SP 2012, 10:30 a.m. MWF, CC 0326

**Instructors:**

Rodica D. Costin, office MW 436

Ulrich Gerlach, office
MW 506

**Office hours:**

Rodica D. Costin: W, F 11:30-12:20 a.m. or by appointment

Ulrich Gerlach:

Professor
Gerlach's online book

Further reading: you may find useful to consult also

Fourier Series and Boundary Value Problems, by
Brown and Churchill

Fourier
analysis: an introduction, by Stein and Shakarchi

Fourier Series,
by Tolstov

Course log:

Monday,
March 26 Lecture 1 notes

Friday, March 30 pages 1 to 5b from the notes here.

The HW assignment handed in in class today is due next Friday April 6, at the beginning of class.

Monday,
April 2 ...continuation

NOTE: the HW is due Monday April 9 (since we are behind with the material taught in class).

Read the proof of the Fourier Theorem, pages C5-C8 in the notes above. Today we do:

The Fourier Integral. Dirac's delta function and other distributions.

Please read these notes (including the material not covered in class) and solve the exercises included.

Friday, April 6 (i) Fourier Integral Theorem and its proof reviewed

(ii) Fourier transform as an isometric transformation

Lecture notes

Monday, Apr. 9 (i) Fourier transform of a periodic function

(ii) Construction of an orthonormal wave packet basis for L^2

(iii) Handout on four applications of O.N. wavepackets.

Wednesday, April 11 Lecture topics were:

- Fourier Series Theorem + Fourier Integral Theorem ==> The set of wave packets form a complete (and O.N.) basis for the space of square integrable (= finite energy) functions on the real line.
- The phase space representation of such functions.
- Parceval's relation as a sum of wave packet energies occupying their respective phase space cells.
- A musical score's (say, Mozart's or Beethoven's sonatas; see e.g. .pdf) aggregate of notes on a 2-D sheet of music represented as wave packet energies occupying their respective 2-D phase space cells.

Homework 2 is due this Friday, of course.

Friday, April 13 The purpose of the lecture is to set the context for Monday, when we will decompose L^2 into the hierarchy of multi resolution subspace, hoping to open the curiosity about multi resolution analysis. Today's material is not yet in the typeset notes, but see the handwritten notes: a quick study to wavelets, notes on wavelets.

Multi Scale Analysis:

a) Translated sinc functions: its application to the Whittaker-Shannon sampling theorem.

b) Translated and dilated sinc functions as a multi resolution basis for L^2 .

c) The "father" wavelet as the fountainhead for these basis functions.

Here is a handout for HW3.

Monday, April 16 Ideas developed today:

- Spaces of wavelets generated by translation and dilation of the father wavelet.
- Dilation hierarchy of father generated wavelets.

- Stepwise Multi Resolution Analysis into bulk ("father
") and detail ("mother") wavelets generated subspaces of L^2;
MRA <--> father wavelet.

- Projection of an L^2 function into these subspaces.

Wednesday, April 18 Today's development consisted of

- Input-output relation as the generalization based on an observed causal source-response relation for a linear system.
- Linear algebra review of the existence and uniqueness of a linear input-output relation expressed equivalently in terms (i) the existence of right and left inverses as prototypes for the unit impulse response (a.k.a. the Green's function) for the system, (ii) onto and one-to-one mapping properties, (iii) spanning and linear independence properties, and hence the solution to the inhomogeneous linear system.
- Definition of the concept of a given operator and its adjoint to include the given and the (in general different) adjoint boundary conditions characterizing their domains.
- (See 4.1.1. in the text and an algebra review.)

Monday, April 23

- The reciprocity theorem: proof and application (end of Section 4.2)
- Poisson's equation for the simple string (Section 4.3.1)
- Continuity and jump conditions for a string imbedded in an elastic medium (Section 4.3.2)

- The Fundamental Theorem of Green's function Theory: discussion of its featues and its proof;
- Uniqueness of the Green's function, if it exists; a preview of the importance of uniqueness.
- Construction of the Green's function for separated boundary
condition stated as a theorem; geometrical and analytical reason
for when the construction is successful, and when it isn't.

- Worked example: the amplitude profile of a static simple string

- Solution to the totally inhomogeneous boundary value problem in terms of the system's Green's function.
- Construction of the Green's function for arbitrary homogeneous boundary conditions.
- Solution to the inhomogeneous boundary value problem in terms
of solutions to the Sturm-Liouville eigenvalue problem.

- Review: Solutions to a non-selfadjoint inhomogeneous boundary value problem in terms of the eigenfunctions of the associated given S-L problem and its adjoint, whose eigenfunction are "bi-orthonormal" to those of the given one.
- Integrate the Green's function in the complex lambda plane to obtain the orthonormalized eigenfunction of a self-adjoint boundary value problem
- Worked example of a simple string with free boundary conditions at both ends.

- Comments about the infinite string boundary value problem as the limit of a finite string boundary value problem.
- Review: branches, branch cuts, and Riemann sheets for the square root function
- Handouts: You and Your Research (by Richard Hamming), a guide for doing great work in your chosen field"; Integral equations via Green's functions"

- Harmonically driven semi-infinite string: (i) Its amplitude
profile, (ii) outgoing and incoming boundary conditions, (iii)
Green's function for the amplitude profile, (iv) square
integrability and non-integrability on the two Riemann sheets of
the square root function.

- Comment about section 4.10.4: The infinite string as the limit of one which is finite: the mixed Dirichelet-Neumann conditions become the outgoing or incoming boundary conditions as the string length becomes arbitrarily large.

- Outgoing and square integrability boundary conditions of the
Green's function for a simple string infinite in both
directions.

- Additional comment about section 4.10.4: The infinite string as the limit of one which is finite: the aggregate of simple poles of the coalesce into a branch cut.
- Fourier Sine Theorem via complex integration of the Green's function for a semi-infinite string.
- Students should start reading Chapter 5

- Helmholtz's equation: its plane wave solutions relative to polar coordinates
- Helmholtz's equation as a highly degenerate eigenvalue problem
- Eigenvalues of a complete set of commuting operators as an eigenfunction identifier

Monday, May 14

- Rotation eigenfunction (a.k.a. cylinder harmonic) as a
superposition of plane wave solutions to the the Helmholtz
equation

- The defining and convergence properties of the complex contour integral representations of the two Hankel functions

- Selections from the set of 23 properties of Hankel and Bessel functions:
- Solution to the wave equation on a cylindrically symmetric background in terms of cylinder normal modes (Property 6)
- The asymptotic behaviour and its physical meaning for the
Hankel, Bessel, and Neuman functions (Properties 8-10)

- The Bessel function defined in compliance with the reflection principle of complex variables theory (Property 11)

- Integral representation of Bessl functions of integral order
- "Property 14": Complex contour integral proof of a Hankel function as a linear combination of Bessel functions
- "Property 15-16": Complex conjugates of Hankel and Neumann functions

- The Contiguity Relations (Property 16)

- Plane wave as a superposition of cylinder waves (Property 18)
- The Cylindrical Addition Theorem (Property 19)

- The exterior boundary value ("scattering") problem

- The interior boundary and initial value ("normal modes in a cavity") problem

Watson's lemma.

Stationary phase method.

Steepest descent method.

Additional resources:

Distributions and operators (p.9 revised)

Green's Function; uniqueness

(Note: "fundamental solution" F is for linear operators with constant coefficients, and solves P(d/dx)F=delta.)

Green's function of the adjoint problem (older- revised p.3, 5B,5C,6), and newer (revised). Supplement.

Simpler forms for second order linear differential operators

Review of the resolvent and elements of complex analysis

Non-self-adjoint boundary value problems (p.2B, 7B added)

More web resources: recall the dual space (p.4), the adjoint (p.15), for the adjoint boundary conditions (p.20), the Green's function for non-selfadjoint problems (p. 21)

Shannon's Sampling Theorem - the case of a compact interval.