Introduction to Fourier Analysis

ABOUT THE COURSE: This course will cover the fundamental theory of Fourier series and Fourier integrals, as well as Fourier series for finite abelian groups. Some applications of Fourier series in differential equations, the isoperimetric problem, and number theory will be explored. Basic calculus and Riemann integrals will be required as prerequisites. The book “Introduction to Fourier Analysis” by E. Stein and R. Shakarchi will serve as the basis for this course.INTENDED AUDIENCE: UndergraduatePREREQUISITES: Basic Calculus, Riemann IntegrationINDUSTRY SUPPORT: This topic holds significant importance in mathematics and is considered a central theme that every mathematics student should be familiar with.

Week 1 : Introduction and Overview, Derivation of heat equation periodic function and Review of Riemann Integration, Definition of Fourier CoefficientsWeek 2 : Computation of Fourier series for some functions, Partial sum as convolution with Dirichlet Kernel Properties of ConvolutionWeek 3 : Riemann Lebesgue Lemma Decay for differentiable functions Fej´er’s Theory and applicationsWeek 4 : Abel Summability Pointwise convergence localization Gibb’s PhenomenaWeek 5 : Basic Inner product spaces Convergence in 2-mean Trigonometric series which are not Fourier seriesWeek 6 : Applications Isoperimetric Problem Weyl EquidistributionsWeek 7 : Improper Integral Fourier Transform on R Computations of Fourier transformWeek 8 : Schwartz Space Approximate Identity Fourier InversionWeek 9 : Plancherel Formula Applications Poisson summation formulaWeek 10 : Heisenberg Uncertainity Principle Heat EquationWeek 11 : Fourier Analysis on Z(N) Fast Fourier TransformWeek 12 : Fourier Analysis on Finite Abelian Groups Applcations