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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 R^d, Definition of X-ray transform in R² and Radon transform in R³, Connection with Fourier Transform, Uniqueness

Suggested Books:

Texts:

• E.M.Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Univ Press, 2003
• (For topics marked with a*) W. Rudin, Functional Analysis, 2nd Ed, Tata McGraw-Hill, 2006

References:

• J. Douandikoetxea, Fourier Analysis (Graduate Studies in Mathematics), AMS, 2000
• L. Grafakos, Classical Fourier Analysis (Graduate Texts in Mathematics), 2nd Ed, Springer, 2008

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