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What is Applied Harmonic Analysis?

The purpose of this paper is to serve as an introduction into the new field of Applied Harmonic Analysis, which is nowadays already one of the major research area in Applied Mathematics. 1. Data, Data, Data,... Today we are living in a data-drenched world, in which we are challenged to not only provide the methodology to process various different types of data, but – especially as mathematicians – to also analyze the accuracy of such methods and to provide a deeper understanding of the underlying structures. There is a pressing need for those tasks coming from various fields as diverse as air traffic control, digital communications, seismology, medical imaging, and cosmology. As diverse as those fields are the characteristics of the data themselves, where data are usually modeled as functions f : X → Y or just collections of points in X. Here X can, for instance, be Z or R for arbitrarily large n, a compact subset Ω ⊂ R, or a general Riemannian manifold, and Y can be similarly diverse concerning its mathematical structure. Let us take a quick look at some intriguing examples of such modern data. • High-dimensional data can be found in various applications where the most prominent one might be the internet. Here usually the task of search engines is to organize webpages in the widest sense. One common model for such data is to regard a collection of attributes (characterizing words, etc.) for each single webpage as one point in R usually with n > 10.000. • New technology also generates new types of data not encountered before, such as manifold-valued data. Here we would like to mention an example coming from air plane control, where a time series of aircraft orientations (pitch, roll, yaw) can be modeled as ranging over SO(3). • Certainly, also ‘common’ types of data such as 2-D images appear in new fashions, for instance, astronomical images from galaxies, medical images from MRI machines, surveillance images from military aircrafts, and so forth, each having its own specifics in, for instance, how the data are measured and what their main features are. The task which we now face is manifold, starting from how to measure data in the most efficient way, especially where time is a factor such as when collecting MRI data. Since data are never ‘clean’, usually the next task is to denoise the data, which requires a suitable model for the noise. If data are missing, we face the task of inpainting, which certainly calls The author would like to thank the Department of Statistics at Stanford University and the Department of Mathematics at Yale University for their hospitality and support during her visits. She acknowledges support by Deutsche Forschungsgemeinschaft (DFG) Heisenberg-Fellowship KU 1446/8-1.