Abstract. Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.In this paper, we focus on cone-adapted shearlet systems which -accounting for stability questions -are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images.

Abstract. Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far.In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L 2 (R 2 ), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such a oversampled shearlet systems and coshearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.

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In this paper, the invariant subspace method is used to solve the time-and pricing-fractional Black-Scholes equations, in which the fractional derivatives are described in the Caputo and Weyl sense, respectively. We introduce invariant subspaces for time and pricing differential operators. Applying an appropriate invariant subspace will reduce the fractional Black-Scholes equation to a system of ordinary fractional differential equations. Finally, using point symmetries of the obtained system, we construct the explicit solutions of the fractional Black-Scholes equations.

In this paper, the invariant subspace method is used to solve the time-and pricing-fractional Black-Scholes equations, in which the fractional derivatives are described in the Caputo and Weyl sense, respectively. We introduce invariant subspaces for time and pricing differential operators. Applying an appropriate invariant subspace will reduce the fractional Black-Scholes equation to a system of ordinary fractional differential equations. Finally, using point symmetries of the obtained system, we construct the explicit solutions of the fractional Black-Scholes equations.