Is there a continuous, or smooth, generalization of the iterated logarithm $\log^* n$?

Question

I was wondering about the properties of certain classes of functions related to algorithmic runtimes in this post.

So I know that $n!$ has a continuous/smooth generalization to the gamma function. Is there a continuous/smooth generalization of the iterated logarithm $\log_*n$?

Answer 1

Let $f(x)$ be smooth s.t. $f(0) = 1$, $f(x) = 0$ if $|x| > 1$ - bump function, for example $f(x) = \exp(-\frac{1}{1 - x^2})$ if $|x| < 1$ and $f(x) = 0$ otherwise.

Then $g(x) = \sum\limits_{n=1}^\infty \log^*n \cdot f(x - n)$ (this series has at most one non-zero term for any given $x$, so convergence isn't an issue) is smooth extension of iterated logarithm: $g(n) = \log^* n$. It's quite useless, however.

You may also be interested in super logarithm - it has the same asymptotic, $\log^*n = \lceil \operatorname{slog}n\rceil$, and it has analytic extension. For details, check Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point by Trappman, Kouznetsov and references from it.