Undergraduate Course: Linear and Fourier Analysis (MATH10081)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||In this course, we will introduce students to techniques and tools in modern analysis which have important uses in a variety of areas of analysis, including the study of partial differential equations.
We will achieve this in the context of linear and fourier analysis, introducing normed linear, inner product spaces and their completions, Banach and Hilbert spaces. The structure and geometry of these spaces will be studied as well as bounded linear operators acting on them.
A rigorous treatment of Fourier series and related topics will be given.
- Inner product spaces and normed spaces.
- Completeness and completions of spaces with concrete realisations of standard examples. Lp spaces, Holder and Minkowski inequalities.
- Geometric and metric properties of Hilbert spaces, including orthonormal bases and generalised Fourier series.
- Bounded linear functionals, operators and duality,
- Test functions and distributions.
- Fourier series, fourier coefficients, trigonometric polynomials and orthogonality.
- Properties of fourier coefficients; Bessel's inequality, Parseval's identity and the Riemann-Lebesgue lemma.
- Various notions of convergence of Fourier series, including pointwise, uniform and mean square convergence. Summability methods, convolution and Young's inequality.
- Fourier Analysis in broader contexts; for example, fourier integrals, fourier expansions in groups, Schwartz spaces and tempered distributions.
Information for Visiting Students
|High Demand Course?
Course Delivery Information
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| 1. Facility with the interplay between analysis, geometry and algebra in the setting of Banach and Hilbert spaces, both abstractly and in specific examples.
2. Ability to use orthogonality arguments in a variety of theoretical and concrete situations.
3. Capacity to work with the classes of normed linear spaces appearing in the course, particularly specific calculations around Hilbert spaces and operators acting on them.
4. Facility with Fourier series and their coefficients.
5. Ability to use the main ideas of Fourier Analysis, in both the proof of structural properties and in concrete situations.
6. Capacity to work with theoretical and concrete concepts related to Fourier series and their coefficients.
7. Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
8. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.
|1. An Introduction of Hilbert Space, by N. Young, Cambridge Mathematical Textbooks.|
2. Introduction to Hilbert Space, by S. Berberian, Oxford University Press.
3. Fourier Analysis: An Introduction, by E.M. Stein and R. Shakarchi, Princeton University Press.
4. Fourier Series and Integrals, by H. Dym and H. McKean, Academic Press.
5. Fourier Analysis, by T.W. Korner, Cambridge University Press
|Graduate Attributes and Skills
|Course organiser||Prof Jim Wright
Tel: (0131 6)50 8570
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045