Shearlet Smoothness Spaces Demetrio Labate 1 , Lucia Mantovani 2 and Pooran Negi 3 November 27, 2012 Abstract The shearlet representation has gained increasingly more prominence in recent years as a flexible mathe- matical framework which enables the efficient analysis of anisotropic phenomena by combining multiscale analysis with the ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure. Key words and phrases: atomic decompositions, Banach frames, Besov spaces, decomposition spaces, shearlets. AMS Mathematics Subject Classification: 22D10, 42B35, 42C15, 46E35, 47B25. 1 Introduction Over the past twenty years, wavelets and multiscale methods have been extremely successful in applications from harmonic analysis, approximation theory, numerical analysis and image processing. However, it is now well established that, despite their remarkable success, wavelets are not very efficient when dealing with multidimensional functions and signals. This limitation is due to their poor directional sensitivity and limited capability in dealing with the anisotropic features which are frequently dominant in multidimensional applications. To overcome this limitation, a variety of methods have been recently introduced to better capture the geometry of multidimensional data, leading to reformulate wavelet theory and applied Fourier analysis within the setting of an emerging theory of sparse representations. It is indicative of this change of perspective that the latest edition of the classical wavelet textbook by S. Mallat was titled “A wavelet tour of signal processing. The sparse way.” Among the new methods emerged in recent years to overcome the limitations of traditional multiscale systems and wavelets, shearlets, originally introduced by one of the authors and his collaborators in [22], offer a unique combination of very useful properties. Similar to the curvelets of Donoho and Cand` es [3], the elements of the shearlet system form a pyramid of well localized waveforms ranging not only across various scales and locations, like wavelets, but also at various orientations and with highly anisotropic shapes. This makes the shearlet approach particularly efficient for capturing the anisotropic and directional features of multidimensional data [24]. Thanks to these properties, shearlets provide optimally sparse representations, in a precise sense, for a large class of images and other multidimensional data where wavelets are suboptimal 1 Department of Mathematics, University of Houston, Houston, Texas 77204, USA. E-mail: dlabate@math.uh.edu 2 Dipartimento di Matematica, Universita’ di Genova, 16146 Genova, ITALY. E-mail: luciamanto@dima.unige.it 3 Department of Mathematics, University of Houston, Houston, Texas 77204, USA. E-mail: psnegi@math.uh.edu 1