Page 43 – E. Kowalski's blog

Orthodoxy and Bessel functions

While working on a recent paper, I’ve had the occasion to browse a bit through the book of Watson on Bessel functions. This was first to look for references for various facts, and then, lazily, more out of curiosity, on the lookout for some of these “more recondite investigations” that English books of analysis of the time (1922) seem to enjoy.

I thus learnt that integral equations had a sulfurous reputation back then:

[…] On the other hand, researches based on the theory of integral equations are liable to give rise to uneasy feelings in the mind of the ultra-orthodox mathematician. (p. 578)

[Amusing typographical note: it’s not clear if Watson wants to spell “ultraorthodox” or “ultra-orthodox”: there is a hyphenated end-of-line just at the right place to make this ambiguous; or is there some accepted typographical rule to distinguish between the two possibilites?]

Although integral equations are presumably not going to create much unease nowadays, that does not mean that orthodoxy has relented. Somewhere in Godement’s opinionated course of analysis (the perfect gift to annoy your mathematically-minded neo-con nephew), there is a rather stern warning against exposing young minds to tables of integrals of Bessel functions. Presumably, all such functions must be understood only as matrix coefficients of representations of SL(2) (except when it is clearly better to see them as having to do with D-modules.)

There’s of course a lot of truth in thinking this way, but personally I feel that sometimes one should enjoy mathematics outside of the big picture. Here are two fun facts about Bessel functions that I didn’t know a few days ago. Precisely, they concern the J Bessel functions, which are defined either as integrals
$J_{\nu}(x)=\frac{1}{\pi}\int_0^{\pi}{\cos(\nu\theta-x\sin\theta)d\theta}- \frac{\sin(\nu \pi)}{\pi}\int_0^{+\infty}{e^{-\nu t -x\sinh(\theta)}dt},$
or as power series
$J_{\nu}(x)=\sum_{m\geq 0}{\frac{(-1)^m(z/2)^{\nu+2m}}{m!\Gamma(\nu+m+1)}}.$

For our purpose, let’s assume that ν is a non-negative real number and x also (of course these functions extend to much larger domains).

(1) First fact: what is the size of
$J_{\nu}(\nu)?$

This is a type of uniformity question which is often quite important in analytic number theory. The answer (which can be found by the stationary phase method) is due to Cauchy:
$J_{\nu}(\nu)\sim \frac{\Gamma(1/3)}{2^{2/3}3^{1/6}\pi \nu^{1/3}}$
as ν tends to infinity. (Page 231 in Watson’s book.)

(2) Second fact: one can try to represent functions by series of Bessel functions in various ways. If we consider the Schlömilch series for ν=0, namely
$f(x)=\frac{a_0}{2}+\sum_{m\geq 1}{a_m J_0(mx)},$
something funny happens: although every function f which is regular enough on the closed interval [0,π] can be represented in this manner for suitable coefficients a_m, and although one can even write formulas for these coefficients
$a_0=2f(0)+\frac{2}{\pi}\int_0^{\pi}\int_0^{\pi/2}uf'(u\sin\phi)d\phi du,$
$a_n=\frac{2}{\pi}\int_0^{\pi}\int_0^{\pi/2}uf'(u\sin\phi)\cos mud\phi du,$
there is no unicity! Precisely, the formula
$\frac{1}{2}+\sum_{m\geq 1}{(-1)^m J_0(mx)}=0$
which is valid for all
$x\in ]0,\pi[$
(Watson, page 634), gives a non-trivial expression of the zero function as a Schlömilch series on this open interval. However, Watson goes on to prove that this is, up to scaling, the only representation of the zero function. In other words, there is some kind of canonically split exact sequence associated with the problem of representing (smooth enough) functions as Schlömilch series.

The existence of the series representation above (both for general f, and the exceptional one for the zero function) are not very difficult, using the integral representation of J₀. In the former, one starts by showing that
$g(x)=f(0)+x\int_0^{\pi/2}{f'(x\sin \phi)d\phi}$
is a solution of the integral equation
$f(x)=\frac{2}{\pi}\int_0^{\pi}{g(x\sin\theta)d\theta},$
and then expand g into a Fourier series (which is possible if f is smooth enough, e.g., C²).

Watson’s determination of the Schlömilch series representing zero is also interesting: it is more or less similar to Riemann’s approach to trigonometric series. I think the latter is not as well-known today as it was some decades ago; I only remember reading something about it from Zygmund’s classical book on trigonometric sums (another very — in fact, even more — delightful book; I’ve heard a very well-known mathematician say during a lecture that it would be on his short-list of books to bring to a desert island). Very roughly speaking, one of the issues is, given a trigonometric series
$\frac{a_0}{2}+\sum_{m\geq 1}{(a_m\cos mx+b_m\sin mx)}$
(in old-fashioned writing…), with the only assumption that it converges pointwise at all points in [0,2π], and that the sum is zero, to show that every coefficient is zero. The basic idea of Riemann is to consider the series obtained by integrating formally twice; the latter is absolutely convergent everywhere (having gained a factor 1/n²), and represents a continuous function F; one shows then that some kind of generalized second derivative of F is everywhere zero, and one can deduce from this that it is a linear function. The linear term having been shown to vanish, one is left with a uniformly convergent trigonometric series vanishing identically, and multiplying by cosines and sines and integrating, one gets the fact that every coefficient is zero. What is nice and surprising is how little of the property of orthogonality is used. This explains probably why it can be adapted somehow to Schlömilch series, since the corresponding functions are not orthogonal!

Of course, notwithstanding my disparaging comments on “big pictures”, I’d be delighted to learn that these two facts are related to matrix coefficients or to D-modules. And at least I am pretty certain to incorporate them one day in a course.

Positions in Zürich

Now seems a good time for the annual update on the postdoctoral positions at the Mathematics Department at ETH. Basic information, and a link to the online application form is here. While there’s not much to add this year to what I wrote two years ago about these positions, I’ll just say that the fairly confusing ticket machines in the tramway stations are being replaced with newer ones which are much more self-explanatory (and take credit cards in addition to cash)… The first deadline for application is November 22.

While I’m at it, here is a link to the lectureship/postdoc positions at the University of Zürich. In particular, one position there is funded by the Swiss National Science Foundation project of Ulrich Derenthal, and is therefore in the direction of arithmetic geometry.

And finally, there are two assistant professor positions (non tenure-track) currently advertised at ETH. One of them is in pure mathematics (deadline December 31), the other in applied mathematics (deadline December 15).

The amusing distribution of refereeing requests

I am sure that I am not the only mathematician to have observed how the distribution of refereeing requests during the year is very far from uniform. In the last few days, I have been experiencing a peak of almost one request per day, and I could probably use this to obtain a rough guess of the editorial efficiency of various journals, since it seems more than likely (e.g., because this is what I often do as an author…) that these papers were submitted formally just before the beginning of the new semester.

It is somewhat interesting to wonder about the possible consequences of this; clearly, if someone gets ten refereeing requests in a few days instead of them being spread over three or four months, many more will be declined, and that is unlikely to lead to an improvement of the mathematics literature…

Comment on “Esperantism expands”

My co-authors (J. Ellenberg and C. Hall) and myself have received some interesting remarks from P. Sarnak concerning our joint work discussed a few days ago. With his permission, it follows below (with minimal editing to clean-up the list of references at the end).

One important point he makes is that the result of Abramovich, who gave a lower bound for the gonality of modular curves using Selberg’s 3/16 lower-bound for the eigenvalue of the Laplace operator on these curves and an inequality of Li-Yau, had been anticipated by P. Zograf in a little-known paper in Russian (see here for the English translation; it is Theorem 5 at the very end of that paper). We will soon update the arXiv version of our paper to add a reference to this result (the PDF from my home page is already updated).

Dear Jordan,Chris and Emmanuel,

I was pointed to your recent posting with the above
title.I particularly like the application in Corollary 4.
It would be nice if you supplemented it with a similar explicit application
which requires the full force of the blossoming number/combinatorial theory of
what I like to call “thin subgroups of SL(n,Q)”.By the latter I mean a finitely
generated such group which is not of finite index in the S-integral points of
the Zariski closure of the group.
My reason for writing is to point out a couple of papers connected to
parts of your paper,which are not commonly known and which should be.The first
is a paper of Zograf [Z] and the second
is result of Solovay ,called the Solovay-Kitaev theorem (see[ C-N ]),
which was pointed out to me some years ago by Dorit Aharonov.
Zograf’s paper should have been cited in the paper of Abramovich
that you cite as a key to your paper.I feel partially responsible for this
mistake as I communicated Abramovich’s paper.I was
aware of the results in Zograf which give a lower bound for
the genus of a curve in terms of its volume and lambda_1 ,
but I was not aware that he also gave the lower bound on the gonality.
Anyway his genus lower bound is what you use in (5) page 8 of your paper and
attribute to Kelner and the gonality lower bound is
the same as that in Abramovich. Note that his bound is based
on the earlier paper of Yang and Yau [Y-Y] rather than Li and Yau.
Since Abramovich’s paper there have been improvements in
bounds towards Selberg’s Conjecture and thus improvements
in the gonality lower bound.The recent paper of Blomer
and Brumley [B-B] gives the best such bounds in the case
of general Shimura curves.For me it is an interesting
question as to whether for modular curves,the ratio gonality/genus tends to a
limit as the genus goes to infinity? If so is this constant the same as what
one gets for the same ratio,gonality/genus,of a random curve of large genus?
Solovay’s theorem is concerned with diameter and expansion
in the group SU(2). Let A and B be members of SU(2) such that
the group generated by them is Zariski dense in SL(2,C).
The issue is how quickly can one approximate any element in
SU(2),by words in A and B.Given any g in SU(2) and e>0 there is a word w in A
and B such that d(w,g)<e [in some some fixed metric on SU(2)].Let l(e,g)
be the length of the shortest such w and let l(e) be the maximum of l(e,g) as g
varies over SU(2).If A and B form an “expander” in
the sense that the left convolution operator corresponding
to A+A^(-1)+B +B^(-1) on L^2(SU(2),haar) has a spectral gap,
then l(e)=O(log (1/e)).This “diameter” bound is optimal.
In [G-J-S] a substitute in this setting was
found for the argument exploiting the high dimension of any
nontrivial representation of G(F_p) [G a Chavelley group],that
is used in all proofs of expansion in this finite simple group setting.
This led to many examples of such expanders in SU(2) and it was
conjectured there that every such pair A,B gives an expander.
[B-G] have the best result towards this conjecture showing that
it is true if A and B have entries in Q bar.In terms of
technical difficulty this is probably the most difficult case
of expansion that is known.The proof of expansion for SL_2(F_p)
relies indirectly on the proof in [E-M] of the Erdos-Volkman
ring conjecture ,while this SU(2) case relies on the
proof in [B] of the more difficult quantitive local version
of this ring conjecture.
What Solovay proves is the analogue of your “esperante” property,
though he didnt give it any such colorful name. He shows that
l(e)=O(log(1/e)^2),for any given A and B as above.So
in this setting we have the small diameter property but
unlike your case ,where as you note a proof of the expansion property
is close to being proven,for SU(2) it remains wide open.Solovay proves this
polynomial diameter bound in an appendix
to his paper with Yao[S-Y], where it is used as one of the ingredients
to analyse how quantum computation classes can simulate classical ones up to
polynomial slowdown [whether the converse
is true or false is far from clear].For this purpose they need not
only that the diameter is small,but also that one can find a short
word quickly, and Solovay’s proof provides such a short word
in more or less the same number operations.The expansion proof
of an optimally small diameter goes via a counting argument and
it offers no means of finding a short word (or path in the
graph case).Even in the case of optimal expansion in the
graph setting,that is of Ramanujan Graphs,no fast algorithm for
finding a short path between vertices is known,and I have long
thought of this as a basic problem. In [L],Larsen resolves this
problem when one relaxes the diameter bound to something slightly bigger
than O(log) and also allowing a random element into the
algorithm,making it a probablistic algorithm.For computer science applications
what he does is surely good enough.
I know that you guys are avid bloggers,so feel free to include this letter in
your discussions if you feel it is appropriate.

Best regards
Peter

References:

[B] J. Bourgain, GAFA 13 (2003), 334-365.

[B-B] V. Blomer and F. Brumley, “On the Ramanujan Conjecture over
Number Fields”, arXiv:1003.0559.

[B-G] J.Bourgain and A.Gamburd, Invent Math. 171 (2008), 83-121.

[E-M] G.Edgar and C.Miller, Proc. AMS 131 (2003) 1121-1129.

[J-G-S] D.Jakobson, A.Gamburd and P.Sarnak, Journal European
Math. Soc. 1 (1999), 51-85.

[L] M.Larsen, International Math. Res. Notices (2003) No 27, 1455-1471.

[C-N] M. Nielsen and I. Chuang, “Quantum computation and
Quantum Information” CUP, Cambridge 2000.

[S-Y] R.Solovay and A.Yao, preprint 1996.

[Y-Y] P.Yang and S.T.Yau, Ann Scuola Norm. Super. Pisa, 7 (1980), 55-63.

[Z] P.Zograf, “Small eigenvalues of automorphic Laplacians in spaces
of cusp forms”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov
(LOMI) 134 (1984), 157-168; translation Journal of Math. Sciences 36,
Number 1, 106-114, DOI: 10.1007/BF01104976

Reading Burnside (and thanking Noether)

In 1905, the famous rower W. Burnside (then aged 52) proved one of the results known as Burnside’s Theorem (the other one being, usually, the striking result that finite groups of order divisible by at most two primes are solvable):

Let k be an algebraically closed field, and let
$G\subset GL_n(k)$
be a subgroup of the invertible matrices of size n over k. Let k[G] be the span of G in the matrix algebra M(n,k) of size n. Then G acts irreducibly on kⁿ if and only if k[G]=M(n,k).

Here, recall that irreducibility (a notion apparently first introduced by Burnside himself) means that there is no proper non-zero subspace

$W\subset k^n,\quad 0\not=W\not=k^n,$

such that G leaves W invariant (globally).

This result turns out to play a role in a current research project (with O. Dinai), and since I had never looked properly at the proof(s) before, I’ve been a bit curious about it, and tried recently to understand it. There are very simple proofs known, but the shortest ones seem to be typically not very enlightening when it comes to understand why the result is true. They’re the kind of arguments you might feel you could find once you knew the result, but why would you think of proving it first? So — it was vacation time! — I had a look at Burnside’s original paper. This can be found here; if you do not have access to the Proceedings of the L.M.S, here is a fairly representative extract of the style:

As far as I’m concerned, this is barely recognizable as meaningful mathematics, and almost unreadable. I say almost, because (vacation effect) I took it as an intellectual challenge to try to reformulate Burnside’s argument in more modern terms, and I believe that I succeeded. It was a big help that the paper is only four pages long; it turns out that the one page from which the extract is taken, although I can’t explain it in any reasonable way, contains the last step of Burnside’s argument. From the fact that he needed seventeen lines to prove the “obvious” half of his theorem, there was therefore every chance that whatever is done here in one page should not be too difficult to figure out with some thought.

So here is a sketch of my reading of his proof of the non-trivial direction (that, for an irreducible action, we have k[G]=M(n,k); for full details, see this short note). We denote

$V=k^n,\quad E=M(n,k),\quad E^\prime=M(n,k)^\prime=Hom(E,k),\quad\text{the dual of } E,$

and then we define

$R=\{\phi\in E^\prime\,\mid\, \phi(g)=0\text{ for all } g\in G\}\subset E^\prime,$

which is the linear space of all linear relations satisfied by the matrices in the group. By linearity and duality, we see that the goal is to show that, if G is irreducible, then the space R is zero. The steps for this are:

R is a subrepresentation of E’ for a natural action of G on E’ given by
$\langle g\cdot \phi,A\rangle=\langle \phi,g^{-1}A\rangle,\quad\quad g\in G,\ \phi\in E^{\prime},\ A\in E$
in duality-bracket notation (this is not the same as the usual tensor product of the tautological representation of G on V and its contragredient);

We now attempt to analyze this representation on E’, and the next three steps do this (they are completely independent of the problem at hand); first, the representation on E’ is isomorphic to a sum of n copies of the (irreducible) contragredient of the tautological representation;

Hence any irreducible subrepresentation in R is obtained as the image of a G-equivariant embedding
$V^\prime\rightarrow E^\prime;$

But all such maps
$\alpha\,:\, V^\prime\rightarrow E^\prime$
are of the form
$\langle \alpha(\lambda),A\rangle=\lambda(Av)$
for some fixed vector v in V;

And we now come back to the problem of understanding the relations R; if R were non-zero, it would contain the image of a map α of this form, for some non-zero vector v;

But if we then specialize the definition of α with A being the identity matrix — which is in G –, we find, from the fact that the image of α is in R, that
$0=\alpha(\lambda)(1)=\lambda(v),$
for all λ, and this is a contradiction with the condition that v be non-zero… Hence we must have R=0.

It is not obvious here where it is necessary to use the fact that k is algebraically closed, but this is hidden in an application of Schur’s Lemma. Interestingly, it seems that Schur published this result also in 1905. Since Burnside also uses it without any comment (or hint of proof), it must have been known (at least to him) before. It is also amusing to note that, in fact, there is no mention whatsoever of a base field in Burnside’s paper.

I like this proof, in part because it would make sense to try to proceed in this way, even if the result turned out to be different (say, a characterization of the relation module R instead of a proof that it is zero). Also, I may be influenced by the similarity with the study of relations between roots of polynomials that can also be done using elementary representation theory of the Galois group, as discussed in this old blog post.

But as I said, there is a part of Burnside’s paper I really don’t understand, even if I suspect it is equivalent or very similar to what I did. And I am forever thankful that Emmy Noether came along some years later to put algebra on a more reasonable track than endless talk of “successive sets of symbols with the same second suffix” (which sounds almost like one of those alliterative exercises used to detect drunkenness…)