| Credits: 20 | Convenor: Dr J. Levesley | Semester: 1 |
| Prerequisites: | essential: MC144, MC145, MC146, MC147, MC240, MC241 | desirable: MC243 |
| Assessment: | Continual assessment (30% Notebook, 10% worksheets): 40% | Three hour exam: 60% |
| Lectures: | 36 | Classes: | 10 |
| Tutorials: | none | Private Study: | 104 |
| Labs: | none | Seminars: | none |
| Project: | none | Other: | none |
| Total: | 150 |
and L2 norms.
We will start by looking at interpolation by polynomials and piecewise
polynomials, discussing algorithms and error estimates. Following a
section on normed linear spaces, we will consider periodic
approximation using classical Fourier series. Here we will meet best
approximation in the L2 sense. We will generalise these ideas to
more general orthogonal polynomials on an interval. We will also
consider uniform approximation using generalised Fourier series. In
the final section we will prove the celebrated Weierstrass theorem,
which says that every continuous function can be uniformly
approximated by polynomials. This proof will be done via Korovkin's
theorem and the use of Bernstein polynomials. We will end by
considering some fascinating results characterising best
approximations in the sense.
To understand and be able to use the main results and proofs of this course.
To improve mathematical writing skills.
To improve skills in solving problems in approximation theory.
Normed linear spaces: basic properties, compactness and convexity, inner product spaces, Hilbert spaces, and the Gram-Schmidt process.
Periodic approximation using Fourier series: real and complex Fourier series, square integrable functions and the convergence of Fourier series, the Dirichlet kernel and divergent Fourier series, the Fejer kernel and the convergence of Cesaro means, Jackson and Bernstein theorems.
Approximation on an interval using orthogonal polynomials: Legendre polynomials and other examples. Three term recurrence relation, orthogonal projection, generalised Fourier series, best approximation, zeros of orthogonal polynomials, Gaussian quadrature, interpolation using a basis of orthogonal polynomials, Christoffel Darboux formula and pointwise convergence of Fourier series, decay rates of Fourier coefficients and uniform convergence of Fourier series.
Approximation of continuous functions: Korovkin's theorem, Bernstein polynomials and Weierstrass' theorem, polynomial approximation alternation theorems. Approximation using a general family of functions and the Haar condition.
E. W. Cheney, An Introduction to Approximation Theory, McGraw-Hill, 1966.
M.J.D. Powell, Approximation Theory and Methods, Cambridge 1981.
P.J. Davis, Interpolation and Approximation, Dover Publications, 1975.
The coursework for the continual assessment consists of about 5 pieces of written work, consisting of problems from the problem sheets. Students will have the opportunity to resubmit the work in order to improve their mark. The average of the two marks will be taken as the final mark.
Each student must keep a notebook, in which a perfect version of the lectures is maintained. The lectures themselves will be somewhat terser than normal, and students will be expected to provide a full account of the details of sketched arguments in their notebooks. Initially, lectures will leave very few details unattended, but there will be a progression towards omission of easy arguments which the student can be reasonably expected to fill in.