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The structure of shift-invariant subspaces of $$L^2(\mathbb{R}^n)$$. (English) Zbl 0986.46018
The author of this note investigates the structure of shift invariant spaces in $$L^2(\mathbb{R}^n)$$ under the action of some lattice $$\Gamma= P\mathbb{Z}^n$$, where $$P$$ is a nonsingular $$n$$ by $$n$$ real matrix. He treates $$P$$ as the unit matrix, since general $$P$$ case follows by standard arguments.
The proofs are followed by the idea from H. Helson’s book “Lectures on invariant subspaces”, New York/London (1964; Zbl 0119.11303) and it also can reproduce the former results by R. Ron and Z. Shen of $$L_2(\mathbb{R}^d)$$, Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016).
The typical theorem is in the following:
Theorem. Suppose $$V\subset L^2(\mathbb{R}^n)$$ is shift invariant and $$J$$ is its range function. For every shift preserving operator $$L: V\to L^2(\mathbb{R}^n)$$ there exists a measurable range operator $$R$$ on $$J$$ such that $({\mathcal T}\circ L) f(x)= R(x)({\mathcal T}f(x))\quad\text{for a.e. }x\in\mathbb{T}^n,\quad f\in V,\tag{$$*$$}$ where $${\mathcal T}: L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n, \ell^2(\mathbb{Z}))$$ defined for $$f\in L^2(\mathbb{R}^n)$$ by ${\mathcal T}f: \mathbb{T}^n\to \ell^2(\mathbb{Z}^n),\quad{\mathcal T}f(x)= (\widehat f(x+ k))_{k\in \mathbb{Z}^n},$ is an isometric isomorphic between $$L^2(\mathbb{R}^n)$$ and $$L^2(\mathbb{T}^n, \ell^2(\mathbb{Z}^n))$$. Conversely, given a measurable range operator $$R$$ on $$J$$ with $$\text{sup ess}_{x\in\mathbb{T}^n}\|R(x)\|< \infty$$ there is a bounded shift preserving operator $$L: V\to L^2(\mathbb{R}^n)$$ such that $$(*)$$ holds. The correspondence between $$L$$ and $$R$$ is one-to-one under the convention that the range operators are identified if they are equal a.e. Moreover, we have $$\|L\|= \text{sup ess}_{x\in\mathbb{T}^n}\|R(x)\|$$.

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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##### References:
 [1] de Boor, C.; DeVore, R.A.; Ron, A., The structure of finitely generated shift-invariant spaces in L2($$R$$d), J. funct. anal., 119, 37-78, (1994) · Zbl 0806.46030 [2] de Boor, C.; DeVore, R.A.; Ron, A., Approximation from shift-invariant subspaces of L2($$R$$d), Trans. amer. math. soc., 341, 787-806, (1994) · Zbl 0790.41012 [3] Benedetto, J.J.; Li, S., The theory of multiresolution analysis frames and applications to filter banks, Appl. comput. harmon. anal., 5, 389-427, (1998) · Zbl 0915.42029 [4] M. Bownik, Z. Rzeszotnik, and, D. Speegle, A characterization of dimension functions of wavelets, preprint, 1999. · Zbl 0979.42018 [5] Daubechies, I., Ten lectures on wavelets, (1992), Soc. Indust. & Appl. Math Philadelphia · Zbl 0776.42018 [6] Helson, H., Lectures on invariant subspaces, (1964), Academic Press New York/London · Zbl 0119.11303 [7] Papadakis, M., On the dimension function of orthonormal wavelets, Proc. amer. math. soc., (1999) [8] Ron, A.; Shen, Z., Frames and stable bases for shift-invariant subspaces of L2($$R$$d), Canad. J. math., 47, 1051-1094, (1995) · Zbl 0838.42016 [9] Ron, A.; Shen, Z., Weyl – heisenberg frames and Riesz bases in L2($$R$$d), Duke math. J., 89, 237-282, (1997) · Zbl 0892.42017 [10] Ron, A.; Shen, Z., Affine systems in L2($$R$$d): the analysis of the analysis operator, J. funct. anal., 148, 408-447, (1997) · Zbl 0891.42018 [11] Ron, A.; Shen, Z., Affine systems in L2($$R$$d). II. dual systems, J. Fourier anal. appl., 3, 617-637, (1997)
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