Jeffrey Schenker's Generals Exam
October 21, 1998

Committee: Fefferman (chair), Stalker, Trotter.
Topics: Functional Analysis
  Singular Integrals and Harmonic Analysis on R^n

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Fefferman asked me with which topic I would like to start.  I replied,
"anything but algebra," and we settled on functional analysis:

T: Consider L^2[0,1].  Are the operations of multiplication by x and
multiplication by x^2 unitarily equivalent on this space?

I couldn't decide an answer to this one right away, but they helped me
to the answer which I then proved.

T: Consider L^2[-1,1].  Is the operation of multiplication by x^2 on this
space unitarily equivalent to multiplication by x^2 on L2[0,1]?

I mentioned that the multiplicities seemed to be different in each case,
and Trotter suggested that I consider cyclic vectors.

At this point, Stalker mentioned that he would like to ask me a question
which Trotter had asked of him when he took generals!

S: Consider the functions 1/(1 + x^2) and all of its translates.  Prove
that the closed linear span of these functions is all of L^1.

I had a lot of difficulty with this one, but they helped me through an
argument using the Fourier transform, and expanding functions of compact
support as Fourier series.  In the process, I had to quote the fact that
the Fourier transform of 1/(1 + x^2) is exp(-2 pi |s|). This led into
complex analysis:

S: Use contour integration to find the Fourier transform of 1/(1 + x^2).

S: How can you construct an entire function with zeros only at certain
specified points?

S: There's a famous function with zeros at the negative integers, how
can you express this function as a product?

I wasn't sure how to find the factors that force the product to
converge, so they helped me with that.

S: Suppose an entire function is real on the unit circle, what does this
say about the function?

I answered this in a convoluted fashion, after much floundering, by
saying that the behavior of the function at infinity would have to be
the same as the behavior at zero, so it would have to map the sphere to
the plane holomorphically, so it would have to be constant.  This is
technically correct, but it also led into real analysis, because
Fefferman pointed out that you could prove this much more easily with
the following line of questions:

F: What is the real part of an analytic function?

F: What is the imaginary part of an analytic function?

F: So if an entire function is real on the boundary of the disc, what do
you know about the imaginary part on the inside of the disc?

Fefferman described the following question as real analysis, but I
suspect that it was on the exam because I took harmonic analysis as a
special topic.

F: Discuss the Poisson kernel.

This led to an extended discussion, in which I proved results such as Lp
convergence at the boundary for Poisson integrals in the upper half
plane.  Along the way I mentioned that you could use the maximal
function to prove almost everywhere convergence. This led, of course, to
the following questions, which made up the singular integrals part of
the exam.

F: Define the maximal function.

F: What is the basic fact about the maximal function?

F: Use the maximal function to prove the almost everywhere convergence
of Poisson integrals.

Now we switched to algebra.

T: Does L1 have a natural multiplication with which it becomes an
algebra?

T: Consider a translation invariant subspace of L1, what can you say
about its relation to L1 as a convolution algebra?

This was interesting, because once I figured out what was going on, I
had to talk about Riemann integration of Banach space valued functions
to prove that such a subspace is an ideal.

T: Consider a commutative ring, R, and a maximal ideal, I, what can you
say about the structure of R/I?

T: Suppose I were prime?

T: Talk about the possible subgroups of Z+Z+Z.

This led to an extended discussion of what I knew about free groups and
free Abelian groups.  I didn't really know much, but this didn't seem to
upset the committee.

T: Now suppose that you have a subgroup of Z+Z+Z, what theorem tells you
something about the structure of the quotient group?

He was trying to get me to quote the structure theorem for finitely
generated Abelian groups, which I eventually did.

T: This theorem has a generalization to another algebraic structure,
what is that generalization?

T: Name an application for the structure theorem for modules over a PID.

Of course I mentioned Jordan canonical form for matrices.  Trotter said
that he didn't want to see another proof of Jordan form, so we moved on.

T: Is the additive group of the reals isomorphic to the multiplicative
group of the positive reals?

T: Is the same result true with reals replaced by rationals?

This was the last question on the exam, and Fefferman eventually handed
me the answer.  I went out into the hall and after a few minutes,
Fefferman came out and said that I had passed.  Then I got some coffee
in the common room.

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Jeffrey H. Schenker
schenker@princeton.edu