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Interpolation in wavelet spaces and the HRT-conjecture. (English) Zbl 1497.42055

In the present paper, the author shows that, for a squared integrable representation \(\pi:G\to\mathcal U(\mathcal H_\pi)\) with \(G\) being a locally compact group, the wavelet spaces \(\mathcal W_g(\mathcal H_\pi)\), where \(\mathcal W_gf(x):=\langle f, \pi(x)g\rangle\) are rigid. This means that for two squared integrable representations \(\pi\) and \(\rho\) of the same group, if \(\mathcal W_g(\mathcal H_\pi)\cup\mathcal W_h(\mathcal H_\rho)\neq\emptyset\), then \(\mathcal W_g(\mathcal H_\pi)=\mathcal W_h(\mathcal H_\rho)\).
Additional, it is investigated the interpolation problem: given \(\{x_1, \dots, x_n\}\subset G\) and \(\{\lambda_1, \dots, \lambda_n\}\subset \mathbb C\) find a function \(F\in \mathcal W_g(\mathcal H_\pi)\) such that \(F(x_i)=\lambda_i\), for \(1\leq i\leq n\). When this problem is always solvable, \(\mathcal W_g(\mathcal H_\pi)\) is said to be fully interpolating. The author proves that for compact or abelian groups, wavelet spaces can not be fully interpolating.
Finally, it is considered the problem of whether all the wavelet spaces \(\mathcal W_g(\mathcal H_\pi)\) of a locally compact group \(G\) collectively exhaust the ambient space \(L^2(G)\). About this, the author gives some partial results.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] Abreu, LD; Pereira, JM, Measures of localization and quantitative Nyquist densities, Appl. Comput. Harmon. Anal., 38, 3, 524-534 (2015) · Zbl 1404.94012 · doi:10.1016/j.acha.2014.08.002
[2] Aronszajn, N., Theory of reproducing kernels, Trans. Am. Math. Soc., 68, 3, 337-404 (1950) · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7
[3] Austad, A.; Enstad, U., Heisenberg modules as function spaces, J. Fourier Anal. Appl., 26, 2, 1-28 (2020) · Zbl 1442.42068 · doi:10.1007/s00041-020-09729-7
[4] Balan, R.; Krishtal, I., An almost periodic noncommutative Wiener’s lemma, J. Math. Anal. Appl., 370, 2, 339-349 (2010) · Zbl 1211.46043 · doi:10.1016/j.jmaa.2010.04.053
[5] Bekka, B.; de La Harpe, P.; Valette, A., Kazhdan’s Property (T) (2008), Cambridge: Cambridge University Press, Cambridge · Zbl 1146.22009 · doi:10.1017/CBO9780511542749
[6] Berlinet, A.; Thomas-Agnan, C., Reproducing Kernel Hilbert Spaces in Probability and Statistics (2011), Berlin: Springer, Berlin · Zbl 1145.62002
[7] Cheney, EW; Light, WA, A Course in Approximation Theory (2009), New York: American Mathematical Society, New York · Zbl 1167.41001 · doi:10.1090/gsm/101
[8] Currey, B.; Oussa, V., Translates of functions on the Heisenberg group and the HRT conjecture, Can. Math. Bull., 63, 1-12 (2020) · Zbl 1456.42038 · doi:10.4153/S0008439520000107
[9] Dahlke, S.; De Mari, F.; Grohs, P.; Labate, D., Harmonic and Applied Analysis: From Groups to Signals (2015), New York: Birkhäuser, New York · Zbl 1329.42001 · doi:10.1007/978-3-319-18863-8
[10] Daubechies, I., Ten Lectures on Wavelets (1992), New York: SIAM, New York · Zbl 0776.42018 · doi:10.1137/1.9781611970104
[11] de Vries, J., The local weight of an effective locally compact transformation group and the dimension of \({L}^2({G})\), Colloq. Math., 39, 2, 319-323 (1978) · Zbl 0403.43003 · doi:10.4064/cm-39-2-319-323
[12] Deitmar, A.; Echterhoff, S., Principles of Harmonic Analysis (2014), Berlin: Springer, Berlin · Zbl 1300.43001 · doi:10.1007/978-3-319-05792-7
[13] Duflo, M.; Moore, CC, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal., 21, 2, 209-243 (1976) · Zbl 0317.43013 · doi:10.1016/0022-1236(76)90079-3
[14] Enstad, U., The Balian-Low theorem for locally compact abelian groups and vector bundles, J. Math. Pures Appl., 139, 143-176 (2019) · Zbl 1508.42034 · doi:10.1016/j.matpur.2019.12.005
[15] Feichtinger, H.G., Gröchenig, K.: A unified approach to atomic decompositions via integrable group representations. In: Function Spaces and Applications, pp. 52-73. Springer, Berlin (1988) · Zbl 0658.22007
[16] Feichtinger, HG; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal., 86, 2, 307-340 (1989) · Zbl 0691.46011 · doi:10.1016/0022-1236(89)90055-4
[17] Feichtinger, HG; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions. II, Monat. Math., 108, 2-3, 129-148 (1989) · Zbl 0713.43004 · doi:10.1007/BF01308667
[18] Folland, GB, A Course in Abstract Harmonic Analysis (2016), London: Chapman and Hall, London · Zbl 1342.43001 · doi:10.1201/b19172
[19] Führ, H., Abstract Harmonic Analysis of Continuous Wavelet Transforms (2005), Berlin: Springer, Berlin · Zbl 1060.43002 · doi:10.1007/b104912
[20] Ghandehari, M., Taylor, K.F.: Images of the continuous wavelet transform. In: Operator Methods in Wavelets, Tilings, and Frames, pp. 55-65. American Mathematical Society, New York (2014) · Zbl 1312.42039
[21] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), London: Springer, London · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1
[22] Gröchenig, K., Linear independence of time-frequency shifts?, Monat. Math., 177, 1, 67-77 (2015) · Zbl 1318.42036 · doi:10.1007/s00605-014-0637-z
[23] Gröchenig, K., Romero, J.L., Rottensteiner, D., van Velthoven, J.T.: Balian-Low type theorems on homogeneous groups. Preprint arXiv:1908.03053 (2019) · Zbl 1463.22007
[24] Gröchenig, K.; Rottensteiner, D., Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups, J. Funct. Anal., 275, 12, 3338-3379 (2018) · Zbl 1400.22009 · doi:10.1016/j.jfa.2018.06.011
[25] Grossmann, A.; Morlet, J.; Paul, T., Transforms associated to square integrable group representations. I. General results, J. Math. Phys., 26, 10, 2473-2479 (1985) · Zbl 0571.22021 · doi:10.1063/1.526761
[26] Heil, C.: Linear independence of finite Gabor systems. In: Harmonic Analysis and Applications, pp. 171-206. Springer, Berlin (2006) · Zbl 1129.42421
[27] Heil, C.; Ramanathan, J.; Topiwala, P., Linear independence of time-frequency translates, Proc. Am. Math. Soc., 124, 9, 2787-2795 (1996) · Zbl 0859.42023 · doi:10.1090/S0002-9939-96-03346-1
[28] Heil, C., Speegle, D.: The HRT conjecture and the zero divisor conjecture for the Heisenberg group. In: Excursions in Harmonic Analysis, vol. 3, pp. 159-176. Springer, Berlin (2015) · Zbl 1415.42026
[29] Hutníková, M.; Hutník, O., An alternative description of Gabor spaces and Gabor-Toeplitz operators, Rep. Math. Phys., 66, 2, 237-250 (2010) · Zbl 1242.42016 · doi:10.1016/S0034-4877(10)80029-1
[30] Jakobsen, M.S., Luef, F.: Duality of Gabor frames and Heisenberg modules. Preprint arXiv:1806.05616 (2018)
[31] Kreisel, M., Letter to the editor: Linear independence of time-frequency shifts up to extreme dilations, J. Fourier Anal. Appl., 25, 6, 3214-3219 (2019) · Zbl 1427.42036 · doi:10.1007/s00041-019-09699-5
[32] Kutyniok, G., Linear independence of time-frequency shifts under a generalized Schrödinger representation, Arch. Math., 78, 2, 135-144 (2002) · Zbl 0997.43006 · doi:10.1007/s00013-002-8227-z
[33] Linnell, P., Von Neumann algebras and linear independence of translates, Proc. Am. Math. Soc., 127, 11, 3269-3277 (1999) · Zbl 0937.46054 · doi:10.1090/S0002-9939-99-05102-3
[34] Luef, F., Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces, J. Funct. Anal., 257, 6, 1921-1946 (2009) · Zbl 1335.46064 · doi:10.1016/j.jfa.2009.06.001
[35] Luef, F., The Balian-Low theorem and noncommutative tori, Expos. Math., 36, 2, 221-227 (2018) · Zbl 1396.46053 · doi:10.1016/j.exmath.2018.03.003
[36] Luef, F., Skrettingland, E.: A Wiener Tauberian theorem for operators and functions. Preprint arXiv:2005.04160 (2020) · Zbl 1444.81024
[37] Nicola, F., Trapasso, S.I.: A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform. Preprint arXiv:1911.12241 (2019) · Zbl 1447.42012
[38] Okoudjou, KA, Extension and restriction principles for the HRT conjecture, J. Fourier Anal. Appl., 25, 4, 1874-1901 (2019) · Zbl 1417.42036 · doi:10.1007/s00041-018-09661-x
[39] Paulsen, VI; Raghupathi, M., An Introduction to the Theory of Reproducing Kernel Hilbert Spaces (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1364.46004 · doi:10.1017/CBO9781316219232
[40] Romero, J.L., van Velthoven, J.T., Voigtlaender, F.: On dual molecules and convolution-dominated operators. Preprint arXiv:2001.09609 (2020)
[41] Wong, MW, Wavelet Transforms and Localization Operators (2002), Berlin: Springer, Berlin · Zbl 1016.42017 · doi:10.1007/978-3-0348-8217-0
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