What are the "local degrees of freedom" in the space of smooth functions?

Let $$C^k$$ be the set of $$k$$th-order smooth real functions $$f:\mathbb{R}\to\mathbb{R}$$, and $$C^\infty$$ the set of smooth real functions. One can specify an $$f\in C^k$$ by specifying all its derivatives at a point, giving the $$k$$th derivative $$f^{(k)}$$ at all points (which may be an arbitrary $$C^0$$ function), and then integrating. So there's a "free choice" at every point of the $$k$$th derivative. On the other hand, to choose an analytic ($$C^\omega$$) function, it already suffices to give all its derivatives at one point; there's no further free choice.

My question is: is there an analogous way of understanding $$C^\infty$$ functions? Can a $$C^\infty$$ function be given by specifying all its derivatives at one point, and some further data at all other points? Informally, how can we understand the "free choices" that can be made at each point when choosing a smooth function, or, how does one "steer" a smooth function? I'd appreciate any references (especially if they address the intuitive high-level question).

More detail:

Another motivation: It would be cool if there were a natural notion of a random smooth walk. We can say, a random $$C^k$$ walk comes from integrating a Weiner process $$k$$ times. Is there some analogous way to define random smooth walks? It would be good to have an "independent increments" property like: if we condition on all the derivatives of $$f$$ at $$x$$, then $$f(x+\epsilon)-f(x)$$ and $$f(x)-f(x-\epsilon)$$ are (conditionally) i.i.d.. (Perhaps this can be shown impossible by the theory around Lévy processes?)

We can define the ring $$G^\infty_x$$ of germs of $$C^\infty$$ functions at some point $$x$$, i.e. equivalence classes of $$C^\infty$$ under the equivalence "are equal on some neighborhood of $$x$$". (Since all the $$G^\infty_x$$ are isomorphic we can discuss $$G^\infty$$.) We can likewise define the ring of germs of analytic functions $$G^\omega$$, and we have the obvious injection $$G^\omega \hookrightarrow G^\infty$$. And, we can define the ring of "right half-germs" $$G^\infty_R$$ by the relation "are equal on all $$y\ge x$$ in some neighborhood of $$x$$". Likewise define "left half-germs" $$G^\infty_L$$. Is it the case that $$\dfrac{G^\infty}{G^\omega} = \dfrac{G^\infty_L}{G^\omega} \oplus \dfrac{G^\infty_R}{G^\omega}$$ as $$G^\omega$$ modules? Is it the case that specifying $$G^\infty_R$$ at 0 as well as $$G^\infty_R/G^\omega$$ for all $$x>0$$, determines a unique smooth function $$f:\mathbb{R}^{\ge 0} \to \mathbb{R}$$? What's known about the structure of $$G^\infty$$? (Does sheaf theory shed any light on these questions? What about the theory of jet spaces?) Is there a natural notion of "random continuous walk on $$G^\infty$$" (or on, say, $$G^\infty_R/G^\omega$$)? (Is there a general theory of random walks on topological vector spaces or similar?)

Dave L. Renfro's "Essay on nowhere analytic c-infinity functions" part 1 part 2 and Ralph P. Boas's "A primer of real functions" (pgs. 186-193) give interesting info about smooth functions:

• Rosenthal: for any sequence $$\{a_n\} \in \mathbb{R}$$, there's a smooth $$f$$ with $$\forall n: f^{(n)}(0) = a_n$$. (This is far from true for $$C^\omega$$ functions.)
• Ramsamujh: on a comeager subset of $$C^\infty([0,1])$$, every point of $$f$$ is Pringsheim, i.e. at any point the Taylor series of $$f$$ at that point has null radius of convergence.
• Bernstein, McHugh: if each $$f^{(n)}(x)$$ has a fixed sign on an interval $$I$$, then $$f$$ is analytic on $$I$$. (Hence, if $$f$$ is smooth and non-analytic at $$p$$, then on some sequence $$\{p_i\} \to p$$, for each $$i$$ some $$f^{(n_i)}(x)$$ changes sign at $$p_i$$; and for Baire-most smooth $$f$$, on a dense set some derivative of $$f$$ changes sign.)
• Bernal-González: for any sequence of positive numbers $$\{a_n\}$$, Baire-most smooth $$f$$ will at every $$x\in \mathbb{R}$$ have $$\limsup_{n \to \infty} a_n|f^{(n)}(x)| = \infty$$ and $$\liminf_{n \to \infty} a_n|f^{(n)}(x)| = 0$$.

These facts paint a picture of functions with crazily swinging derivatives. This vaguely makes intuitive sense by the heuristic reasoning that the lower derivatives are determined by the higher derivatives, so the "steering" has to "bubble down from infinity through the higher derivatives". What else can we say about this? Is there some way to characterize what sequences $$\{f_n\}$$ can be the derivatives of some $$f \in C^\infty$$ (beyond just saying, we must have $$\forall n: f_{n+1}^{(1)}=f_n$$), probably in terms of the behavior of $$\{f_n\}$$ as $$n\to \infty$$?

This question benefited from discussions and research with Sam Eisenstat (errors are mine).

(These citations are copied from Renfro and from Boas.)

Luis Bernal-González, "Funciones con derivadas sucesivas grandes y pequeñas por doquier" [Functions with successive derivatives everywhere large or small], Collect. Math. 38 (1987), 117-122. [MR 90c:26013; Zbl 661.26009]

Taje I. Ramsamujh, "Nowhere analytic C-infinity functions", J. Math. Analysis Appl. 160 (1991), 263-266. [MR 92j:26014; Zbl 751.26014]

Ralph P. Boas, A primer of real functions, 4'th edition (revised and updated by Harold P. Boas), Carus Mathematical Monographs #13, Mathematical Association of America, 1996, xiv + 305 pages. [MR 97f:26001; Zbl 865.26001]

A. Rosenthal, On functions with infinitely many derivatives, Proceedings of the American Mathematical Society 4 (1953), 600-602

J. A. M. McHugh, A proof of Bernstein's theorem on regularly monotonic functions, Proceedings of the American Mathematical Society 47 (1975), 358-360.

• Your question seems a little vague and ill-defined. Isn't the question about local degrees of freedom answered by characterising the topological vector space of $C^\infty$-functions? If not, why not -- what isn't sufficiently local? The details of your question seem to be more about an orthogonal issue of interpolation. Commented Aug 18, 2021 at 7:26
• @RyanBudney Thanks for your response. Yes, I tried to be as precise as I could, but the question is still ill-defined. I don't follow what you're saying, unfortunately. Could you clarify how the structure of the TVS $C^\infty$ is orthogonal to the question of what data determines a smooth function? In the $C^k$ case, for example, it seems enlightening on both counts to notice (what I think is) the isomorphism $C^k \cong C^0$ given by integrating (/differentiating) $k$ times. Commented Aug 18, 2021 at 17:36
• Um, or rather, $C^k \cong C^0 \oplus \mathbb{R}^k$, or similar (for the initial conditions). Commented Aug 18, 2021 at 18:06