Unit I

Fourier series and integrals – Definitions and easy results – The Fourier transform – Convolution – Approximate identities – Fejer’s theorem – Unicity theorem – Parselval relation – Fourier Stieltjes Coefficients – The classical kernels.

Unit II

Summability – Metric theorems – Pointwise summability – Positive definite sequences – Herglotz;s theorem – The inequality of Hausdorff and Young.

Unit III

The Fourier integral – Kernels on R. The Plancherel theorem – Another convergence theorem – Poisson summation formula – Bachner’s theorem – Continuity theorem.

Unit IV

Characters of discrete groups and compact groups – Bochners’ theorem – Minkowski’s theorem.

Unit V

Hardy spaces- Invariant subspaces – Factoring F and M. Rieza theorem – Theorems of Szego and Beuoling.
Text Book: Henry Helson, Harmonic Analysis, Hindustan Book Agency, Chapters 1.1 to 1.9,
to 3.5 and 4.1 to 4.3.

CO1- Understand the general concept of weak solution and the criterion of having weak solution for hyperbolic equation.
CO2- Able to model the basic diffusion processes and understand the mathematical methods that are useful in studying the structure of their solutions.
CO3-Understand the existence and uniqueness of traveling wave solutions solutions.
CO4-Understand the concept of nonlinear eigenvalue problem the stability of equilibrium solutions for reaction-diffusion equation.
CO5-Understand the formulation of system of PDEs and their applications.