2019/20 Taught Postgraduate Module Catalogue

MATH5216M Hilbert Spaces and Advanced Fourier Analysis

20 creditsClass Size: 40

Module manager: Dr Vladimir Kisil
Email: V.Kisil@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2019/20

Pre-requisite qualifications

MATH2016, or MATH2017, or equivalent. MATH3211 or MATH5211M are useful, but not required.

This module is mutually exclusive with

 MATH3216 Hilbert Spaces and Fourier Analysis

This module is not approved as an Elective

Module summary

The theory of Hilbert spaces is one of the most striking examples of what mathematical abstraction can achieve in very concrete problems. A Hilbert space is an infinite-dimensional analogue of Euclidean space. It possesses the structure of a vector space together with notions of orthogonality and distance. Expansions of vectors into orthonormal bases are called Fourier-Bessel series.Fourier series are a special case of Fourier-Bessel series. They are at the heart of modern applied and pure mathematics and have revolutionised the way we think about solutions of the wave- and the heat-equation. Fourier series resolve complicated wave forms into frequencies and phases. From those frequencies and phases the original wave form can be recaptured.The properties of Fourier and Fourier-Bessel series derive from the abstract theory of Hilbert spaces in an elegant manner. The development of this abstract theory and its application to Fourier series will constitute the first two thirds of the module.Fourier integrals are continuous versions of Fourier series that also apply to functions that are not necessarily periodic. The theory of the Fourier transform is fundamental in any modern treatment of partial differential equations and real analysis. The mapping properties of the Fourier transform and the relation to Sobolev spaces will be discussed.The last third of the module is devoted to the theory of linear operators, which are the natural generalisations of matrices and mappings on finite-dimensional spaces. We shall examine the role eigenvalues play in this setting and see how they generalise the notion of spectrum.

Objectives

On completion of this module, students should be able to:

- Calculate the Fourier coefficients of certain elementary functions
- Compute the Fourier transform of elementary functions on R
- Perform a range of calculations involving orthogonal expansions in Hilbert spaces and Fourier integrals
- Apply functional analytic techniques to the study of Fourier series and Fourier integrals
- Give the definitions and basic properties of various classes of operators (including the classes of compact, self-adjoint, and unitary operators) on a Hilbert space, and use them in specific examples
- Prove results related to the theorems in the course.

Learning outcomes
- Demonstrate a broad understanding of the concepts, information, practical competencies and techniques of Hilbert space theory and its relation to Fourier series and Fourier integrals.
- Appreciate the coherence, logical structure and broad applicability of Hilbert spaces and the Fourier transform.
- Use Fourier-Bessel series and the Fourier transform to initiate and undertake problem solving.
- Solve advanced mathematical problems using the Fourier transform and the theory of Hilbert spaces.

Syllabus

(1) Real and complex Fourier series. The vibrating string.
(2) Banach spaces.
(3) Hilbert spaces: Subspaces. Linear spans and orthogonal complements. Fourier-Bessel series. Bessel's inequality. The Riesz-Fischer theorem.
(4) Applications to Fourier series. Fejer's theorem. Parseval's formula. Sums of numerical series. The Weierstrass approximation theorem.
(5) Dual space of a normed space. Self-duality of Hilbert spaces.
(6) Linear operators. Adjoint. Self-adjoint, unitary and normal operators. B(H) as a Banach space. The spectrum of an operator on a Hilbert space. The spectral radius formula.
(7) Compact operators. Hilbert-Schmidt operators.
(8) The spectral theorem for compact self-adjoint operators.
(9) Lp-spaces and their duals. Weighted L2-spaces. Schwartz space.
(10) The Fourier transform and its mapping properties
(11) Sobolev Spaces

Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

Private study

Studying and revising of course material. Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular exercise sheets

Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated