Module Identifier MAM4520  
Academic Year 2003/2004  
Co-ordinator Professor Russell Davies  
Semester Semester 2  
Other staff Dr Robert J Douglas  
Pre-Requisite MA25110 , MA30210  
Course delivery Lecture   20 x 1hour lectures  
  Seminars / Tutorials   7 x 1hour seminars  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module students should be able to:

Brief description

The expansion of a function in terms of an infinite sequence of orthogonal functions underlies many numerical methods of approximation. The accuracy of the approximations and the efficiency of their implementation are important factors when determining the applicability of these methods in scientific computations. In this module, we study representations of functions both in terms of orthogonal and non-orthogonal bases, and examine what modifications are needed in representing experimental data.


  1. To introduce students to key issues in the numerical approximation of a given function;
  2. To demonstrate the power of certain orthogonal functions to produce rapidly convergent approximations to smooth functions;
  3. To introduce students to the smoothing of experimental data.


  1. APPROXIMATION IN A HILBERT SPACE: Introduction. Definitions. Best L^{2} approximation theorem. Complete orthonormal sequences in Hilbert spaces. Parseval's Theorem. Reproducing Kernel Hilbert Spaces.
  2. APPROXIMATION BY TRIGONOMETRIC POLYNOMIALS: Continuous Fourier expansions. Rate of decay of Fourier coefficients. Error estimates. Discrete Fourier series. Trigonometric interpolating polynomials. Differentiation of Fourier series. Fast Fourier transform.
  3. APPROXIMATION BY ORTHOGONAL POLYNOMIALS: Generation of orthogonal polynomials. Properties of orthogonal polynomials. Gauss-type quadrature rules. Use of Chebyshev and Legendre polynomials.
  4. SMOOTHING OF EXPERIMENTAL DATA: Global smoothing methods versus spline smoothing methods.

Reading Lists

** Recommended Text
N Young (1988) An Introduction to Hilbert space Cambridge University Press 0521337178
E Kreyszig (1978) Introductory Functional Analysis with Applications Wiley 047103729X
P Lancaster & K Salkauskas (1986) Curve and Surface Fitting: an introduction Academic Press 0124360610
E V Shikin & A I Plis (1995) Handbook on Splines for the User CRC Press 084939404X


This module is at CQFW Level 7