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A Note on the Structure of Affine Subspaces of L2(Rd)

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ABSTRACT
This paper investigates the structure of general affine subspaces of L2(Rd) <span "="">. For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a purely non-reducing subspace, and every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces when |detA| = 2.
Cite this paper
Zhou, F. and Xu, X. (2015) A Note on the Structure of Affine Subspaces of L2(Rd). Advances in Pure Mathematics, 5, 62-70. doi: 10.4236/apm.2015.52008.
 
References
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[2]Dai, X., Diao, Y., Gu, Q. and Han, D. (2002) Frame Wavelets in Subspaces of . Proceedings of the American Mathematical Society, 130, 3259-3267. http://dx.doi.org/10.1090/S0002-9939-02-06498-5
 
[3]Zhou, F.Y. and Li, Y.Z. (2010) Multivariate FMRAs and FMRA Frame Wavelets for Reducing Subspaces of . Kyoto Journal of Mathematics, 50, 83-99. http://dx.doi.org/10.1215/0023608X-2009-006
 
[4]Dai, X., Diao, Y. and Gu, Q. (2002) Subspaces with Normalized Tight Frame Wavelets in . Proceedings of the American Mathematical Society, 130, 1661-1667. http://dx.doi.org/10.1090/S0002-9939-01-06257-8
 
[5]Dai, X., Diao, Y., Gu, Q. and Han, D. (2003) The Existence of Subspace Wavelet Sets. Journal of Computational and Applied Mathematics, 155, 83-90.
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[7]Li, Y.Z. and Zhou, F.Y. (2010) Affine and Quasi-Affine Dual Frames in Reducing Subspaces of . Acta Mathematica Sinica (Chinese Edition), 53, 551-562.
 
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