Illinois Journal of Mathematics

Linear independence of Parseval wavelets

Marcin Bownik and Darrin Speegle

Full-text: Open access


We establish several results yielding linear independence of the affine system generated by $\psi$ in exchange for conditions on the space $V(\psi)$ of negative dilates. A typical assumption yielding linear independence is that the space $V(\psi)$ is shift-invariant. In particular, the affine system generated by a Parseval wavelet is linearly independent. As an illustration of our techniques, we give an alternative proof of the theorem of Linnell (see Proc. Amer. Math. Soc. 127 (1999), 3269–3277) on linear independence of Gabor systems.

Article information

Illinois J. Math., Volume 54, Number 2 (2010), 771-785.

First available in Project Euclid: 14 October 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C40: Wavelets and other special systems


Bownik, Marcin; Speegle, Darrin. Linear independence of Parseval wavelets. Illinois J. Math. 54 (2010), no. 2, 771--785. doi:10.1215/ijm/1318598681.

Export citation


  • B. Behera, An equivalence relation on wavelets in higher dimensions, Bull. London Math. Soc. 36 (2004), 221–230.
  • C. de Boor, R. A. DeVore and A. Ron, The structure of finitely generated shift-invariant spaces in ${L}\sb2({\mathbb{R}}^d)$, J. Funct. Anal. 119 (1994), 37–78.
  • M. Bownik, The structure of shift-invariant subspaces of $L\sp2({\mathbb R}^n)$, J. Funct. Anal. 177 (2000), 282–309.
  • M. Bownik, Baggett's problem for frame wavelets, Representations, wavelets and frames: A celebration of the mathematical work of Lawrence Baggett, Birkhäuser, 2007, pp. 153–173.
  • M. Bownik and Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J. 51 (2003), 387–414.
  • M. Bownik and Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, Math. Ann. 332 (2005), 705–720.
  • M. Bownik and E. Weber, Affine frames, GMRA's, and the canonical dual, Studia Math. 159 (2003), 453–479.
  • P. Casazza, O. Christensen, A. Lindner and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005), 1025–1033.
  • P. Casazza, G. Kutyiniok, D. Speegle and J. Tremain, A decomposition theorem for frames and the Feichtinger Conjecture, Proc. Amer. Math. Soc. 136 (2008), 2043–2053.
  • P.G. Casazza and J.C. Tremain, The Kadison–Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. 103 (2006), 2032–2039.
  • O. Christensen and A. Lindner, Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets, Lin. Alg. Appl. 355 (2002), 147–159.
  • G. Edgar and J. Rosenblatt, Difference equations over locally compact abelian groups, Trans. Amer. Math. Soc. 253 (1979), 273–289.
  • C. Heil, Linear independence of finite Gabor systems, Harmonic Analysis and Applications, Birkhäuser, Boston, 2006, pp. 171–206.
  • C. Heil, J. Ramanathan and P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc. 124 (1996), 2787–2795.
  • P. Linnell, von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc. 127 (1999), 3269–3277.
  • A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of ${L}\sb2({\mathbb R}\sp d)$, Canad. J. Math. 47 (1995), 1051–1094.
  • J. Rosenblatt, Linear independence of translations, J. Austral. Math. Soc. Ser. A 59 (1995), 131–133.
  • J. Rosenblatt, Linear independence of translations, Int. J. Pure Appl. Math. 45 (2008), 463–473.
  • H. Šikić and D. Speegle, Dyadic PFW's and $W\sb o$-bases, Functional analysis IX, Various Publ. Ser. (Aarhus), vol. 48, Univ. Aarhus, Aarhus, 2007, pp. 85–90.
  • P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982.