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Indian Institute of Science Education and Research Bhopal


MTH 610: Fourier Analysis on the Real Line (4)

Pre-requisites: MTH 404 Measure and Integration, MTH 503 Functional Analysis: Normed linear spaces, completeness, Uniform boundedness principle, MTH 405 Partial Differential Equations: Basic knowledge of Laplacian, Heat and Wave equations

The vibrating string, derivation and solution to the wave equation, The heat equation

Definition of Fourier series and Fourier coefficients, Uniqueness, Convolutions, good kernels, Cesaro/Abel means, Poisson Kernel and Dirichlet’s problem in the unit disc

Mean-square convergence of Fourier Series, Riemann-Lebesgue Lemma, A continuous function with diverging Fourier Series

Applications of Fourier Series : The isoperimetric inequality, Weyl’s equidistribution Theorem, A continuous nowhere-differentiable function, The heat equation on the circle

Schwartz space*, Distributions*, The Fourier transform on R: Elementary theory and definition, Fourier inversion, Plancherel formula, Poisson summation formula, Paley-Weiner Theorem*, Heisenberg Uncertainty principle, Heat kernels, Poisson Kernels

(If time permits/possible project topic) Definition of Fourier transform on Rd, Definition of X-ray transform in R2 and Radon transform in R3, Connection with Fourier Transform, Uniqueness

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