# Continuous Mathematics

*Lecturer: Dr J.G. Daugman*
(`jgd1000@cl.cam.ac.uk`)

*This course is a prerequisite for Computer Vision *(*Part II
and Diploma*)*, Information Theory and Coding *(*Part II*)*
and Neural Computing *(*Part II*)*.*

#### Aims

The aims of this course are to review some key concepts and operations
defined in continuous mathematics involving real- and complex-valued
functions of real variables. Focus is on the use and implementation
of these notions in the discrete spaces we enter when computing.
Topics include: expansions and basis functions; orthogonality and
projections; differential equations and their computational solution;
linear operators and their eigenfunctions; wavelets and Fourier
analysis.
#### Lectures

**Review of analysis.**
Real and complex-valued functions of a real variable.
Power series and transcendental functions. Expansions
and basis functions. Smoothness, continuity, limits.

**Linear vector spaces and decompositions.**
Orthogonality, independence, and orthonormality.
Linear combinations. Projections, inner products and completeness.
Linear subspaces. Useful expansion bases for continuous functions.

**Differential and integral operators in computation.**
The infinitesimal calculus. Taylor series. Numerical integration.
Differential equations and computational ways to solve them.
Complex exponentials. Introduction to Fourier analysis in one
and two dimensions; useful theorems. Convolution and filtering.

**Signals and systems.**
Eigenfunctions of linear operators. Fourier analysis and series;
continuous Fourier Transforms and their inverses. Representation in
non-orthogonal functions, and wavelets. The degrees-of-freedom in
a signal. Sampling theorem. How to operate on continuous signals
computationally in order to extract their information.

#### Objectives

At the end of the course students should

- understand how data or functions can be represented in terms of
their projections onto other groups of functions

- be fluent in the use of, and properties of, complex variables

- be able to implement and use, in discrete computational form, such
continuous notions as differentiation, integration, and convolution

- grasp key properties and uses of Fourier analysis, transforms,
and wavelets

#### Reference books

Kaplan, W. (1992). *Advanced Calculus*. Addison-Wesley (4th ed.).

Oppenheim, A.V. & Willsky, A.S. (1984). *Signals and Systems*.
Prentice-Hall.

**Lecturer:** Dr John Daugman (jgd1000@cl.cam.ac.uk)

**Taken by:** Part IB, Part II (General), Diploma

**Number of lectures:** 4

**Lecture location:** Heycock Room

**Lecture times:** 11:00 on MWF starting 24-Nov-99

IB | II(G) | Dip