Fourier Analysis and Its Applications

Fourier Analysis continues to be an active area of research in
Mathematics with numerous applications in the field of both pure and
applied mathematics. The principles of Fourier analysis are applied in
diverse areas of physics and engineering such as celestial mechanics, wave
propagation, image processing and modulation problems to name a few.
Thus Fourier analysis forms an essential component in the tool-kit of
scientists and engineers.

In this course we shall focus on the basic theoretical aspects of the
subjects relegating computational issues to exercises. The latter are
adequately covered in most standard curricula while issues of convergence
are seldom addressed. Here we shall prove rigorously some of the basic
theoretical results that are accessible with a rudimentary knowledge of
mathematical analysis and linear algebra.

The novel feature of the course is the type of applications presented
coming from diverse areas of mathematics. We shall look at three
applications namely the isoperimetric theorem in geometry, a problem in
celestial mechanics namely, the inverting of the Kepler's equation in
planetary orbit theory. The third application is a proof of Weyl's
equi-distribution theory that is of immense use in number theory. We shall
also derive the transformation formula for the Jacobi theta function which
is another "avataar" of the famous functional equation of Riemann for the
function bearing his name. It must be mentioned here that one of the most
celebrated open problems today in mathematics concerns the zeta function
of Riemann.


We shall be looking closely at the Fourier transform proving its basic
properties and its applications to differential equations. We shall derive
an integral form for the solution of Airy's equation which plays an
important role in optics. For problems involving radial symmetry the
Fourier transform of radial functions come into play. The transform then
reduces to a one variable Bessel transform. We shall end the course by
proving an important formula on Fourier transform of the square of the
absolute value of the gamma function along vertical lines in the right
half plane. This formula was originally obtained by Srinivasa Ramanujan
but we shall provide here an alternate proof via Fourier analysis.