# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 20 x 1hour lectures |

Seminars / Tutorials | 7 x 1hour seminars |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 100% |

Supplementary Assessment | 2 Hours (written examination) | 100% |

### Learning Outcomes

On completion of this module students should be able to:

1. explain the importance of numerical approximation;

2. prove the least squares approximation theorem;

3. calculate inner products, and demonstrate orthogonality;

4. demonstrate the properties of complete orthonormal sequences;

5. state and use Parseval's Theorem;

6. define the concept of Fourier series;

7. calculate the Fourier coefficients for some simple functions;

8. describe the relationship between the smoothness and periodicity properties of a function and the rate of decay of its Fourier coefficients;

9. estimate the error incurred inapproximating a function by its Fourier series;

10. construct a trigonometric polynomial which interpolates a given set of data;

11. compute the discrete Fourier coefficients of a given function;

12. differentiate continuous and discrete Fourier expansions;

13. construct the entries of the Fourier collocation derivative matrix;

14. compute the discrete Fourier coefficients of a function using the Fast Fourier Transform (FFT);

15. generate a set of orthogonal polynomials;

16. construct Gauss-type quadrature rules;

17. prove the rate of decay of the coefficients of a function expanded in a series of orthogonal polynomials;

18. prove elementary properties of Chebyshev and Legendre polynomials;

19. generate smoothing splines for experimental data;

20. determine optimal representations for experimental data by linear regression.

### 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.

### Aims

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

### Content

- 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.
- 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.
- APPROXIMATION BY ORTHOGONAL POLYNOMIALS: Generation of orthogonal polynomials. Properties of orthogonal polynomials. Gauss-type quadrature rules. Use of Chebyshev and Legendre polynomials.
- SMOOTHING OF EXPERIMENTAL DATA: Global smoothing methods versus spline smoothing methods.

### Notes

This module is at CQFW Level 7