# Fourier Analysis and Its Applications

By Gopala Krishna Srinivasan   |   Indian Institute of Technology Bombay
Learners enrolled: 1029
```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.
```

Summary
 Course Status : Completed Course Type : Elective Duration : 16 weeks Category : Annual Refresher Programme in Teaching (ARPIT) Credit Points : 5 Level : Undergraduate/Postgraduate Start Date : 01 Dec 2020 End Date : 31 Mar 2021 Enrollment Ends : 31 Dec 2020 Exam Date : 10 Apr 2021 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

The course will broadly be in two parts: Fourier series and Fourier Transforms.

consist of the following modules:

1) Basic convergence theorem of Fourier series and its applications

2) Mean convergence

3) Cesaro Summability - Fejer's theorem and its applications.

4) The space S of rapidly decreasing functions

5) Fourier transform as an operator from S to S and the Plancherel's theorem

6) Sturm-Liouville problems and its applications

7) Application to PDEs.

### Books and references

1) G. B. Folland, Fourier Analysis and its applications, Wadsworth and Brooks/Cole, California 1992.

### Instructor bio

Short Resume of Gopala Krishna Srinivasan (Professor) Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076 Personal Details: • Gender: Male • Age: 52 • Phone number: 9819807454 (Mobile), 022 256 7454 (Office) Academic Qualifications: • Bachelor of Science, University of Bombay, 1987. • Master of Science, University of Bombay, 1989. • Doctor of Philosophy, Mathematics, University of Minnesota, Minneapolis, in 1995. Thesis: WTC Expansions and Painlevé Analysis. Areas of Specialization/interest: 1. Partial Differential Equations, Shock waves in Hyperbolic Systems of Conservation Laws. 2. Dynamical systems. 3. Classical Analysis, Special Functions and History of Mathematics. Publications: 1. S. Kichenassamy and G. K. Srinivasan, The Structure of WTC Expansions and Applications, Jour. Physics - A, 28 (2005) 1977-2004. 2. N. Joshi and G. K. Srinivasan,Well-Posedness of Painlevé Expansions, Nonlinearity, 10 (1997) 71-79. 3. G. K. Srinivasan and V. D. Sharma, Modulation Equations for Weakly nonlinear Geometrical Optics in Media Exhibiting Mixed Nonlinearity, Studies in Applied Mathematics, 110 (2003) 103-122. 4. G. K. Srinivasan and V. D. Sharma, A Note on the Jump Conditions For Systems of Conservation Laws, Studies in Applied Mathematics, 110 (2003) 391-396. 5. G. K. Srinivasan and V. D. Sharma, On Weakly Nonlinear Waves in Media Exhibiting Mixed Nonlinearity, Journal of Mathematical Analysis and Applications, 285 (2003) 629-641. 6. G. K. Srinivasan and V. D. Sharma, Energy Dissipated Across Shocks in Weak Solutions of a Scalar conservation Law, Studies in Applied Mathematics, 112 (2004) 281-291. 7. G. K. Srinivasan and V. D. Sharma, Implosion-Time for Converging Spherical and Cylindrical Shells, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004) 974-982. 8. V. D. Sharma and G. K. Srinivasan, Wave Interaction in a Non-equilibrium Flow, International Journal of Non-Linear Mechanics, 40 (2005) 1031-1040. 9. G. K. Srinivasan, The gamma function - an eclectic tour, American Mathematical Monthly 114 (2007) 297-315. 10. G. K. Srinivasan, A Note on Lagrange’s Method of Variation of Parameters, Missouri Journal of Mathematical Sciences, 19 (2007) 11-14. 11. G. K. Srinivasan, V. D. Sharma and B. S. Desale, An Integrable System of ODE Reductions of the stratified Boussinesq Equations, Compt. Math. Appl, 53 No. 2, (2007) 296-304. 12. B. S. Desale and G. K. Srinivasan, Singular analysis of the system of ODE reductions of the stratified Boussinesq equations, IAENG International journal of applied mathematics, 38 (2008) no. 4, 184-191. 13. G. K. Srinivasan and P. Zvengrowski, On the horizontal monotonicity of |Γ(s)|, Canadian Math. Bulletin, 54 (2011), no. 3, 538-543. 14. G. K. Srinivasan, Dedekind’s proof of Euler’s reflection formula via ODEs, Mathematics Newsletter (published by Ramanujan Mathematical Socity) 21 (2011) no. 3, 82-83. 15. D. Chakrabarty and G. K. Srinivasan, On a remarkable formula of Ramanujan, Arch. Math. (Basel) 99 (2012), 125-135. 16. G. K. Srinivasan, A unified approach to the integrals of Mellin–Barnes–Hecke type, To appear in Expositiones Mathematicae. Ph.D student: 1. Title of Thesis: An Integrable System of ODE Reductions of the stratified Boussinesq Equations. Defended in January 2007. 2> sunil Kumar Yadav. Books I have completed a web-book on Algebraic Topology under the NPTEL Scheme. The link is http://nptel.iitm.ac.in/courses/111101002/ Proficiency in foreign languages: (i) Have completed four levels of basic German at the prestigious Göethe Institute (Max Muller Bhavan) upto Mittelstuffe - I. (ii) Completed the first two levels of French at Alliance Française de Mumbai. Awards and Recognitions: 1. Award for Excellence in Teaching, Indian Institute of Technology, Bombay (Sept 5, 2011) 2. Award for Excellence in Teaching, Indian Institute of Technology, Bombay (Sept 5, 2007) 3. Award for Excellence in Teaching, Indian Institute of Technology, Bombay (Sept 5, 2002) 4. Citation For Excellence in Teaching, awarded by the Department of Mathematics at the University of Minnesota, Minneapolis (1995). 5. Shri Pandharinath Moroba Mungre Prize, for the year 1989 for passing MSc examination with highest number of marks on the aggregate. 6. The Late Shri Balkrishna Waman Deshpande Prize, for the year 1989 for passing MSc examination with highest number of marks on the aggregate. 7. Principal V. K. Joag Memorial Prize, for the year 1989 for passing MSc examination with highest number of marks on the aggregate. 8. Professor Laxman Vasudeo Gurjar Prize, for the year 1989 for passing MSc examination with highest number of marks on the aggregate. 9. Certificate of Merit for Outstanding Performance and Being Among the Top 5% of Candidates Qualifying in C.S.I.R - U.G.C Fellowship in Mathematics (1988)