Chapter 9 — Fourier calculus and distributions

Summary

This chapter introduces the Fourier transform, first in the classical sense (as a unitary operator on \(L^2(\mathbb{R})\)). Then, the main concepts from the theory of distributions is introduced, which provides the mathematical framework for working with ‘generalised functions’ such as the Dirac delta ‘function’. Within the setting of distributions, we can significantly extend the concept of derivatives, limits, series and Fourier transforms beyond their classical meaning, and we provide several examples of this. Finally, we revisit the different types of Fourier transforms and introduce a fourth type that combines very naturally with the three types that we have already seen. Then, we find various relations between these different types, in which the use of distributions plays a prominent role.

Not covered in class

Subsections 9.2.11 and Section 9.4 have not been covered at all, whereas section 9.3 was only mentioned very briefly. None of these need to be known.

Important concepts

• Fourier transform and its elementary properties, convolution, Fourier transform as unitary operator on \(L^2(\mathbb{R})\), Parseval and Plancherel relation for Fourier transform
• Fourier transform of Gaussian distribution, characteristic function
• Test function, compact support, distribution, regular versus singular distribution, Dirac-delta distribution (and its derivatives), Heaviside function/distribution, Cauchy principal value, distributional derivative, distributional limit, distributional Fourier series and Fourier transform

Not really covered this year:

• Fourier transforms on different domains: discrete Fourier transform, Fourier series, discrete-time Fourier transform, (continuous-time) Fourier transform.
• Sampling, Nyquist rate, reconstruction via sinc (Whittaker–Shannon interpolation formula).

For applications / exercises

No exercises on Fourier transforms and distributions this year.