysis, but couldn't find any. Here also, maybe my readers will be more clever than me.

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