Harmonic Analysis at its Boundaries

In June 2016 we, Christine Grathwohl, Fabian Hornung and Luca Hornung, had the opportunity to participate in the summer school “Harmonic Analysis at its Boundaries” in Nantes. The aim of this event was to present recent advances in abstract and applied harmonic analysis in depth through three minicourses and to give an overview about connected topics through accompanying talks.

The origins of the harmonic analysis go back to the end of the 18th century. One of the founders of this theory was the French mathematician Joseph Fourier, who developed it during his attempts to solve differential equations. Nowadays applications are found in various fields of science and technology, reaching from acoustics and optics over signal processing till cryptography.

Consider for example a music song that you want to record as an MP3 file in such a way that you need as low memory capacity as possible. All the information on frequencies that are not in the frequency range that can be heard by us humans is not needed. It is impossible to delete this information directly from the signal caused by the air pressure in the microphone. Instead of this, one computes the Fourier transform of a signal that divides the information on the different frequencies. Now the very high and very low frequencies can be cut off easily and a retransform gives the desired result.

In the following, we want to give an insight into the topics of the minicourses mentioned above. A little cautionary reminder: this short report requires knowledge of some advanced mathematics.

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