This paper investigates the structure of general affine subspaces of L2(Rd) . 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| =
Cite this paper
Zhou, F. and Xu, X. (2015) A Note on the Structure of Affine Subspaces of L2
). Advances in Pure Mathematics
, 62-70. doi: 10.4236/apm.2015.52008
 Weiss, G. and Wilson, E.N. (2001) The Mathematical Theory of Wavelets. In: Byrnes, J.S., Ed., Twentieth Century Harmonic Analysis-A Celebration//Proceedings of the NATO Advanced Study Institute, Kluwer Academic Publishers, Dordrecht, 329-366.
 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
 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
 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
 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. http://dx.doi.org/10.1016/S0377-0427(02)00893-2
 Lian, Q.F. and Li, Y.Z. (2007) Reducing Subspace Frame Multiresolution Analysis and Frame Wavelets. Communications on Pure and Applied Analysis, 6, 741-756. http://dx.doi.org/10.3934/cpaa.2007.6.741
 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.
 Gu, Q. and Han, D. (2009) Wavelet Frames for (Not Necessarily Reducing) Affine Subspaces. Applied and Computational Harmonic Analysis, 27, 47-54. http://dx.doi.org/10.1016/j.acha.2008.10.006
 Gu, Q. and Han, D. (2011) Wavelet Frames for (Not Necessarily Reducing) Affine Subspaces II: The Structure of Affine Subspaces. Journal of Functional Analysis, 260, 1615-1636. http://dx.doi.org/10.1016/j.jfa.2010.12.020
 Zhou, F.Y. and Li, Y.Z. (2013) A Note on Wavelet Frames for Affine Subspaces of . Acta Mathematica Sinica (Chinese Edition), 33A, 89-97.