Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-09T22:55:46.418Z Has data issue: false hasContentIssue false
  • English
  • Français

On Parseval Wavelet Frames via Multiresolution Analyses in $H_{G}^{2}$

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  06 December 2019

A. San Antolín
A. San Antolín*
Affiliation:
Departamento de Matemáticas, Universidad de Alicante, 03080 Alicante, Spain Email: angel.sanantolin@ua.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

Keywords

dilation matrixextension principleFourier transformrefinable functiontight wavelet frame

MSC classification

Primary: 42C40: Wavelets and other special systems
Secondary: 42C15: General harmonic expansions, frames

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 157 - 172
Copyright
© Canadian Mathematical Society 2019

References

Atreas, N., Papadakis, M., and Stavropoulos, T., Extension principles for dual multiwavelet frames of L 2(ℝs) constructed from multirefinable generators. J. Fourier Anal. Appl. 22(2016), 854–877.CrossRefGoogle Scholar
Atreas, N., Melas, A., and Stavropoulos, T., Affine dual frames and extension principles. Appl. Comput. Harmon. Anal. 36(2014), 51–62.CrossRefGoogle Scholar
Baggett, L. W., Medina, H. A., and Merril, K. D., Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝn. J. Fourier Anal. Appl. 5(1999), 563–573.CrossRefGoogle Scholar
Bakić, D., Semi-orthogonal parseval frame wavelets and generalized multiresolution analyses. Appl. Comput. Harmon. Anal. 21(2006), 281–304.CrossRefGoogle Scholar
Benedetto, J. J. and Li, S., The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5(1998), 389–427.CrossRefGoogle Scholar
Benedetto, J. J. and Treiber, O. M., Wavelet frames: multiresolution analysis and extension principles. In: Wavelet Transforms and Time Frequency Signal Analysis. ed. Debnath, L.Birkhäuser, 2001, pp. 3–36.CrossRefGoogle Scholar
Bownik, M., Riesz wavelets and generalized multiresolution analyses. Appl. Comput. Harmon. Anal. 14(2003), 181–194.CrossRefGoogle Scholar
Bownik, M. and Garrigós, G., Biorthogonal wavelets, MRA’s and shift-invariant spaces. Studia Math. 160(2004), 231–248.CrossRefGoogle Scholar
Bownik, M. and Rzeszotnik, Z., Construction and reconstruction of tight framelets and wavelets via matrix mask functions. J. Funct. Anal. 256(2009), 1065–1105.CrossRefGoogle Scholar
de Boor, C., DeVore, R. A., and Ron, A., On the construction of multivariate (pre)wavelets. Constructive Approximation 9(1993), 123–166.CrossRefGoogle Scholar
Bruckner, A., Differentiation of real functions. Lecture Notes in Math. 659, Springer, Berlin, 1978.CrossRefGoogle Scholar
Calogero, A. and Garrigós, G., A characterization of wavelet families arising from biorthogonal MRA’s of multiplicity d. J. Geom. Anal. 11(2001), 187–217.CrossRefGoogle Scholar
Christensen, O., An introduction to frames and Riesz bases. Birkhäuser, Boston, 2003.CrossRefGoogle Scholar
Chui, C. K., He, W., and Stöckler, J., Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmonic Anal. 13(2002), 224–262.CrossRefGoogle Scholar
Chui, C. K., He, W., Stöckler, J., and Sun, Q., Compactly supported tight affine frames with integer dilations and maximum vanishing moments. Adv. Comput. Math. 18(2003), 159–187.CrossRefGoogle Scholar
Dai, X., Diao, Y., Gu, Q., and Han, D., Frame wavelets in subspaces of L 2(ℝd). Proc. Amer. Math. Soc. 130(2002), 3259–3267.CrossRefGoogle Scholar
Daubechies, I., Han, B., Ron, A., and Shen, Z. W., Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14(2003), 1–46.CrossRefGoogle Scholar
Duffin, R. and Schaeffer, A., A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72(1952), 341–366.CrossRefGoogle Scholar
Gripenberg, G., A necessary and sufficient condition for the existence of a father wavelet. Studia Math. 114(1995), 207–226.CrossRefGoogle Scholar
Gröchening, K. and Madych, W. R., Multiresolution analysis, Haar bases and self-similar tillings of R n. IEEE Trans. Inform. Theory 38(1992), 556–568.CrossRefGoogle Scholar
Han, B., Nonhomogeneous wavelet systems in high dimensions. Appl. Comput. Harmon. Anal. 32(2012), 169–196.CrossRefGoogle Scholar
Han, B., Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space. Appl. Comput. Harmon. Anal. 29(2010), 330–353.CrossRefGoogle Scholar
Han, B., On dual wavelet tight frames. Appl. Comput. Harmon. Anal. 4(1997), 380–413.CrossRefGoogle Scholar
Hernández, E. and Weiss, G., A first course on Wavelets. CRC Press Inc., 1996.CrossRefGoogle Scholar
Jia, H.-F. and Li, Y.-Z., Refinable function-based construction of weak (quasi-)affine bi-frames. J. Fourier Anal. Appl. 20(2014), 1145–1170.CrossRefGoogle Scholar
Jia, R. Q. and Shen, Z., Multiresolution and Wavelets. Proc. Edinburgh Math. Soc. (2) 37(1994), 271–300.CrossRefGoogle Scholar
Kazarian, K. S. and 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, NJ, 2008, pp. 118–140.CrossRefGoogle Scholar
Kim, H. O., Kim, R. Y., Lee, Y. H., and Lim, J. K., On Riesz wavelets associated with multiresolution analyses. Appl. Comput. Harmon. Anal. 13(2002), 138–150.CrossRefGoogle Scholar
Kim, H. O., Kim, R. Y., and Lim, J. K., Characterizations of biorthogonal wavelets which are associated with biorthogonal multiresolution analyses. Appl. Comput. Harmon. Anal. 11(2001), 263–272.CrossRefGoogle Scholar
Krivoshein, A., Protasov, V., and Skopina, M., Multivariate wavelet frames. Industrial and Applied Mathematics, Springer, Singapore, 2016.Google Scholar
Li, Y.-Z. and Lian, Q.-F., Reducing subspace frame multiresolution analysis and frame wavelets. Commun. Pure Appl. Anal. 6(2007), 741–756.Google Scholar
Li, Y.-Z. and Zhang, J. P., Extension principles for affine dual frames in reducing subspaces. Appl. Comput. Harmon. Anal. 46(2019), 177–191.CrossRefGoogle Scholar
Li, Y.-Z. and Zhang, L., An embedding theorem on reducing subspace frame multiresolution analysis. Kodai Math. J. 35(2012), 157–172.CrossRefGoogle Scholar
Li, Y.-Z. and Zhou, F.-Y., Generalized multiresolution structures in reducing subspaces of L 2(ℝd). Sci. China Math. 56(2013), 619–638.Google Scholar
Li, Y.-Z. and Zhou, F.-Y., GMRA-based construction of framelets in reducing subspaces of L 2(ℝd). InternatJ̇. Wavelets Multiresolut. Inf. Process 9(2011), 237–268.CrossRefGoogle Scholar
Mallat, S., Multiresolution approximations and wavelet orthonormal bases for L 2(R). Trans. Amer. Math. Soc. 315(1989), 69–87.Google Scholar
Meyer, Y., Ondelettes et opérateurs. I. Hermann, Paris, 1996 [English Translation: Wavelets and operators, Cambridge University Press, 1992].Google Scholar
Nathanson, I. P., Theory of functions of a real variable. London, 1960.Google Scholar
Papadakis, M., Frames of translates in abstract Hilbert spaces and the generalized frame multiresolution analysis. In: Trends in approximation theory (Nashville, TN, 2000). Innov. Appl. Math., Vanderbilt University Press, Nashville, TN, 2001, pp. 353–362.Google Scholar
Papadakis, M., On the dimension function of orthonormal wavelets. Proc. Amer. Math. Soc. 128(2000), 20432–049.CrossRefGoogle Scholar
Petukhov, A., Explicit construction of framelets. Appl. Comput. Harmon. Anal. 11(2001), 313–327.CrossRefGoogle Scholar
Ron, A. and Shen, Z. W., Affine systems in L 2(ℝd). II. Dual systems. Dedicated to the memory of Richard J. Duffin. J. Fourier Anal. Appl. 3(1997), 617–637.CrossRefGoogle Scholar
Ron, A. and Shen, Z. W., Affine systems in L 2(ℝd): The analysis of the analysis operator. J. Funct. Anal. 148(1997), 408–447.CrossRefGoogle Scholar
San Antolín, A., On translation invariant multiresolution analysis. Glas. Mat. Ser. III 49(2014), 377–394.CrossRefGoogle Scholar
San Antolín, A., Characterization of low pass filters in a multiresolution analysis. Studia Math. 190(2009), 99–116.CrossRefGoogle Scholar
Soto-Bajo, M., Closure of dilates of shift-invariant subspaces. Cent. Eur. J. Math. 11(2013), 1785–1799.Google Scholar
Stavropoulos, T., The geometry of extension principles. Houston J. Math. 38(2012), 833–853.Google Scholar
Zalik, R. A., Riesz bases and multiresolution analyses. Appl. Comput. Harmon. Anal. 7(1999), 315–331.CrossRefGoogle Scholar