MATH 325 Homepage (Spring 2016)


Lecturer: Alexandru Ionescu, aionescu@math.princeton.edu


Handouts

Information.
Problem Set 1. Due in class on Thursday , February 15.
Problem Set 2. Due in class on Thursday , February 22.
Problem Set 3. Due in class on Thursday , March 1.
Problem Set 4. Due in class on Thursday , March 8.
Practice exercises for the midterm exam.
Problem Set 5. Due in class on Thursday , March 29.
Problem Set 6. Due in class on Thursday , April 5.
Problem Set 7. Due in class on Thursday , April 12.
Problem Set 8. Due in class on Thursday , April 19.
Problem Set 9. Due in class on Thursday , April 26.
Problem Set 10. Due in class on Thursday , May 3.
Practice exercises for the final exam.


Syllabus

Feb. 06: Introduction to Fourier series.
Feb. 08: Summability of Fourier series.
Feb. 09: Review: Riemann integrability.
Feb. 13: Good kernels, Cesaro and Abel summability.
Feb. 15: Mean-square convergence of Fourier series.
Feb. 16: Hilbert spaces, Parseval's identity, proof of mean square convergence.
Feb. 20: Pointwise convergence revisited.
Feb. 22: Applications of Fourier series: equidistributed sequences.
Feb. 23: A continuous but nowhere differentiable function.
Feb. 27: The isoperimetric inequality.
Mar. 01: The Fourier transform on $R$: main definitions.
Mar. 02: The Schwartz class.
Mar. 06: The Fourier inversion formula, the Plancherel theorem.
Mar. 08: Convolutions, the Weierstrass approximation theorem, the Poisson summation formula.
Mar. 09: Review: Chapters 2, 3, 4, 5.
Mar. 13: Applications of the Poisson summation formula, the Heisenberg uncertainty principle.
Mar. 15: MIDTERM EXAM in class.
Mar. 16: Applications to PDEs.
Mar. 27: The higher dimensional Fourier transform.
Mar. 29: The Schwartz class and the Fourier inversion formula.
Mar. 30: Convolutions, good kernels, the Plancherel theorem.
Apr. 03: Polar coordinates, the wave equation in $R^d$.
Apr. 05: The wave equation in 3 and 2 dimensions, the Huygens principle.
Apr. 06: The Fourier transform of radial functions, Bessel functions.
Apr. 10: Fourier analysis on the group Z(N).
Apr. 12: Characters, Fourier analysis on finite abelian groups.
Apr. 13: Products of abelian groups, subgroups, quotient groups, the structure of finite abelian groups.
Apr. 17: Dirichlet's theorem: preliminaries.
Apr. 19: The zeta function and the Euler product formula.
Apr. 20: Dirichlet characters modulo $q$, L-functions.
Apr. 24: Proof of Dirichlet's theorem: the main reduction.
Apr. 26: The hypebolic method, Dirichlet's ivisor theorem.
Apr. 27: Non-vanishing of the L-function for complex characters.
May 01: Non-vanishing of the L-function for real characters.
May 03: Review.
May 04: Practice problems.