A fundamental, and well-known, fact in harmonic analysis is that a function
or signal f(t) can be analyzed both in physical space and in frequency
space, with the two viewpoints being connected by the Fourier
transform.
Here is an example of a function, described in physical space (the physical
space variable being time, measured in seconds):
And here is the same function, now described in frequency space (time-frequency
being measured in Hertz. Only intensity is shown; phase is suppressed):
It is slightly less well known that one can combine both the physical and
frequency variables and analyze a function in time-frequency space,
also known as the phase plane. The precise representation
of functions on the phase plane is a little fuzzy, because of the Uncertainty
Principle, however one can perform phase plane analysis rigorously and
to good effect to problems in analysis.
Actually, the idea of a phase space representation of sound waves predates
that of a physical space or frequency space representation. Here
is an old example of a phase space representation:
In this course we shall develop some intuition for the phase plane, and
then show how one can exploit this intuition to obtain results concerning
Basic linear and multilinear estimates (Sobolev embedding, fractional Leibnitz
rule, div-curl lemma, averaging lemma, etc.);
Pseudo-differential operators;
Propagation of singularities;
Solvability of linear operators (the Levy example);
Fourier summation; and the analysis of highly singular integrals (such
as the bilinear Hilbert transform);
Other topics as time permits.
We will also show how such standard tools as Littlewood-Paley theory and
wavelet decomposition can be viewed as ways to decompose the phase plane
into manageable pieces. We shall also study the Walsh phase plane,
a discrete analogue of the Fourier phase plane.
There is no formal prerequisite for this course, but it is recommended
that you have already taken 245A or an equivalent course. In particular
you should be comfortable with the Fourier transform.
There is no text; printed notes will be distributed during the course and
will also be available on the web. There are no examinations for
this course, but homework will be assigned weekly, and will play a part
in determining the final grade.