Abstract
Anisotropic representation systems such as curvelets and shearlets have had a significant impact on applied mathematics in the last decade. The main reason for their success is their superior ability to optimally resolve anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. By now, a large variety of such anisotropic systems have been introduced, for instance, second-generation curvelets, bandlimited shearlets, and compactly supported shearlets, all based on a parabolic dilation operation. These systems share similar approximation properties, which are usually proven on a case-by-case basis for each different construction. The novel concept of parabolic molecules, which was recently introduced by two of the authors, allows for a unified framework encompassing all known anisotropic frame constructions based on parabolic scaling. The main result essentially states that all such systems share similar approximation properties. One main consequence is that at once all the desirable approximation properties of one system within this framework can be deduced virtually for any other system based on parabolic scaling. This paper motivates and surveys recent results in this direction.
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Grohs, P., Keiper, S., Kutyniok, G., Schäfer, M. (2014). Parabolic Molecules: Curvelets, Shearlets, and Beyond. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_9
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DOI: https://doi.org/10.1007/978-3-319-06404-8_9
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