**MATHEMATICS 5102: Linear Mathematics in Infinite Dimensions
**

Info for the 9:10 a.m. section of SP 2013

** Instructor:** Dr. Rodica D. Costin

**Office:** 436 Math Tower

**Office hours:** MWF after class (10:15-11 a.m.) or
by appointment

**e-mail:** rcostin"at"math"dot"ohio-state"dot"edu

Here are lecture notes for the first two chapters. I am planning post revised notes as the semester advances.

Hilbert Spaces

Sturm-Liouville problems

Topics studied in each lecture and general announcement are posted below.

Monday Jan. 7 We defined the space l^2 (Hilbert Spaces notes, Sec. 1.4)

Wed Jan. 9 Hilbert Spaces notes, Sec. 1.5, 2.1, 2.3, started 2.4.

Homework 1 - due Fri Jan 16.

You may work in teams: each team submits only one write-up, signed by all the members of the team.

Fri Jan. 11 - up to Sec. 3.2 (including)

Monday Jan. 14 Orthonormal bases. Revised notes

Further references for Hilbert Spaces:

M. Reed, B. Simon,

(see the handout).

Wed Jan. 16 Revised notes

Homework 2 (due Fri Jan 25)

Fri Jan. 18 re-Revised notes

Wed Jan. 23

Homework 3

Fri Jan. 25 Revised notes for Jan. 23 and 25

Monday Jan. 24 We finished the chapter on Hilbert spaces, see here revised notes and we started Sturm-Liouville chapter, Sec. 1.1

Wed Jan. 30 Revised notes

Homework 4

Fri Feb 1st Revised notes

For Fourier series and integrals, see also the Stanford lectures of Prof. Osgood, available on Youtube.

See also a

Mon Feb 4 Revised notes

Wed Feb 6 Dirichlet Kernel

Homework 5

Fri Feb 8 The Fourier Theorem;

Gibbs phenomenon. Smoothness versus decay of Fourier coefficients

Discrete convolution

Recall the Rayleigh's quotient for self-adjoint matrices. Warning: disregard the comments about existence of min/max in finite dimensions; in infinite dimensions extra care is needed. In the (infinite-dimensional) case we will study on Monday, R will have a min but not a max.

Mon Feb 11 Completeness. Shannon's sampling Theorem (compact interval case)

Wed Feb 13 Sturm-Liouville Oscillation Theorems; the wave equation; the vibrating rod (Sec. 6, 9.1. 9.3 in the Sturm-Liouville notes posted on top of this page)

Homework 6 (due March 1st)

Mon Feb 18 Fourier Integral and Examples of Fourier integrals

Wed Feb 20

Mon Feb 25 Wave packets - Prof. Ulrich Gerlach

Homework 7 (due March 8 -

Wed Feb 27 Distributions (see also the

Fri March 1 Fourier transform of distributions

Mon March 4 Fundamental sequences for Dirac's delta-function. Green's function

See the hand-outs on Carmen from Hilbert & Courant's book. It was written before the introduction of distributions, and you can see the physical intuition, beautifully written and developed. See pages 351-362, and 371-376. Must read!

Wed March 6 Green's Function, continued

Fri March 8 Uniqueness, Fredholm Alternative We only discussed Theorem 8. Please read Theorems 9 and 10 as well, and use them to solve Homework 8.

A simple example of Green's function, corrected.

Homework 8 due March 22

Recall some elements of complex functions from last semester.

March 18 - April 12 Helmholtz equation - lectures by Prof. Gerlach

Translation as exp(d/dx)

Homework 9 due Fri March 29

Homework 10 due Fri Apr 5

The revised chapter on Distributions, Green's function, Adjoint Problems. Please read!

Revised notes: Hilbert Spaces, Sturm-Liouville Theory