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


General information

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. (Actually, Fri Jan 18th...)
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, Methods of Modern Mathematical Physics, vol. I
(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 handout on Carmen.

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

Midterm: Wed Feb 20th

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)

Fri Feb 15 Prufer coordinates: existence of eigenvalues for Sturm-Liouville problems (Sec. 7. in the Sturm-Liouville notes)

Mon Feb 18 Fourier Integral  and Examples of Fourier integrals
 
Wed Feb 20 Midterm

Fri Feb 22  Experiments: See the Fourier transform!. Wave packets - Guest speaker, Prof. Ulrich Gerlach

Mon Feb 25 Wave packets - Prof. Ulrich Gerlach

Homework 7 (due March 8 -  only problems 3,4,8)

Wed Feb 27 Distributions (see also the hand-out on Carmen)


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


Pick up the final exam on Monday Apr 22 after class, to be returned in my office, according the the official schedule seen below.

Final exam scheduled for Tuesday Apr 30 8:00am-9:45am, see Registrar Final Examination Schedule