Continuous Mathematics

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

No. of lectures: 4

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.