**MTH 506: 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

**R**and Radon transform in

^{2}**R**, Connection with Fourier Transform, Uniqueness

^{3}*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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