MAS296 Fourier Theory (NJTech)

Semester 2, 2016/17 10 Credits
Lecturer: Dr Fionntan Roukema Reading List
Aims Outcomes Assessment Full Syllabus

We introduce inner product spaces, including the space of continuous periodic functions. This allows us to define distances and angles between functions, by analogy with distances and angles between vectors in R3. We then reinterpret the theory of Fourier series in these terms. We also discuss adjoints of operators. We show that any self-adjoint operator admits an orthonormal basis of eigenvectors, and that the eigenvalues are all real numbers.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.


Outline syllabus

  • Inner products and the Cauchy-Schwartz inequality.
  • Projections and the Gram-Schmidt procedure.
  • Adjoints, and diagonalisation of self-adjoint operators.
  • Fourier theory and the L2 convergence theorem.



Aims

  • Introduce the abstract theory of inner product spaces.
  • Reinterpret Fourier theory in terms of inner product spaces.
  • Familiarise students with abstract and axiomatic mathematics.

Learning outcomes

  • Students should understand the main concepts of inner product spaces: the Cauchy-Schwartz inequality, orthonormal sequences, orthogonal complements, projectors, Parseval's inequality, the Gram-Schmidt procedure.
  • Students should be familiar with a range of examples of these concepts, including the space of continuous periodic functions.
  • Students should understand how this relates to Fourier theory, including the statement of the L2 convergence theorem.

16 lectures, 16 tutorials

Assessment

One formal 2 hour written examination. All questions compulsory.

Full syllabus

1. Inner products

Definitions and examples.
2. Elementary results
The Cauchy-Schwartz inequality, Pythagoras' theorem, the parallelogram law.]
3. Orthogonality
Definition, orthogonal complements of sets, orthogonal and orthonormal bases.
4. Projections and the Gram-Schmidt procedure
The definition of a projection, the Gram-Schmidt method of orthogonalisation.
5. Adjoints of linear maps
Definition, proof of existence and uniqueness.
6. Diagonalisation of self-adjoint operators
Self-adjoint operators have real eigenvalues, and admit an orthonormal basis of eigenvectors.
7. Fourier theory
Finite Fourier series are orthogonal projections onto spaces of trigonometric polynomials. Parseval's inequality. Statement of the L2 convergence theorem (with a proof for continuous 2π periodic functions outlined in a non-examinable appendix).

Reading list

Type Author(s) Title Library Blackwells Amazon
A Friedberg, Insel, Spence Linear Algebra Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.