MATH 1270 home page

MATH 1270, Fall 2008. Topics in Functional Analysis.

MWF 11:00--11:50 am, Barus & Holley 165

Instructor: Prof. Sergei Treil,
office: 216 Kassar, ph.: x1122
e-mail: treil<at>math.brown.edu
Web: http://www.mth.brown.edu/~treil/

Office hours: MWF 1:00-2:00 pm and by appointment.

Additional materials:

Book Theory of Functions of real variable (2 Meg PDF), by Shlomo Sternberg, available from his website
Handout "Sets, relations, functions" available on "My Courses"

Syllabus. (pdf format)

Outline of the course and homework assignments.

Lecture Notes taken by students (pdf format).

9/12: Bounded linear operators
9/15: $\epsilon/3$ -theorem and the uniform boundedness principle.
9/17: Hilbert spaces. Cauchy-Schwarz inequality, parallelolgramm and polarization identities.
9/19: Orthonormal bases. Abstract Fourier series.
9/22: Orthogonal projection and best approximation.
9/24: Fourier series. Definition, motivation. Dirichlet kernel.
9/26: Convergence of Fourier series; convergence in $L^2$ . Divergence at a point. Fejer means.
9/29: Uniform convergence of Fejer means.
10/1: Convergence of Fejer means in $[; L^p ;]$ . Properties of convolution, convolution theorem. Dini condition, Riemann--Lebesgue lemma and pointwise convergence of Fourier series.
10/3: Riesz Representation theorem.
10/6: Adjoint operator. Self-adjoint, normal and unitary operators.
10/8: Invertible operators. Relation between kernels and ranges.
10/10: Kernels and ranges. Banach inverse map theorem. Spectrum.
10/15: Spectrum is non-empty and compact.
10/17: Spectral mapping theorem. Gelfand formula for spectral radius.
10/20: Volterra integral operator. Application to differential equations. Continuous functions of self-adjoint operators.
10/22: Mollification. Proof of Weierstrass approximation theorem.
10/24: Proof of Weirstrass approxiamtion theorem (continued). $[; \mathcal C^\infty ;]$ mollifiers and glue.
10/27: Fourier transform.
10/29: Proof of Plancherel formula and Fourier inversion theorem Shannon_Kotelnikov sampling theorem.
10/31: Physical meaning of Sannnon--Kotelnikov theorem. Poisson summation formula.
11/3: Heisenberg uncertainty principle. Discrete Fourier transform.
11/5: Fast Fourier transform.
11/7: Fourier tarnsform on groups. Tempered distributions.
11/10: Differentiating distributions. Dual spaces and Hahn--Banach theorem.
11/12: Duals of $l^p$ and $L^p$ . Banach limit. Dual extremal problems.
11/14. Dual extremal problems (continued). Minimal systems, biorthogonal system. Basis.
11/17: Uniqueness of dual system. Banach basis theorem. Riesz bases.
11/19: Minimal system and bases: some examples. Frames.
11/21. Frames. Feightinger anf paving conjectures.
12/01. Introduction to wavelets. Gabor bases. Haar wavelets.
12/03: Multiresolution analysis. Getting wavelets from MRA.
12/05: Getting "mother wavelet" from MRA.
12/08: Mother wavelet from MRA. Lemarrie--Meyer wavelets.