Description
The approximation of periodic functions by trigonometric polynomials
is closely tied to the name of the French mathematician Joseph Fourier
(1768-1830). His contribution to this topic started in 1807 when he
wrote his first treatise of the heat equation using trigonometric
series and thus establishing a very powerful tool in the general
theory of partial differential equations. Fourier became famous through his
1822 book ''Theorie analyticque de la chaleur''. The practical
applications of Fourier theory are widespreak.
In this topic, we will consider Fourier series and integrals and the
project students will later focus on different and specific topics
related to Fourier theory, such as
- Poincare summation formula, relating sums over the Fourier
coefficients of a function with its values at certain lattice
points. This has also a spectral interpretation via the Laplace
eigenfunctions on flat tori.
- Gibbs Phenomenon, an ''overshoot'' phenomenon of the
trigonometric approximation at discontinuities of a periodic function,
discovered by H. Wilbraham (1848) and J. W. Gibbs (1899). The same
overshoot phenomenon appears also in other families of orthogonal
polynomials.
- Discrete and Fast Fourier Transform (DFT and FFT), which are
important in speech analysis or image processing. The
discovery of fast algorithms to calculate Fourier coefficients
had consequences in numerous scientific areas.
- Related transforms like the Funk transform or the Radon- and
X-ray transform. The latter transformation has even practical medical
applications in X-ray analysis.
Many other exciting topics could be added to this list.
Corequisite
The module Analysis III/IV is recommended.
Resources
Some recommendable Books covering the general theory are
- M. C. Pereyra, L. A. Ward: Harmonic Analysis, From Fourier to
Wavelets , American Mathematical Society, 2012
- H. Dym, H. P. McKean: Fourier Series and Integrals ,
Academic Press, London, 1972
- E. M. Stein, R. Shakarchi: Fourier Analysis, An Introduction , Princeton University Press, Princetonm 2003
After specialisation, the students will read and study original
research articles related to their topics.
email: W
Klingenberg
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