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Nov 24 2009:
Homework set 6 is available for download from the right-most column of the class schedule. It is due in class on Tuesday, December 8.

Have a great Thanksgiving break! When we return we will finish Chapter 7 with a discussion of weak convergence and then move on to highlights from Chapters 8 and 9 to finish the semester.

Jump to class schedule »

Here are some notes and solutions:

  • Notes on the two lectures on Fourier series.
  • Notes on fundamental solutions to Laplace's equation.
  • Notes on fundamental solutions to the wave equation.
  • A solution to Griffel's problem 3.18 that was part of Homework set 3 (and was closely related to question 4 on the midterm).
  • Solutions to the midterm exam.

Home Page for Math 556

Textbook:

Applied Functional Analysis, by D. H. Griffel, Dover Publications, Mineola, New York, 2002.

Class Meetings:

Tuesdays and Thursdays 11:40 AM - 1 PM in 3088 East Hall.

Office hours:

Tuesdays, Wednesdays, and Thursdays, 1-2 PM.

Grading and Course Policies:

Students will be evaluated on the basis of
Schedule
Week Meeting Date In Class Homework
Week 1 Lecture 1 Tuesday, September 8 Course Introduction. Initial and boundary value problems for partial differential equations as models for physical processes.
Lecture 2 Thursday, September 10 Chapter 1. The Dirac delta "function". Basic distribution theory. Test functions. Regular and singular distributions.
Week 2 Lecture 3 Tuesday, September 15 Comparison of distributions and functions: equality of distributions on open sets and approximation of distributions by smooth functions. Basic operations with distributions: (i) linear combinations, (ii) products of smooth functions and distributions, (iii) translations of distributions, (iv) dilations of distributions, (v) differentiation of distributions.
Lecture 4 Thursday, September 17 Convergence of distributions. Distributions describing the regularization of divergent integrals.
Week 3 Lecture 5 Tuesday, September 22 Review of classical Fourier series. Orthogonality of harmonics, formula for Fourier coefficients, Bessel's inequality, and smoothness of a function versus decay of its Fourier coefficients. Uniform convergence of Fourier series with rapidly decaying coefficients (to something...).  
Lecture 6 Thursday, September 24 Classical Fourier series, continued. Pointwise and uniform convergence of the Fourier series of a continuous and piecewise smooth function to the function itself. Distributional Fourier series and the Poisson summation formula.
Week 4 Lecture 7 Tuesday, September 29 Chapter 2. Introduction to differential equations in distributions. Formal adjoints. Distributional, weak, and classical solutions of linear differential equations. The simplest differential equation: u'=f. Introduction to Green's functions and fundamental solutions. Homework Set 1 Due
Lecture 8 Thursday, October 1 Existence of Green's functions. Use of Green's functions to solve inhomogeneous problems.
Week 5 Lecture 9 Tuesday, October 6 Chapter 3. The classical theory of Fourier transforms.  
Lecture 10 Thursday, October 8 Distributions of slow growth (AKA tempered distributions). Generalized Fourier transforms.
Week 6 Lecture 11 Tuesday, October 13 Applications to partial differential equations. Homework Set 2 Due
Lecture 12 Thursday, October 15 Chapter 4. Basic concepts of vector spaces. Normed linear spaces. Examples.  
Week 7   Tuesday, October 20 FALL BREAK  
Lecture 13 Thursday, October 22 Topological notions. Convergence and completeness. Equivalence of norms. Banach spaces.  
Week 8 Lecture 14 Tuesday, October 27 Chapter 5. Linear and nonlinear operators on vector spaces.  Homework Set 3 Due
Lecture 15 Thursday, October 29

Contraction operators and The Contraction Mapping Theorem. Neumann series for operator inverses.

Midterm Exam: 6-8 PM, 3088 East Hall

  
Week 9 Lecture 16 Tuesday, November 3 Applications. Newton's Method. Picard iteration.  
Lecture 17 Thursday, November 5 Applications continued. Solution of boundary-value problems for partial differential equations by using the "wrong" Green's function. Nonlinear diffusive equilibrium.
Week 10 Lecture 18 Tuesday, November 10 Chapter 7. Inner product spaces. Hilbert spaces. Orthogonal bases and the Gram-Schmidt process. Homework Set 4 Due
Lecture 19 Thursday, November 12 Orthogonal expansions. The Bessel, Parseval, and Riesz-Fischer Theorems.
Week 11 Lecture 20 Tuesday, November 17 Linear functionals, weak convergence in Hilbert space, and the Riesz Representation Theorem.  
Lecture 23 Thursday, November 19 Chapter 12. Construction of L2 as a distributional Hilbert space without Lebesgue integration theory. Sobolev spaces.
Week 12 Lecture 22 Tuesday, November 24 Chapter 8. The algebra of bounded operators on normed spaces. Homework Set 5 Due
Thursday, November 26 THANKSGIVING BREAK
Week 13 Lecture 23 Tuesday, December 1 Selfadjoint operators and the eigenvalue problem. Sturm-Liouville theory.  
Lecture 24 Thursday, December 3 Compact operators. Examples.
Week 14 Lecture 25 Tuesday, December 8 Chapter 9. The Spectral Theorem for compact selfadjoint operators. Application to Sturm-Liouville problems. Homework Set 6 Due
Lecture 26 Thursday, December 10 The Fredholm alternative.
Week 15   Wednesday, December 16 FINAL EXAM