Continuous Mathematics

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

No. of lectures: 4  

Review of analysis.

Real and complex-valued functions of a real variable. Smoothness, continuity, limits. Taylor series. The differential and integral calculus for scalar and vector functions. Methods of numerical integration; partial differential equations.

Linear operators and their eigenfunctions.

Orthogonality, inner products, expansions, and basis functions. Complex exponentials. Fourier analysis and series; continuous Fourier Transforms and their inverses. Duals. Projections onto non-orthogonal functions.

The degrees-of-freedom in a signal.

Sampling, interpolation, and signal approximation at a certain scale. Spectra, statistics, and information. Functional expansion spaces and bases. How to operate on continuous signals computationally in order to extract their structure.

Reference books:

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

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