Abstract
In this paper, firstly, in order to establish our main techniques we give a direct proof for the existence of the dilations for pairs of dual group frames. Then we focus on proving the uniqueness of such dilations in certain sense of similarity and giving an operator parameterization of the dilations of all pairs of dual group frames for a given group frame. We show that the operators which transform different dilations are of special structured lower triangular.
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Benedetto, J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal., 5(4), 389–427 (1998)
Chui, C. K.: An Introduction to Wavelets, Acad. Press, New York, 1992
Casazza, P. G., Kovačević, J.: Equal-norm tight frames with erasures. Adv. Comput. Math., 18(2–4), 387–430 (2003)
Casazza, P. G., Han, D., Larson, D. R.: Frames for Banach spaces. Contemp. Math., 247, 149–182 (1999)
Duffin, R. J., Shaffer, A. C.: A class of nonharmonic Fourier Series. Trans. Amer. Math. Soc., 72, 341–366 (1952)
Daubechies, I.: Ten Lectures on Wavelets, CBMS 61, SIAM, Philadelphia, PA, 1992
Dai, X., Larson, D.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Amer. Math. Soc., 134(640) (1998)
Eldar, Y., Blcskei, H.: Geometrically uniform frames. IEEE Trans. Inform. Theory, 49(4), 993–1006 (2003)
Gabardo, J. P., Han, D.: Frame representations for group-like unitary operator systems. J. Operator Theory, 49, 1–22 (2003)
Guo, X.: Wandering r-tuples for unitary systems. J. Math. Anal. Appl., 374, 722–728 (2011)
Guo, X.: Multi-frame vectors for unitary systems. Indian J. Pure Appl. Math., 43(4), 391–409 (2012)
Hernandez, E., Weiss, G.: A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996
Han, D., Larson, D.: Bases, frames and group representations. Mem. Amer. Math. Soc., 147(697) (2000)
Han, D.: Wandering vectors for irrational rotation unitary systems. Trans. Amer. Math. Soc., 350, 309–320 (1998)
Han, D., Larson, D.: Wandering vector multipliers for unitary groups. Trans. Amer. Math. Soc., 353, 3347–3370 (2001)
Han, D. G., Wu J., Larson, D., et al.: Dilation of dual frame pairs in Hilbert C*-modules. Results. Math., 63(1), 241–250 (2013)
Han, D., Kornelson, K., Larson, D., et al.: Frames for Undergraduates. Student Mathematical Library, 40. American Mathematical Society, Providence, RI, 2007
Mallat, S.: Multiresolution approximations and wavelet orthonormal basis of L2(R). Trans. Amer. Math. Soc., 315, 69–87 (1989)
Papadakis, M.: On the dimension function of a wavelet. Proc. Amer. Math. Soc., 128(7), 2043–2049 (2000)
Papadakis, M.: Generalized frame multiresolution analysis of abstract Hilbert spaces, sampling, wavelets, and tomography. Appl. Numer. Harmon. Anal., 179–223 (2004)
Sz-Nagy, B.: Expansion theorems of Paley–Wiener type. Duke Math. J., 14, 975–978 (1947)
Young, R. M.: An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980
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Guo, X.X. Similarity and parameterizations of dilations of pairs of dual group frames in Hilbert spaces. Acta. Math. Sin.-English Ser. 33, 1671–1683 (2017). https://doi.org/10.1007/s10114-017-7078-2
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DOI: https://doi.org/10.1007/s10114-017-7078-2