November  2021, 4(4): 281-297. doi: 10.3934/mfc.2021019

Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $

Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Leganés, 28911, Spain

Received  April 2021 Revised  August 2021 Published  November 2021 Early access  September 2021

Fund Project: The author is supported by the grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO)

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $ \Lambda $ in $ \mathbb{R}^{2d} $. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.

Citation: Antonio G. García. Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $. Mathematical Foundations of Computing, 2021, 4 (4) : 281-297. doi: 10.3934/mfc.2021019
References:
[1]

A. AldroubiQ. Sun and W. S. Tang, Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces, J. Fourier Anal. Appl., 11 (2005), 215-244.  doi: 10.1007/s00041-005-4003-3.

[2]

J. J. Benedetto and G. E. Pfander, Frame expansions for Gabor multipliers, Appl. Comput. Harmon. Anal., 20 (2006), 26-40.  doi: 10.1016/j.acha.2005.03.002.

[3]

O. Christensen, An Introduction to Frames and Riesz Bases, 2$^nd$ edition, Birkhäuser, Basel, 2016.

[4]

J. B. Conway, A Course in Operator Theory, American Mathematical Society, Providence, RI, 2000.

[5]

A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, 2$^nd$ edition, Universitext, Springer, Cham, 2014.

[6]

H. G. Feichtinger, Spline-type spaces in Gabor analysis, Wavelet Analysis (Hong Kong, 2001), Ser. Anal., World Sci. Publ., River Edge, NJ, 1 (2002), 100-122. 

[7]

H. G. Feichtinger, F. Luef and T. Wherter, A guided tour from linear algebra to the foundations of Gabor analysis, Gabor and Wavelet frames, (eds. Say Song Goh et al.), Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 10 (2007), 1–49. doi: 10.1142/9789812709080_0001.

[8]

H. R. Fernández-MoralesA. G. GarcíaM. A. Hernández-Medina and M. J. Muñoz-Bouzo, Generalized sampling: From shift-invariant to $U$-invariant spaces, Anal. Appl., 13 (2015), 303-329.  doi: 10.1142/S0219530514500213.

[9] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton NJ, 1989. 
[10] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton FL, 1995. 
[11]

H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transform, Lecture Notes in Mathematics, 1863, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104912.

[12]

A. G. García, Average sampling in certain subspaces of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, Sampl. Theory Signal Process. Data Anal., 19 (2021), 10. 

[13]

A. G. García and G. Pérez-Villalón, Dual frames in ${L}^2(0, 1)$ connected with generalized sampling in shift-invariant spaces, Appl. Comput. Harmon. Anal., 20 (2006), 422-433.  doi: 10.1016/j.acha.2005.10.001.

[14]

A. G. GarcíaM. A. Hernández-Medina and G. Pérez-Villalón, Generalized sampling in shift-invariant spaces with multiple stable generators, J. Math. Anal. Appl., 337 (2008), 69-84.  doi: 10.1016/j.jmaa.2007.03.083.

[15]

A. G. García, M. A. Hernández-Medina and G. Pérez-Villalón, Convolution systems on discrete abelian groups as a unifying strategy in sampling theory, Results Math., 75 (2020), 20pp. doi: 10.1007/s00025-020-1164-y.

[16]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Basel, 2001.

[17]

K. Gröchenig and C. Heil, Modulation spaces and pseudodifferential operators, Integr. Equ. Oper. Theory, 34 (1999), 439-457.  doi: 10.1007/BF01272884.

[18]

K. Gröchenig and E. Pauwels, Uniqueness and reconstruction theorems for pseudodifferential operators with a bandlimited Kohn-Nirenberg symbol, Adv. Comput. Math., 40 (2014), 49-63.  doi: 10.1007/s10444-013-9297-0.

[19]

M. S. Jakobsen, On a (no longer) new Segal Algebra: A review of the Feichtinger Algebra, J. Fourier Anal. Appl., 24 (2018), 1579-1660.  doi: 10.1007/s00041-018-9596-4.

[20]

F. Krahmer and G. E. Pfander, Local sampling and approximation of operators, Construct. Approx., 39 (2014), 541-572. 

[21]

F. Luef and E. Skrettingland, Convolutions for localizations operators, J. Math. Pures Appl., 118 (2018), 288-316.  doi: 10.1016/j.matpur.2017.12.004.

[22]

G. E. Pfander, Sampling of operators, J. Fourier Anal. Appl., 19 (2013), 612-650.  doi: 10.1007/s00041-013-9269-2.

[23]

G. E. Pfander and D. F. Walnut, Sampling and reconstruction of operators, IEEE Trans, Inform. Theory, 62 (2016), 435-458.  doi: 10.1109/TIT.2015.2501646.

[24]

E. Skrettingland, Quantum harmonic analysis on lattices and Gabor multipliers, J. Fourier Anal. Appl., 26 (2020), 37pp. doi: 10.1007/s00041-020-09759-1.

[25]

T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal., 20 (2006), 237-249.  doi: 10.1016/j.acha.2005.06.003.

[26]

R. F. Werner, Quantum harmonic analysis on phase space, J. Math. Phys., 25 (1984), 1404-1411.  doi: 10.1063/1.526310.

show all references

References:
[1]

A. AldroubiQ. Sun and W. S. Tang, Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces, J. Fourier Anal. Appl., 11 (2005), 215-244.  doi: 10.1007/s00041-005-4003-3.

[2]

J. J. Benedetto and G. E. Pfander, Frame expansions for Gabor multipliers, Appl. Comput. Harmon. Anal., 20 (2006), 26-40.  doi: 10.1016/j.acha.2005.03.002.

[3]

O. Christensen, An Introduction to Frames and Riesz Bases, 2$^nd$ edition, Birkhäuser, Basel, 2016.

[4]

J. B. Conway, A Course in Operator Theory, American Mathematical Society, Providence, RI, 2000.

[5]

A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, 2$^nd$ edition, Universitext, Springer, Cham, 2014.

[6]

H. G. Feichtinger, Spline-type spaces in Gabor analysis, Wavelet Analysis (Hong Kong, 2001), Ser. Anal., World Sci. Publ., River Edge, NJ, 1 (2002), 100-122. 

[7]

H. G. Feichtinger, F. Luef and T. Wherter, A guided tour from linear algebra to the foundations of Gabor analysis, Gabor and Wavelet frames, (eds. Say Song Goh et al.), Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 10 (2007), 1–49. doi: 10.1142/9789812709080_0001.

[8]

H. R. Fernández-MoralesA. G. GarcíaM. A. Hernández-Medina and M. J. Muñoz-Bouzo, Generalized sampling: From shift-invariant to $U$-invariant spaces, Anal. Appl., 13 (2015), 303-329.  doi: 10.1142/S0219530514500213.

[9] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton NJ, 1989. 
[10] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton FL, 1995. 
[11]

H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transform, Lecture Notes in Mathematics, 1863, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104912.

[12]

A. G. García, Average sampling in certain subspaces of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, Sampl. Theory Signal Process. Data Anal., 19 (2021), 10. 

[13]

A. G. García and G. Pérez-Villalón, Dual frames in ${L}^2(0, 1)$ connected with generalized sampling in shift-invariant spaces, Appl. Comput. Harmon. Anal., 20 (2006), 422-433.  doi: 10.1016/j.acha.2005.10.001.

[14]

A. G. GarcíaM. A. Hernández-Medina and G. Pérez-Villalón, Generalized sampling in shift-invariant spaces with multiple stable generators, J. Math. Anal. Appl., 337 (2008), 69-84.  doi: 10.1016/j.jmaa.2007.03.083.

[15]

A. G. García, M. A. Hernández-Medina and G. Pérez-Villalón, Convolution systems on discrete abelian groups as a unifying strategy in sampling theory, Results Math., 75 (2020), 20pp. doi: 10.1007/s00025-020-1164-y.

[16]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Basel, 2001.

[17]

K. Gröchenig and C. Heil, Modulation spaces and pseudodifferential operators, Integr. Equ. Oper. Theory, 34 (1999), 439-457.  doi: 10.1007/BF01272884.

[18]

K. Gröchenig and E. Pauwels, Uniqueness and reconstruction theorems for pseudodifferential operators with a bandlimited Kohn-Nirenberg symbol, Adv. Comput. Math., 40 (2014), 49-63.  doi: 10.1007/s10444-013-9297-0.

[19]

M. S. Jakobsen, On a (no longer) new Segal Algebra: A review of the Feichtinger Algebra, J. Fourier Anal. Appl., 24 (2018), 1579-1660.  doi: 10.1007/s00041-018-9596-4.

[20]

F. Krahmer and G. E. Pfander, Local sampling and approximation of operators, Construct. Approx., 39 (2014), 541-572. 

[21]

F. Luef and E. Skrettingland, Convolutions for localizations operators, J. Math. Pures Appl., 118 (2018), 288-316.  doi: 10.1016/j.matpur.2017.12.004.

[22]

G. E. Pfander, Sampling of operators, J. Fourier Anal. Appl., 19 (2013), 612-650.  doi: 10.1007/s00041-013-9269-2.

[23]

G. E. Pfander and D. F. Walnut, Sampling and reconstruction of operators, IEEE Trans, Inform. Theory, 62 (2016), 435-458.  doi: 10.1109/TIT.2015.2501646.

[24]

E. Skrettingland, Quantum harmonic analysis on lattices and Gabor multipliers, J. Fourier Anal. Appl., 26 (2020), 37pp. doi: 10.1007/s00041-020-09759-1.

[25]

T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal., 20 (2006), 237-249.  doi: 10.1016/j.acha.2005.06.003.

[26]

R. F. Werner, Quantum harmonic analysis on phase space, J. Math. Phys., 25 (1984), 1404-1411.  doi: 10.1063/1.526310.

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