Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2013 | 11 | 10 | 1785-1799
Tytuł artykułu

Closure of dilates of shift-invariant subspaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.
Opis fizyczny
  • Department of Mathematics, Faculty of Sciences, Autonomous University of Madrid, Cantoblanco, 28049, Madrid, Spain,
  • [1] Auscher P., Solution of two problems on wavelets, J. Geom. Anal., 1995, 5(2), 181–236[Crossref]
  • [2] Baggett L.W., Medina H.A., Merrill K.D., Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℂn, J. Fourier Anal. Appl., 1999, 5(6), 563–573[Crossref]
  • [3] Battle G., Phase space localization theorem for ondelettes, J. Math. Phys., 1989, 30(10), 2195–2196[Crossref]
  • [4] de Boor C., DeVore R.A., Ron A., On the construction of multivariate (pre)wavelets, Constr. Approx., 1993, 9(2–3), 123–166[Crossref]
  • [5] de Boor C., DeVore R.A., Ron A., The structure of finitely generated shift-invariant spaces in L 2(ℂd), J. Funct. Anal., 1994, 119(1), 37–78[Crossref]
  • [6] Bownik M., The structure of shift-invariant subspaces of L 2(ℂd), J. Funct. Anal., 2000, 177(2), 282–309[Crossref]
  • [7] Bownik M., Intersection of dilates of shift-invariant spaces, Proc. Amer. Math. Soc., 2009, 137(2), 563–572[Crossref]
  • [8] Bownik M., Rzeszotnik Z., The spectral function of shift-invariant spaces, Michigan Math. J., 2003, 51(2), 387–414[Crossref]
  • [9] Bownik M., Rzeszotnik Z., Speegle D., A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal., 2001, 10(1), 71–92[Crossref]
  • [10] Brandolini L., Garrigós G., Rzeszotnik Z., Weiss G., The behaviour at the origin of a class of band-limited wavelets, In: The Functional and Harmonic Analysis of Wavelets and Frames, San Antonio, January 13–14, 1999, Contemp. Math., 247, American Mathematical Society, Providence, 1999, 75–91[Crossref]
  • [11] Calogero A., Wavelets on general lattices, associated with general expanding maps of ℂd, Electron. Res. Announc. Amer. Math. Soc., 1999, 5, 1–10[Crossref]
  • [12] Cifuentes P., Kazarian K.S., San Antolín A., Characterization of scaling functions in a multiresolution analysis, Proc. Amer. Math. Soc., 2005, 133(4), 1013–1023[Crossref]
  • [13] Cifuentes P., Kazarian K.S., San Antolín A., Characterization of scaling functions, In: Wavelets and Splines, Athens, GA, May 16–19, 2005, Mod. Methods Math., Nashboro Press, Brentwood, 2006, 152–163
  • [14] Curry E., Low-pass filters and scaling functions for multivariable wavelets, Canad. J. Math., 2008, 60(2), 334–347[WoS][Crossref]
  • [15] Dai X., Diao Y., Gu Q., Subspaces with normalized tight frame wavelets in ℝ, Proc. Amer. Math. Soc., 2002, 130(6), 1661–1667[Crossref]
  • [16] Dai X., Diao Y., Gu Q., Han D., Frame wavelets in subspaces of L 2(ℝd), Proc. Amer. Math. Soc., 2002, 130(11), 3259–3267[Crossref]
  • [17] Dai X., Diao Y., Gu Q., Han D., The existence of subspace wavelet sets, J. Comput. Appl. Math., 2003, 155(1), 83–90[Crossref]
  • [18] Dai X., Larson D.R., Speegle D.M., Wavelet sets in ℝd. II, In: Wavelets, Multiwavelets, and their Applications, San Diego, January 1997, Contemp. Math., 216, American Mathematical Society, Providence, 1998, 15–40[Crossref]
  • [19] Dai X., Lu S., Wavelets in subspaces, Michigan Math. J., 1996, 43(1), 81–98[Crossref]
  • [20] Daubechies I., Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992[Crossref]
  • [21] Dobric V., Gundy R., Hitczenko P., Characterizations of orthonormal scale functions: a probabilistic approach, J. Geom. Anal., 2000, 10(3), 417–434[Crossref]
  • [22] Dutkay D.E., Some equations relating multiwavelets and multiscaling functions, J. Funct. Anal., 2005, 226(1), 1–20[Crossref]
  • [23] Gu Q., Han D., On multiresolution analysis (MRA) wavelets in ℝd, J. Fourier Anal. Appl., 2000, 6(4), 437–447[Crossref]
  • [24] Gu Q., Han D., Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames, Proc. Amer. Math. Soc., 2005, 133(3), 815–825[Crossref]
  • [25] Ha Y.-H., Kang H., Lee J., Seo J.K., Unimodular wavelets for L 2 and the Hardy space H 2, Michigan Math. J., 1994, 41(2), 345–361[Crossref]
  • [26] Hernández E., Wang X., Weiss G., Characterization of wavelets, scaling functions and wavelets associated with multiresolution analyses, In: Function Spaces, Interpolation Spaces, and Related Topics, Haifa, June 7–13, 1995, Israel Math. Conf. Proc., 13, Bar-Ilan University, Ramat Gan, 1999, 51–87
  • [27] Hernández E., Weiss G., A First Course on Wavelets, Stud. Adv. Math., CRC Press, Boca Raton, 1996 [Crossref]
  • [28] Jia R.Q., Shen Z., Multiresolution and wavelets, Proc. Edinburgh Math. Soc., 1994, 37(2), 271–300[Crossref]
  • [29] Kazarian K.S., San Antolín A., Characterization of scaling functions in a frame multiresolution analysis in H G2, In: Topics in Classical Analysis and Applications in Honor of Daniel Waterman, World Scientific, Hackensack, 2008, 118–140[Crossref]
  • [30] Kim H.O., Kim R.Y., Lim J.K., On the spectrums of frame multiresolution analyses, J. Math. Anal. Appl., 2005, 305(2), 528–545[Crossref]
  • [31] Kim H.O., Lim J.K., Frame multiresolution analysis, Commun. Korean Math. Soc., 2000, 15(2), 285–308
  • [32] Lian Q.-F., Li Y.-Z., Reducing subspace frame multiresolution analysis and frame wavelets, Commun. Pure Appl. Anal., 2007, 6(3), 741–756[Crossref]
  • [33] Lorentz R.A., Madych W.R., Sahakian A., Translation and dilation invariant subspaces of L 2(ℝ) and multiresolution analyses, Appl. Comput. Harmon. Anal., 1998, 5(4), 375–388[Crossref]
  • [34] Madych W.R., Some elementary properties of multiresolution analyses of L 2(ℝn), In: Wavelets, Wavelet Anal. Appl., 2, Academic Press, Boston, 1992, 259–294
  • [35] Mallat S.G., Multiresolution approximations and wavelet orthonormal bases of L 2(ℝ), Trans. Amer. Math. Soc., 1989, 315(1), 69–87
  • [36] Meyer Y., Ondelettes et Opérateurs. I, Actualites Math., Hermann, Paris, 1990
  • [37] Natanson I.P., Theory of Functions of a Real Variable. II, Frederick Ungar, New York, 1961
  • [38] Révész Sz.Gy., San Antolín A., Equivalence of A-approximate continuity for self-adjoint expansive linear maps, Linear Algebra Appl., 2008, 429(7), 1504–1521[WoS][Crossref]
  • [39] Rzeszotnik Z., Calderón’s condition and wavelets, Collect. Math., 2001, 52(2), 181–191
  • [40] San Antolín A., Characterization of low pass filters in a multiresolution analysis, Studia Math., 2009, 190(2), 99–116[Crossref]
  • [41] San Antolín A., On the density order of the principal shift-invariant subspaces of L 2(ℝn), J. Approx. Theory, 2012, 164(8), 1007–1025[WoS][Crossref]
  • [42] Weiss G., Wilson E.N., The mathematical theory of wavelets, In: Twentieth Century Harmonic Analysis-A Celebration, Il Ciocco, July 2–15, 2000, NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer, Dordrecht, 2001, 329–366[Crossref]
  • [43] Wojtaszczyk P., A Mathematical Introduction to Wavelets, London Math. Soc. Stud. Texts, 37, Cambridge University Press, Cambridge, 1997[Crossref]
  • [44] Zhou F.-Y., Li Y.-Z., Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L 2((ℝn), Kyoto J. Math., 2010, 50(1), 83–99[WoS][Crossref]
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.