C^r smoothness with non-integer r

Stream: maths

[Yury G. Kudryashov]

I want to define (not in the next week or two) times_cont_rdiff to be the real-specific version of times_cont_diff that corresponds to $C^{n+\alpha}$-differentiability. There are a few design choices that I want do discuss before starting to implement this.

1. Should it take nnreal, a pair from nat × Icc 0 1, or a pair from nat × option (Icc 0 1) as an argument? There are two cases when the notation $C^r$ is ambiguous: $C^{n+0}$ (is it $C^n$ or $D^n$+bounded n-th derivative?) and $C^{n+1}$ (is it the usual $C^{n+1}$-smoothness or $C^n$ plus Lipschitz continuity of n-th derivative?). If we want to talk about both boundary cases, then probably we need nat × option (Icc 0 1).
2. On top of the type β for r from the previous part, should we use with_top β (for infinitely smooth functions) or with_top (with_top β) (to allow analytic functions too? With notation $C^\infty$ and $C^\omega$ in the latter case.
3. For times_cont_rdiff_on, should we use Whitney's differentiability, or this should be a separate definition?

[Moritz Doll]

@ 1: there is actually another notion for natural numbers, which is the correct one from an interpolation point of view, called Zygmund spaces, but for the definition one needs Fourier multipliers and tempered distributions (and it is a non-trivial theorem that Zygmund spaces coincide with Hölder spaces for non-integer exponent). For that reason I would say that nnreal would be a bad choice.
I would think the correct thing would be to define Hölder spaces for parameters in nat × Icc 0 1. I don't see any reason to take an option there, times_cont_diff would stiff be still defined for nat?

[Yury G. Kudryashov]

What book/article is a good introduction into Zygmund spaces?

[Heather Macbeth]

Moritz Doll said:

I would think the correct thing would be to define Hölder spaces for parameters in nat × Icc 0 1. I don't see any reason to take an option there, times_cont_diff would stiff be still defined for nat?

This makes sense to me too. Unless there are theorems you're thinking of which genuinely makes sense both for Holder/Lipschitz spaces and for n times continuously differentiable functions? But in the cases I can think of, the Holder/Lipschitz theory develops fairly differently (for example, you'll want to put norms on these functions, and then if you have the option version you'll need to have conditions saying that there's only a norm if the option exists).

[Heather Macbeth]

I'm also not sure I would use an infinity version at all. Because the reason I would want to have Holder/Lipschitz spaces is to get a norm, and this doesn't exist in the infinite case. I think instead I would have no with_top version, but instead a theorem saying that a function which is Holder/Lipschitz for arbitrarily high exponents is times_cont_diff infty.

[Heather Macbeth]

For times_cont_diff_on, there is also the question of whether you want to use the "adapted" version of the Holder norm, with a weight that depends on the distance from the boundary of the set considered ...?

[Moritz Doll]

Yury G. Kudryashov said:

What book/article is a good introduction into Zygmund spaces?

I learned that stuff from Hörmander's nonlinear hyperbolic equations book (you need Zygmund or more generally Besov spacse for nonlinear PDEs), but I think for learning just about the spaces and formalization it might be better to look into the monograph series by Triebel (Theory of Function Spaces 1-3)

[Sebastien Gouezel]

Moritz Doll said:

@ 1: there is actually another notion for natural numbers, which is the correct one from an interpolation point of view, called Zygmund spaces, but for the definition one needs Fourier multipliers and tempered distributions (and it is a non-trivial theorem that Zygmund spaces coincide with Hölder spaces for non-integer exponent).

Isn't it possible to define Zygmund spaces at integer exponents by requiring the modulus of continuity to be O(x/\log x)?

[Sebastien Gouezel]

I agree with everyone that nat x Icc 0 1 seems the right choice (possibly with an additional top element). With the lexicographic order, it even gives directly the right order for the smoothness classes with respect to inclusions.

[Sebastien Gouezel]

As it was already mentioned, there is the question of whether you want the qualitative version (i.e., around each point, you have a bound for the Hölder constant, that may depend on the point), or globally (i.e., spaces on which you will put norms and do analysis with them). In other words, are you looking for an analogue of the space of continuous functions, or of the space of bounded continuous functions?

[Sebastien Gouezel]

Not directly related: I have convinced myself that it would be a good idea to refactor the current times_cont_diff_on, requiring strict differentiability instead of differentiability. For open sets over the reals, it is equivalent, so we would not be changing the definition in the usual cases, but in more exotic cases the current definition of times_cont_diff_on is not really useful because it lacks a local inverse theorem. Note that analytic functions are always strictly differentiable, so analytic functions (over any field) would still satisfy the modified definition.

[Yury G. Kudryashov]

@Sebastien Gouezel What is more useful in case of exotic fields: current definition of times_cont_diff_on with strict differentiability or Whitney differentiability? It seems that Whitney differentiability is a natural generalization of strict differentiability to higher order derivatives and any analytic function is Whitney differentiable.

[Yury G. Kudryashov]

I'm not sure about continuous_map vs bounded_continuous_map.

[Sebastien Gouezel]

That's a good question. I don't have an informed opinion here.