Shearlets and pseudo-differential operators

• Autores: Daniel Vera
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 2, 2017, págs. 279-299
• Idioma: inglés
• DOI: 10.1007/s13348-016-0167-1
• Texto completo no disponible (Saber más ...)
• Resumen

  • Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hörmander definition with x-dependent parameter \delta for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO.

• Referencias bibliográficas
  • Antoine, J.P., Murenzi, R., Vandergheynst, P.: Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns. Appl. Comput....
  • Bamberger, R.H., Smith, M.J.T.: A filter bank for directional decomposition of images: theory and design. IEEE Trans. Signal Process. 40,...
  • Bényi, Á., Bownik, M.: Anisotropic classes of homogeneous pseudodifferential symbols. Stud. Math. 200, 41–66 (2010)
  • Bourdaud, G.: L^p estimates for certain nonregular pseudodifferential operators. Commun. Partial Differ. Equ. 7(9), 1023–1033 (1982)
  • Borup, L., Nielsen, M.: Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13(1), 39–70 (2007)
  • Borup, L., Nielsen, M.: On anisotropic Triebel–Lizorkin type spaces, with applications to the study of pseudo-differential operators. J. Funct....
  • Bownik, M.: Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z. 250(3), 539–571 (2005)
  • Bownik, M., Ho, K.-P.: Atomic and molecular decompositions of anisotropic Triebel–Lizorkin spaces. Trans. Am. Math. Soc. 358(4), 1469–1510...
  • Candès, E.J., Donoho, D.L.: Curvelets–aA surprising effective nonadaptive representation for objects with edges. In: Rabut, C., Cohen, A.,...
  • Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with C^2 singularities. Commun. Pure Appl....
  • Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27, 195–214...
  • Do, M.N., Vetterli, M.: The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image Process....
  • Feichtinger, H.G., Gröbner, P.: Banach spaces of distributions defined by decomposition methods I. Math. Nachr. 123, 97–120 (1985)
  • Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)
  • Frazier, M., Jawerth, B., Weiss, G.: Littlewood–Paley theory and the study of function spaces. In: CBMS Regional Conference Series in Mathematics,...
  • Grohs, P., Kutyniok, G.: Parabolic molecules. Found. Comput. Math. 14, 299–337 (2014)
  • Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. In: Chen, G., Lai,...
  • Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 39, 298–318 (2007)
  • Guo, K., Labate, D.: The construction of smooth Parseval frames of shearlets. Math. Model. Nat. Phenom. 8(1), 82–105 (2013)
  • Guo, K., Lim, W.-Q., Labate, D., Weiss, G., Wilson, E.: Wavelets with composite dilation and their MRA properties. Appl. Comput. Harmon. Anal....
  • Kumano-go, H.: Pseudodifferential Operators. MIT Press, Cambridge (1981). (Translated from the Japanese by the author, Rémi Vaillancourt and...
  • Labate, D., Mantovani, L., Negi, P.S.: Shearlet smoothness spaces. J. Fourier Anal. Appl. 19(3), 577–611 (2013)
  • Labate, D., Weiss, G.: Continuous and discrete reproducing systems that arise from translations. Theory and applications of composite wavelets....
  • Le Pennec, E., Mallat, S.: Sparse geometrical image approximation with bandelets. IEEE Trans. Image Process. 14(4), 423–438 (2004)
  • Marschall, J.: Pseudo-differential operators with non regular symbols in the class S^m_{\rho,\delta }. Commun. Partial Differ. Equ. 12, 921–965...
  • Nagase, M.: The L^p-boundedness of pseudo-differential operators with non-regular symbols. Commun. Partial Differ. Equ. 2, 1045–1061 (1977)
  • Päivärinta, L.: Pseudo differential operators in Hardy–Triebel spaces. Z. Anal. Anwendungen 2(3), 235–242 (1983)
  • Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton...
  • Sugimoto, M., Tomita, N.: Boundedness properties of pseudo-differential and Calderón–Zygmund operators on modulation spaces. J. Fourier Anal....
  • Triebel, H.: Theory of Function Spaces, Monographs in Mathematics, vol. 78. Birkhäuser Verlag, Basel (1983)
  • Triebel, H.: Theory of Function Spaces II, Monographs in Mathematics, vol. 84. Birkhäuser Verlag, Basel (1992)
  • Vera, D.: Triebel–Lizorkin spaces and shearlets on the cone in {\mathbb{R}}^2. Appl. Comput. Harmon. Anal. 35, 130–150 (2013). doi:10.1016/j.acha.2012.08.006
  • Vera, D.: Shear anisotropic inhomogeneous Besov spaces in {\mathbb{R}}^d. Int. J. Wavelets Multiresolut. Inf. Process. 12, 1450007 (2014)....