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2019/20 Undergraduate Module Catalogue
MATH3216 Hilbert Spaces and Fourier Analysis
15 creditsClass Size: 50
Module manager: Dr Arash Ghaani Farashahi
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2019/20
Pre-requisite qualificationsMATH2016, or MATH2017, or equivalent. MATH3211 or MATH5211M are useful, but not required.
This module is mutually exclusive with
|MATH3215||Hilbert Spaces and Fourier Analysis|
|MATH5216M||Hilbert Spaces and Advanced Fourier Analysis|
This module is not approved as a discovery module
Module summaryThe 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.In the last third of the module, we shall develop 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.
ObjectivesOn completion of this module, students should be able to:
- Calculate the Fourier coefficients of certain elementary functions
- Perform a range of calculations involving orthogonal expansions in Hilbert spaces
- Apply functional analytic techniques to the study of Fourier series
- 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.
- Demonstrate a broad understanding of the concepts, information, practical competencies and techniques of Hilbert space theory and its relation to Fourier series.
- Appreciate the coherence, logical structure and broad applicability of Hilbert spaces.
- Use Fourier-Bessel series to initiate and undertake problem solving.
(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.
|Delivery type||Number||Length hours||Student hours|
|Private study hours||117.00|
|Total Contact hours||33.00|
|Total hours (100hr per 10 credits)||150.00|
Private studyStudying and revising course material. Completing of assignments and assessments.
Opportunities for Formative FeedbackRegular exercise sheets
Methods of assessment
|Exam type||Exam duration||% of formal assessment|
|Standard exam (closed essays, MCQs etc)||2 hr 30 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
Reading listThere is no reading list for this module
Last updated: 30/01/2020
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