General 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 make a detailed study of expansions in terms of orthogonal polynomials which yield 'spectrally accurate' approximations (ie those for which the coefficient decays faster than any inverse power of k for certain well-behaved functions).

Aims
The aim of this module is to introduce students to key issues in the numerical approximation of a given function and to demonstrate the power of certain orthogonal functions to produce rapidly convergent approximations to smooth functions.

Learning outcomes
On completion of this module students should be able to:

• explain the importance of numerical approximation;
• prove the best L-squared approximation theorem;
• define the concept of Fourier series;
• calculate the Fourier coefficients for some simple functions;
• describe the relationship between the smoothness and periodicity properties of a function and the rate of decay of its Fourier coefficients;
• estimate the error incurred inapproximating a function by its Fourier series;
• construct a trigonometric polynomial which interpolates a given set of data;
• compute the discrete Fourier coefficients of a given function;
• differentiate continuous and discrete Fourier expansions;
• construct the entries of the Fourier collocation derivative matrix;
• compute the discrete Fourier coefficients of a function using the Fast Fourier Transform (FFT);
• generate a set of orthogonal polynomials;
• prove elementary properties of orthogonal polynomials;
• construct Gauss-type quadrature rules;
• prove the rate of decay of the coefficients of a function expanded in a series of orthogonal polynomials;
• prove elementary properties of Chebyshev and Legendre polynomials;
• differentiate expansions of Chebyshev and Legendre polynomials.

Syllabus
1. APPROXIMATION IN A HILBERT SPACE: Introduction. Definitions. Best L^{2} approximation theorem.
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. Classical polynomials. Rate of convergence of orthogonal series expansions. Chebyshev and Legendre polynomials. Interpolation by Chebyshev polynomials. Differentiation of Chebyshev and Legendre expansions.