Abstract
In this article, we study the supremum cosine angle between two multiplication invariant (MI) spaces and its connection with the closedness of the sum of those spaces. The results obtained for MI spaces are preserved almost everywhere by the corresponding fiber spaces. Also, we provide equivalent conditions for the injectivity of the sampling operator associated with the multiplication generated Bessel system for the union of the sum of finitely generated MI spaces. Employing the Zak transform, we obtain the results for translation invariant spaces on locally compact groups by the action of its closed abelian subgroup as an application.
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References
Aldroubi A, Abry P and Unser M, Construction of biorthogonal wavelets starting from any two multiresolutions, IEEE Trans. Sign. Process. 46(4) (1998) 1130–1133
Anastasio M and Cabrelli C, Sampling in a union of frame generated subspaces, Sampl. Theory Sign. Image Process. 8(3) (2009) 261–286
Bownik M and Ross K A, The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl. 21(4) (2015) 849–884
Bownik M and Garrigós G, Biorthogonal wavelets, MRA’s and shift-invariant spaces, Stud. Math. 160 (2004) 231–248
Bownik M and Iverson J W, Multiplication-invariant operators and the classification of LCA group frames, J. Funct. Anal. 280(2) (2021) Paper No. 108780, 59
Christensen O, An introduction to frames and Riesz bases (2016) (Springer)
Conway J B, A course in functional analysis (1997) (Springer) vol. 96
Broomhead D S and Kirby M, A new approach to dimensionality reduction: Theory and algorithms, SIAM J. Appl. Math. 60 (2000) 2114–2142
Helson H, Lectures on invariant subspaces (1964) (New York: Academic Press)
Iverson J W, Subspaces of \(L^2(G)\) invariant under translation by an abelian subgroup, J. Funct. Anal. 269(3) (2015) 865–913
Kim H O, Kim R Y, Lee Y H and Lim J K, On Riesz wavelets associated with multiresolution analyses, Appl. Comput. Harmon. Anal. 13 (2002) 138–150
Kim H O, Kim R Y and Lim J K, Quasi-biorthogonal frame multiresolution analyses and wavelets, Adv. Comput. Math. 18(2) (2003) 269–296
Kim H O, Kim R Y and Lim J K, The infimum cosine angle between two finitely generated shift-invariant spaces and its applications, Appl. Comput. Harmonic Anal. 19(2) (2005) 253–281
Kim H O, Kim R Y and Lim J K, Characterization of the closedness of the sum of two shift-invariant spaces, J. Math. Anal. Appl. 320(1) (2006) 381–395
Kim H O, Kim R Y and Lim J K, Internal structure of the multiresolution analyses defined by the unitary extension principle, J. Approx. Theory 154(2) (2008) 140–160
Lu Y and Do M, A theory for sampling signals from a union of subspaces, IEEE Trans. Sign. Process. 56(6) (2008) 2334–2345
Sarkar S and Shukla N K, Translation generated oblique dual frames on locally compact groups, Linear Multilinear Algebra (2023) 1–32
Sarkar S and Shukla N K, Characterizations of extra-invariant spaces under the left translations on a Lie group, Adv. Oper. Theory 8(3) (2023) Paper No. 49, 16 pp.
Tang W-S, Oblique projections, biorthogonal Riesz bases and multiwavelets in Hilbert spaces, Proc. Amer. Math. Soc. 128 (2000) 463–473
Acknowledgements
The authors are grateful to the referee for meticulously reading the manuscript and providing several valuable suggestions for improving the manuscript. The research of S Kalra and S Sarkar was supported by a research grant from the University Grant Commission, Ref. No. 191620003953 and CSIR, New Delhi [09/1022(0037)/2017-EMR-I]. N K Shukla gratefully acknowledges the financial support from DST-SERB Project [MTR/2022/000176]. The authors also acknowledge the facilities of the Bhaskaracharya Mathematics Laboratory, IIT Indore, supported by the DST-FIST Project (File No. SR/FST/MS I/2018/26).
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Communicated by Jaydeb Sarkar.
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Kalra, S., Sarkar, S. & Shukla, N.K. An application of the supremum cosine angle between multiplication invariant spaces in \(\varvec{L^2(X; \mathcal H)}\). Proc Math Sci 134, 15 (2024). https://doi.org/10.1007/s12044-024-00785-3
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DOI: https://doi.org/10.1007/s12044-024-00785-3