# The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups

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## Abstract

Let \(\Gamma \) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group \(G\). In this paper we study translation-invariant (TI) subspaces of \(L^2(G)\) by elements of \(\Gamma \). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that \(\Gamma \) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of \(G=\mathbb {R}^n\). We also extend the notion of the spectral function in \(\mathbb {R}^n\) to the LCA setting. It is shown that spectral functions, initially defined in terms of \(\Gamma \), do not depend on \(\Gamma \). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of \(G\) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.

## Keywords

Translation-invariant space LCA group Range function Dimension function Spectral function Continuous frame## Mathematics Subject Classification

42C15 43A32 43A70 22B99 46C05## Notes

### Acknowledgments

The authors are pleased to thank Joey Iverson for reading the manuscript and providing useful comments. We also thank a very careful referee for catching several ambiguities. The first author was partially supported by NSF Grant DMS-1265711.

## References

- 1.Aldaz, J.M.: The weak type \((1,1)\) bounds for the maximal function associated to cubes grow to infinity with the dimension. Ann. Math.
**173**, 1013–1023 (2011)MathSciNetCrossRefGoogle Scholar - 2.Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Continuous frames in Hilbert space. Ann. Phys.
**222**, 1–37 (1993)MathSciNetCrossRefGoogle Scholar - 3.de Boor, C., DeVore, R., Ron, A.: The structure of finitely generated shift-invariant spaces in \(L^2(\mathbb{R}^d)\). J. Funct. Anal.
**119**(1), 37–78 (1994)MathSciNetCrossRefGoogle Scholar - 4.de Boor, C., DeVore, R., Ron, A.: Approximation from shift-invariant subspaces of \(L^2(\mathbb{R}^d)\). Trans. Am. Math. Soc.
**341**, 787–806 (1994)Google Scholar - 5.Bownik, M.: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\). J. Funct. Anal.
**177**, 282–309 (2000)MathSciNetCrossRefGoogle Scholar - 6.Bownik, M.: The structure of shift-modulation invariant spaces: the rational case. J. Funct. Anal.
**244**, 172–219 (2007)MathSciNetCrossRefGoogle Scholar - 7.Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces. Mich. Math. J.
**51**, 387–414 (2003)MathSciNetCrossRefGoogle Scholar - 8.Bownik, M., Rzeszotnik, Z.: The Spectral Function of Shift-Invariant Spaces on General Lattices, Wavelets, Frames and Operator Theory, Contemporary Mathematics, vol. 345, pp. 49–59. American Mathematical Society, Providence, RI (2004)Google Scholar
- 9.Cabrelli, C., Paternostro, V.: Shift-invariant spaces on LCA groups. J. Funct. Anal.
**258**, 2034–2059 (2010)MathSciNetCrossRefGoogle Scholar - 10.Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser Boston Inc., Boston, MA (2003)CrossRefGoogle Scholar
- 11.Diestel, J., Uhl, J.J. Jr.: Vector Measures. With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence, RI (1977)Google Scholar
- 12.Dutkay, D.E.: The local trace function of shift-invariant subspaces. J. Oper. Theory
**52**, 267–291 (2004)MathSciNetGoogle Scholar - 13.Edwards, R.E., Hewitt, E.: Pointwise limits for sequences of convolution operators. Acta Math.
**113**, 181–218 (1965)MathSciNetCrossRefGoogle Scholar - 14.Feldman, J., Greenleaf, F.P.: Existence of Borel transversals in groups. Pac. J. Math.
**25**, 455–461 (1968)MathSciNetCrossRefGoogle Scholar - 15.Folland, G.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)Google Scholar
- 16.Folland, G.: Real Analysis. Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)Google Scholar
- 17.Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York (1964)Google Scholar
- 18.Helson, H.: The Spectral Theorem. Lecture Notes in Mathematics. Springer-Verlag, Berlin (1986)Google Scholar
- 19.Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Springer, New York (1970)Google Scholar
- 20.Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups, Integration Theory, Group Representations, 2nd edn. Springer, Berlin (1979)Google Scholar
- 21.Kaiser, G.: A Friendly Guide to Wavelets. Birkhäuser Boston, Inc., Boston, MA (1994)Google Scholar
- 22.Kamyabi Gol, R.A., Raisi Tousi, R.: The structure of shift-invariant spaces on a locally compact abelian group. J. Math. Anal. Appl.
**340**, 219–225 (2008)MathSciNetCrossRefGoogle Scholar - 23.Kamyabi Gol, R.A., Raisi Tousi, R.: A range function approach to shift-invariant spaces on locally compact abelian groups. Int. J. Wavelets Multiresolut. Inf. Process.
**8**, 49–59 (2010)MathSciNetCrossRefGoogle Scholar - 24.Kamyabi Gol, R.A., Raisi Tousi, R.: Some equivalent multiresolution conditions on locally compact abelian groups. Proc. Indian Acad. Sci. (Math. Sci.)
**120**, 317–331 (2010)MathSciNetCrossRefGoogle Scholar - 25.Kaniuth, E., Kutyniok, G.: Zeros of the Zak transform on locally compact abelian groups. Proc. Am. Math. Soc.
**126**, 3561–3569 (1998)MathSciNetCrossRefGoogle Scholar - 26.Mackey, G.W.: Induced representations of locally compact groups. I. Ann. Math.
**55**, 101–139 (1952)MathSciNetCrossRefGoogle Scholar - 27.Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, Cambridge (1999)Google Scholar
- 28.Reiter, H., Stegeman, J.: Classical Harmonic Analysis and Locally Compact Groups, 2nd edn. London Mathematical Society Monographs. New Series, 22. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
- 29.Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of \(L^2(\mathbb{R}^d)\). Can. J. Math.
**47**, 1051–1094 (1995)MathSciNetCrossRefGoogle Scholar - 30.Ron, A., Shen, Z.: Affine systems in \(L^2(\mathbb{R}^d)\): the analysis of the analysis operator. J. Funct. Anal.
**148**, 408–447 (1997)MathSciNetCrossRefGoogle Scholar - 31.Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, No. 12. Wiley, New York (1962)Google Scholar
- 32.Srinivasan, T.P.: Doubly invariant subspaces. Pac. J. Math.
**14**, 701–707 (1964)CrossRefGoogle Scholar - 33.Stein, E., Strömberg, J.: Behavior of maximal functions in \(\mathbb{R}^n\) for large \(n\). Ark. Mat.
**21**, 259–269 (1983)MathSciNetCrossRefGoogle Scholar - 34.Stroppel, M.: Locally Compact Groups. EMS Textbooks in Mathematics. European Mathematical Society, Zürich (2006)CrossRefGoogle Scholar
- 35.Tao, T., Vu, V.H.: Additive Combinatorics. Cambridge University Press, Cambridge (2009)Google Scholar
- 36.Wiener, N.: Tauberian theorems. Ann. Math.
**33**, 1–100 (1932)MathSciNetCrossRefGoogle Scholar