Continuous Mathematics
Lecturer: Dr John Daugman
This course is a prerequisite for Computer Vision;
Information Theory and Coding;
and Neural Computing.
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.