Abstract
For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.
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Baggett, L.W., Medina, H.M., Merrill, K.D.: Generalized multi-resolution analyses and a construction procedure for all wavelet sets in \(\bold R\sp n\). J. Fourier Anal. Appl. 5(6), 563–573 (1999)
Benedetto, J.J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5(4), 389–427 (1998)
Blu, T., Unser, M.: The fractional spline wavelet transform: definition and implementation. In: Proceedings of the Twenty-Fifth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP2000), pp. 512–515. Istanbul, Turkey, 5–9 June 2000
Bownik, M., Speegle, D.: The wavelet dimension function for real dilations and dilations admitting non-MSF wavelets. In: Chui, C.K. et al. (ed.) Approximation Theory, X. Wavelets, Splines, and Applications, Papers from the 10th International Symposium, St. Louis, Mo, USA, 26–29 March 2001. Innov. Appl. Math., pp. 63–85. Vanderbilt Univ. Press, Nashville, TN (2002)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser Boston, Boston (2003)
Christensen, O.: Pairs of dual Gabor frame generators with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20(3), 403–410 (2006)
Chui, C.K., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9(3), 243–264 (2000)
Chui, C.K., He, W., Stöckler, J.: Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13(3), 224–262 (2002)
Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14(1), 1–46 (2003)
de Boor, C., DeVore, R.A., Ron, A.: Approximation from shift-invariant subspaces of \(L\sb 2(\bold R\sp d)\). Trans. Amer. Math. Soc. 341(2), 787–806 (1994)
Hernández, E., Weiss, G.: A First Course on Wavelets. Studies in Advanced Mathematics. CRC, Boca Raton (1996)
Ron, A., Shen, Z.: Affine systems in \(L\sb 2(\bold R\sp d)\): the analysis of the analysis operator. J. Funct. Anal. 148(2), 408–447 (1997)
Ron, A., Shen, Z.: Compactly supported tight affine spline frames in \(L\sb 2(\bold R\sp d)\). Math. Comp. 67(221), 191–207 (1998)
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Communicated by Qiyu Sun.
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Lemvig, J. Constructing pairs of dual bandlimited framelets with desired time localization. Adv Comput Math 30, 231–247 (2009). https://doi.org/10.1007/s10444-008-9066-7
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DOI: https://doi.org/10.1007/s10444-008-9066-7