Some Properties of Range Operators on LCA Groups
Download PDF
Authors: R. VERMA AND K. TEENA
DOI: 10.46793/KgJMat2307.0995V
Abstract:
In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L2(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L2(ℝd) to L2(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V , in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L2(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R−1(ξ) of range operator on L2(G).
Keywords:
Shift-invariant space, range function, range operator, locally compact Abelian group, shift preserving operator, frame, Parseval frame, normal operator, unitary operator.
References:
[1] A. Aguilera, C. Cabrelli, D. Carbajal and V. Paternostro, Diagonalization of shift-preserving operators, Adv. Math. 389 (2021), Paper ID 107892.
[2] A. Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. Comput. Harmon. Anal. 13 (2002), 151–161.
[3] C. de Boor, R. A. DeVore and A. Ron, The structure of finitely generated shift-invariant spaces in L2(ℝd), J. Funct. Anal. 119(1) (1994), 37–78.
[4] M. Bownik and K. Ross, The structure of translation-invariant spaces on locally compact Abelian groups, J. Fourier Anal. Appl. 21(4) (2015), 849–884.
[5] M. Bownik and Z. Rzeszotnik, The spectral function of shift-invariant spaces on general lattices, wavelets, frames and operator theory, Contemporary Mathematics vol. 345 (2004), 49–59.
[6] D. Bakić, I. Krishtal and E. N. Wilson, Parseval frame wavelets with En(2)-dilations, Appl. Comput. Harmon. Anal. 19 (2005), 386–431.
[7] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389–427.
[8] M. Bownik, The structure of shift-modulation invariant spaces: The rational case, J. Funct. Anal. 244 (2007), 172–219.
[9] M. Bownik, The structure of shift-invariant subspaces of L2(ℝn), J. Funct. Anal. 177(2) (2000), 282–309.
[10] M. Bownik, On characterization of multiwavelets in L2(ℝn), Proc. Amer. Math. Soc. 129 (2001), 3265–3274.
[11] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser Boston Inc., Boston, MA, 2003.
[12] C. Cabrelli and V. Paternostro, Shift-invariant spaces on LCA groups, J. Funct. Anal. 258(6) (2010), 2034–2059.
[13] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transform, Springer Lecture Notes in Mathematics No. 1863, Springer-Verlag, Berlin, 2005.
[14] G. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
[15] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Springer, New York, 1970.
[16] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups, Integration Theory, Group Representations, 2nd Edition, Springer, Berlin, 1979.
[17] H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, London, 1964.
[18] R. A. Kamyabi Gol and R. Raisi Tousi, The structure of shift-invariant spaces on a locally compact Abelian group, J. Math. Anal. Appl. 340 (2008), 219–225.
[19] R. A. Kamyabi Gol and R. Raisi Tousi, A range function approach to shift-invariant spaces on locally compact Abelian groups, Int. J. Wavelets. Multiresolut. Inf. Process. 8 (2010), 49–59.
[20] R. Kamyabi-Gol and R. Raisi Tousi, Shift preserving operators on locally compact Abelian groups, Taiwanese J. Math. 15 (2011). https://10.11650/twjm/1500406415.
[21] R. Raisi Tousi and R. Kamyabi-Gol, shift-invariant spaces and shift preserving operators on locally compact Abelian groups, Iran. J. Math. Sci. Inform. 6 (2011).
[22] R. A. Kamyabi Gol and R. Raisi Tousi, Some equivalent multiresolution conditions on locally compact Abelian groups, Proc. Indian Acad. Sci. (Math. Sci.) 120 (2010), 317–331.
[23] G. Kutyniok, Time frequency analysis on locally compact groups, Ph.D. thesis, Padeborn University, 2000.
[24] A. Ron and Z. Shen, Affine systems in L2(ℝd), the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408–447.
[25] A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(ℝd), Canad. J. Math. 47 (1995), 1051–1094.
[26] Z. Rzeszotnik, Characterization theorems in the theory of wavelets, Ph.D. thesis, Washington University, 2000.
[27] G. Weiss and E.N. Wilson, The Mathematical Theory of Wavelets, Academic Publishers, 2001, 329–366.
Login:
