Multiscaling frame multiresolution analysis and associated wavelet frames. (English) Zbl 1446.42047
Summary: Frame multiresolution analysis (FMRA) in \(L^2(\mathbb{R})\) is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in \(L^2(\mathbb{R}^n)\), which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity \(d\). We characterize wavelet frames and Parseval wavelet frames for \(L^2(\mathbb{R}^n)\) under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions \(\{\psi_k\}_{k=1}^K\) in \(V_1\) to be multiframe generators of \(W_0=V_1\ominus V_0\). Especially, the second condition depends on the multiscaling FMRA and \(\{\psi_k\}_{k=1}^K\) only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.
MSC:
| 42C15 | General harmonic expansions, frames |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
References:
| [1] | Benedetto, J. J. and Li, S., Subband coding and noise reduction in multiresolution analysis frames, Proc. SPIE2303 (1994) 154-165. |
| [2] | Benedetto, J. J. and Li, S., The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal.5 (1998) 389-427. · Zbl 0915.42029 |
| [3] | Benedetto, J. J. and Treiber, O., Wavelet frames: Multiresolution analysis and extension principles, in Wavelet Transforms and Time-Frequency Signal Analysis, ed. Debnath, L. (Birkhäuser, Boston, USA, 2001), pp. 1-36. · Zbl 1036.42032 |
| [4] | de Boor, C., DeVore, R. and Ron, A., On the construction of multivariate (pre) wavelets, Constr. Approx.9 (1993) 123-166. · Zbl 0773.41013 |
| [5] | Bownik, M., The structure of shift-invariant subspaces of \(L^2( \mathbb{R}^n)\), J. Funct. Anal.176 (2000) 282-309. · Zbl 0986.46018 |
| [6] | Cabrelli, C. A. and Gordillo, M. L., Existence of multiwavelets in \(\mathbb{R}^n\), Proc. Amer. Math. Soc.130 (2001) 1413-1424. · Zbl 0988.42025 |
| [7] | Calogero, A. and Garrigós, G., A characterization of wavelet families arising from biorthogonal MRA’s of multiplicity \(d\), J. Geom. Anal.11 (2001) 187-217. · Zbl 0994.42020 |
| [8] | Christensen, O., An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, USA, 2003). · Zbl 1017.42022 |
| [9] | Cifuentes, P., Kazarian, K. S. and San Antolín, A., Characterization of scaling functions, in Splines and Wavelets: Athens 2005, eds. Chen, G. and Lai, M. J. (Nashboro Press, Brentwood, USA, 2005), pp. 152-163. · Zbl 1099.65144 |
| [10] | Dai, X., Diao, Y. and Li, Z., The path-connectivity of \(s\)-elementary frame wavelets with frame MRA, Acta. Appl. Math.107 (2009) 203-210. · Zbl 1175.42019 |
| [11] | Daubechies, I., Ten Lectures on Wavelets (SIAM, Philadelphia, USA, 1992). · Zbl 0776.42018 |
| [12] | Gröchenig, K. H. and Haas, A., Self-similar lattice tilings, J. Fourier Anal. Appl.1 (1994) 131-170. · Zbl 0978.28500 |
| [13] | Kazarian, K. S. and San Antolín, A., Characterization of scaling functions in a frame multiresolution analysis in \(H_G^2\), in Topics in Classical Analysis and Applications in Honour of Daniel Waterman, eds. De Carli, L., Kazarian, K. S. and Milman, M. (World Scientific Publication, Hackensack, USA, 2008), pp. 118-140. · Zbl 1160.42014 |
| [14] | Keinert, F., Wavelets and Multiwavelets (Chapman & Hall/CRC, Boca Raton, USA, 2003). · Zbl 1058.65150 |
| [15] | Kim, H. O., Kim, R. Y. and Lim, J. K., On the spectrums of frame multiresolution analyses, J. Math. Anal. Appl.305 (2005) 528-545. · Zbl 1061.42018 |
| [16] | Kim, H. O., Kim, R. Y. and Lim, J. K., New look at the construction of multiwavelet frames, Bull. Korean Math. Soc.47 (2010) 563-573. · Zbl 1191.42015 |
| [17] | Li, Z. and Shi, X., Parseval frame wavelet multipliers in \(L^2( \mathbb{R}^d)\), Chinese Ann. Math. Ser B.33 (2012) 949-960. · Zbl 1259.42023 |
| [18] | Lian, Q. F. and Li, Y. Z., Reducing subspace frame multiresolution analysis and frame wavelets, Commun. Pure Appl. Anal.6 (2007) 741-756. · Zbl 1141.42024 |
| [19] | Mu, L., Zhang, Z. and Zhang, P., On the higher-dimensional wavelet frames, Appl. Comput. Harmon. Anal.16 (2004) 44-59. · Zbl 1040.42034 |
| [20] | Papadakis, M., Generalized frame multiresolution analysis of abstract Hilbert spaces, in Sampling, Wavelets, and Tomography, eds. Benedetto, J. J. and Zayed, A. (Birkhäuser, Boston, USA, 2003), pp. 179-223. · Zbl 1069.42027 |
| [21] | Ron, A. and Shen, Z., Frames and stable bases for shift-invariant subspaces of \(L_2( \mathbb{R}^d)\), Canad. J. Math.47 (1995) 1051-1094. · Zbl 0838.42016 |
| [22] | Rudin, W., Real and Complex Analysis, 3rd edn. (McGraw-Hill Book Company, New York, USA, 1987). · Zbl 0925.00005 |
| [23] | San Antolín, A., On low pass filters in a frame multiresolution analysis, Tohoku Math. J.63 (2011) 427-439. · Zbl 1229.42034 |
| [24] | Yu, X., Semiorthogonal multiresolution analysis frames in higher dimensions, Acta. Appl. Math.111 (2010) 257-286. · Zbl 1194.42041 |
| [25] | Zalik, R. A., Bases of translates and multiresolution analyses, Appl. Comput. Harmon. Anal.24 (2008) 41-57. · Zbl 1141.42025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
