Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
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|Each topic requires approximately 2 weeks to cover||
|Lp spaces: Basic inequalities of Jensen, Holder, Minkowski; Completeness; Continuous linear functionals and weak convergence; Approximation and the theory of mollification; Dual space of Lp; Integral operators and Young's inequality|
|The Sobolev spaces Hk(Ω) k a non-negative integer: Weak derivatives; Completeness of Hk(Ω); Approximation by smooth functions; Sobolev embedding theorem; Morrey's inequality and the Gagliardo-Nirenberg-Sobolev inequality; Extension and trace theorems; Rellich's theorem and weak compactness|
|The Fourier transform: L1(Rn) Fourier transform and its inversion; Schwartz functions of rapid decay; The Gaussian; Extension of Fourier transform to L2(Rn) and Plancheral's theorem; Tempered distributions S' and extension of Fourier transform to S'; Examples of the use of Fourier transform to Poisson, heat, and wave equations|
|The Sobolev spaces Hs, s real: Hs(Rn) via the Fourier transform: Negative-order spaces as distributions. Hs(Tn): Fourier series revisited|