## Pairs of dual wavelet frames from any two refinable functions

Starting from any two compactly supported refutable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2 (R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function φ in L2 (R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates φ(d -k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments. We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.

### Duke Scholars

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## Related Subject Headings

- Numerical & Computational Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0104 Statistics
- 0103 Numerical and Computational Mathematics
- 0102 Applied Mathematics

### Citation

*Constructive Approximation*,

*20*(3), 325–352. https://doi.org/10.1007/s00365-004-0567-4

*Constructive Approximation*20, no. 3 (January 1, 2004): 325–52. https://doi.org/10.1007/s00365-004-0567-4.

*Constructive Approximation*, vol. 20, no. 3, Jan. 2004, pp. 325–52.

*Scopus*, doi:10.1007/s00365-004-0567-4.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Numerical & Computational Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0104 Statistics
- 0103 Numerical and Computational Mathematics
- 0102 Applied Mathematics