A Note on the Structure of Affine Subspaces of L²(R^d)

Fengying Zhou^*, Xiaoyong Xu

School of Science, East China Institute of Technology, Nanchang, China.

DOI: 10.4236/apm.2015.52008                

Abstract

This paper investigates the structure of general affine subspaces of L²(R^d) . 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.

Keywords

Affine Subspace, Reducing Subspace, Shift Invariant Subspace, Orthogonal Sum

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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.
[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. http://dx.doi.org/10.1016/S0377-0427(02)00893-2
[6]	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
[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.
[8]	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
[9]	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
[10]	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.