Nagoya Mathematical Journal

Annihilator, completeness and convergence of wavelet system

Kwok-Pun Ho

Full-text: Open access


We show that if $\{ \varphi \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is a frame and $\{ \psi_{Q} \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is its dual frame (for the definition of $\mathcal{M}_{\alpha}(\mathbb{R}^{n})$, see Definition 2.1), where $\mathcal{Q}$ is the collection of dyadic cubes, then for any $f \in \mathcal{S}'(\mathbb{R}^{n})$, there exists a sequence of polynomials, $P_{L, L', L''}$, such that

\lim_{L, L', L'' \to \infty} \biggl\{ \sum_{-L' \le i \le L} \sum_{|k| \le \delta(i)2^{L''}} \langle f, \psi_{Q_{i, k}} \rangle \varphi_{Q_{i, k}}-P_{L, L', L''} \biggr\} = f

in the topology of $\mathcal{S}'(\mathbb{R}^{n})$, where $\delta(i) = \max(2^{i}, 1)$. We prove this result by explicitly constructing the polynomials $P_{L, L', L''}$. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

Article information

Nagoya Math. J., Volume 188 (2007), 59-105.

First available in Project Euclid: 17 December 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis 42C15: General harmonic expansions, frames 42C40: Wavelets and other special systems
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47B38: Operators on function spaces (general)

Frames Wavelets Annihilator and Schwartz distribution


Ho, Kwok-Pun. Annihilator, completeness and convergence of wavelet system. Nagoya Math. J. 188 (2007), 59--105.

Export citation


  • R. Coifman and Y. Meyer, Wavelets: Calderón-Zygmund and Multilinear Operators, Cambridge studies in adv. math., #48, Cambridge Univ. Press, 1997.
  • J. Conway, A Course in Functional Analysis, Graduate texts in mathematics, #96, Springer-Verlag, 1990.
  • I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF regional conference series in applied mathematics #61, Society for Industrial and Applied Mathematics, 1992.
  • G. Fix and G. Strang, A Fourier analysis of the finite element variational method, Construct. Aspects of Funct. Anal. (1971), 796–830.
  • M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math., 34 (1985), 777–799.
  • M. Frazier and B. Jawerth, A Discrete Transform and Decomposition of Distribution Spaces, J. Funct. Anal., 93 (1990), 34–170.
  • M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Ser., #79, American Math. Society, 1991.
  • E. Hernández and G. Weiss, A first Course on Wavelets, CRC Press, 1996.
  • K.-P. Ho, Frame associated with Expansive Matrix Dilation, Collect. Math., 54 (2003), 217–254.
  • K.-P. Ho, Remarks on Littlewood-Paley analysis, Canad. J. Math., to appear.
  • S. Jaffard and Y. Meyer, Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions, Mem. Amer. Math. Soc., #123, 1996.
  • S. Kelly, M. Kon and L. Raphael, Pointwise convergence of wavelet expansions, Bull. Amer. Math. Soc. (N.S.), 30 (1994), no. 1, 87–94.
  • S. Kelly, M. Kon and L. Raphael, Local convergence for wavelet expansions, J. Funct. Anal., 126 (1994), no. 1, 102–138.
  • S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999.
  • Y. Meyer, Wavelets and Operators, Cambridge studies in adv. math., #37, Cambridge Univ. Press, 1992.
  • Y. Meyer, Wavelets, Vibrations and Scalings, CRM Monograph Series, #9, AMS, 1998.
  • Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, #22, AMS, 2001.
  • J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series #1, Mathematics Depatrment, Duke University, 1976.
  • I. Singer, Bases in Banach spaces I, Springer-Verlag, 1970.
  • F. Treves, Topological Vector Spaces, Distributions and Kernels, Pure and Applied Maths., #25, Academic Press, 1967.
  • D. Walnut, An Introduction to Wavelet Analysis, Birkhauser, 2002.
  • G. Walter, Pointwise convergence of wavelet expansions, J. Approx. Theory, 80 (1995), no. 1, 108–118.
  • G. Walter, Wavelets and generalized functions. Wavelets: A Tutorial in Theory and Application, Wavelet Anal. Appl., #2, Academic Press, 1992, pp. 51–70.
  • P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, 1997.
  • R. Young, An introduction to nonharmonic Fourier series, Academic Press, 2001.
  • A. Zayed, Pointwise convergence of a class of non-orthogonal wavelet expansions, Proc. Amer. Math. Soc., 128 (2000), no. 12, 3629–3637.