## Problems from the archive

(The title of this post is based on WBGO‘s nice late-Sunday show “Jazz from the archive”, which I used to follow when I was at Rutgers; the physical archive in question was that of the Institute of Jazz Studies of Rutgers University; the post itself is partly motivated by seeing Gil Kalai’s examples of his own “early” problems…)

The following problem of classical analysis of functions of one-complex variable has puzzled me for a long time (between 15 and 20 years), though I never really spent a lot of time on it — it has never had anything to do with my “real” research:

Let f,g be entire functions, and assume that for all r>0 we have
$\max_{|z|=r}{|f(z)|}=\max_{|z|=r}{|g(z)|}$
What can we say about f and g? Specifically, do there exist two real numbers a and b such that
$g(z)=e^{ia}f(e^{ib}z)$
for all z?

(Actually, the “natural” conclusion to ask would be: does there exist a linear operator T, from the space of entire functions to itself, such that Tf=g and such that

$\max_{|z|=r}{|T(\phi)|}=\max_{|z|=r}{|\phi(z)|}$

for all entire functions φ; indeed, the function g=Tf satisfies the condition any such “joint isometry” for the vector space of entire functions equipped with the sup-norms on all circles centered at the origin; but I have the impression that it is known that any such operator is of the form stated above).

My guess has been that the answer is Yes, but I must say the evidence is not outstandingly strong; mostly, the analogue is known when one uses L2 norms on the circles, since

$\frac{1}{2\pi}\int_0^{2\pi}{|f(re^{it})|^2dt}=\sum_{n\geq 0}{|a_n|^2r^n}$

if an are the Taylor coefficients of f. If those coincide with the same values for another entire function g (with coefficients bn) we get by unicity of Taylor expansions that

$|a_n|=|b_n|,\ so\ b_n=e^{i\theta_n}a_n$

for all n, and the linear operator

$\sum_{n\geq 0}{c_nz^n}\mapsto \sum_{n\geq 0}{c_ne^{i\theta_n}z^n}$

is of course a joint isometry for the L2 norms on all circles, mapping f to g.

The only result I’ve seen that seems potentially helpful (though I never looked particularly hard; it was mentioned in an obituary notice of the Bulletin of the LMS, I think, that I read rather by chance around 1999) is a result of O. Blumenthal — who is today much better known for his work on Hilbert modular forms, his name surviving in Hilbert-Blumenthal abelian varieties. In this paper from 1907, Blumenthal studies the structure of the sup norms in general, and computes them explicitly for (some) polynomials of degree 2. Although the uniqueness question above is not present in his paper, one can deduce by inspection that it holds for these polynomials at least.

(I wouldn’t be surprised at all if this question had in fact been solved around that period of time, where complex functions were extensively studied in France and Germany in particular; that the few persons to whom I have mentioned it had not heard of it is not particularly surprising since this is one mathematical topic where what was the height of fashion is now become very obscure).