The course introduces students to some of the techniques of modern analysis which are indispensable tools to the present-day mathematician. The expansion of functions in Fourier series (if the function is defined on a bounded interval or periodic) or Fourier integrals is a very efficient method for solving a variety of problems in pure and applied mathematics – compared to power series expansion, it works under very weak assumptions on the regularity of the function. Indeed, even discontinuous functions can reasonably be expanded in a Fourier series, an observation which led to the modern definition of the concept of a function and to the development of mathematical analysis during the 19^th and 20^th centuries. The desire to give a satisfactory answer to the question which functions have a Fourier expansion, and in what sense, led to the abstract notions of normed vector spaces and Hilbert spaces, which have become the foundation of modern analysis and are used in all areas of mathematics. The fundamental idea is to try and extend the framework of linear algebra (matrix theory) to the study of more complicated linear operators, such as differential operators. This requires an infinite-dimensional setting, and ideas of analysis such as convergence and continuity become important. The aim of the course is to study Fourier series and integrals, with emphasis on conditions ensuring their pointwise, uniform or mean convergence, and to give an introduction to the more general theory of functional analysis, illustrated with some further applications.

Prerequisite Modules: MA0212 Linear Algebra, MA0221 Analysis III

• Use correctly basic concepts of functional analysis, including vector spaces, norms, metrics, open and closed sets, inner products, orthogonal complements, linear operators and functionals, bounded and compact operators, symmetric operators, eigenvalues and eigenvectors; Fourier coefficients, Fourier series and integrals in the exponential and trigonometric forms, generalised Fourier transform, mean convergence, convolution
• Explain the logical foundations and mathematical ideas in the development of basic functional analysis, including knowledge of the precise formulations of the main theorems and of their proofs
• Apply the theoretical results  to advanced mathematical problems such as fixed-point equations, multiplication operators, ordinary and partial differential equations

54 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 100 hours private study including preparation of solutions for given exercises.

• Ability to understand and assimilate extended and abstract logical and mathematical arguments
• Ability to apply techniques of proofs from a larger toolbox, appreciating the fact that generally there is no single approach covering all problems
• Appreciation of the necessity to prove existence and/or uniqueness of the solution of advanced mathematical problems, and ability to provide such proof
• Ability to apply theoretical results to concrete cases
• Use of mathematical language in a logically and rhetorically correct way
• Tackling unseen mathematical challenges

Formative assessment is carried out by means of regular homework exercises. Feedback to students on their progress is provided by marked homework and during examples lectures.

The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from six equally weighted questions.

• Fourier series
  • pointwise, uniform convergence, application to differential equations, convolution
  • normed vector spaces
  • norm equivalence
  • convergence and completeness
  • norm topology
  • Banach's fixed point theorem
  • inner product spaces, orthonormal sets and bases, Bessel's inequality, properties of L² spaces

• Linear operators and functionals
  • the projection theorem and the Riesz representation theorem
  • bounded and compact operators
  • adjoint operator
  • symmetric operators
  • spectral theorem for compact symmetric operators, generalised Fourier series

• Fourier transform
  • absolutely integrable functions
  • rapidly decreasing functions
  • square-integrable functions
  • inverse Fourier transform

Elements of the theory of functions and Functional Analysis I - Metric and Normed Spaces, Kolmogorov, A. M., & Fomin, S. V., Graylock Press

Introduction to Hilbert Space, Berberian, S. K., OUP

Linear Operators in Hilbert Spaces, Weidmann, J., Springer

Fourier analysis, Safarov, Yu., Lecture notes, King’s College London, http://www.mth.kcl.ac.uk

Discourse on Fourier Series, Lanczos, C., Oliver & Boyd

Fourier Analysis, Hsu, H. P., Simon and Schuster

Fourier Series and Integrals, Carslaw H S, MacMillan

An Introduction to Fourier Analysis, Stuart R D, Methuen/Wiley

Fourier Series, Sneddon, I. N., Dover

Fourier Series, Kufner, A., & Kadlec, J., Iliffe Books