Math 205C : Syllabus

This is a rough syllabus, with references. Further references might be added. The actual schedule and precise list of topics will be adjusted the progress in class.

• Reminder: distributions, Sobolev spaces

• Inequalities: Sobolev, Rellich-Kondrashov, Poincar'e, Nash, Log-Sobolev

• Beyond Lebesgue measure: Connections to heat semigroups

• Marcinkiewicz interpolation

References: Lieb, Loss, Analysis; Duoandikoetxea, Fourier Analysis.

• Pointwise convergence of Fourier polinomials; Fejer's theorem

• Jackson's theorem

• Degree of approximation for functions with given modulus of continuity

• Kolmogorov widths

• Lower bounds via metric entropy

• Approximation by ridge function (aka neural networks); Barron spaces

• Separation results for multilayer networks

References: Lorentz, Approximation theory; Cheney, Light, A course in approximation theory, Pinkus, Approximation theory of the MLP model in neural networks

• Fourier analysis of Boolean functions

• Hypercontractivity

• Threshold phenomena

• Connections to learning theory

References: O'Donnell Analysis of Boolean functions; Boucheron, Lugosi, Massart, Concentration inequalities

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