Toeplitz operators for the Gabor spherical mean transform. (English) Zbl 1517.94026
Summary: We investigate the Gabor spherical mean transform and demonstrate the quantitative Shapiro dispersion uncertainty principle and the umbrella theorem specifically tailored for this transform. Next, we define the Toeplitz operators \({\mathcal{L}}^{g_1, g_2}_{{\mathcal{S}}}\) in connection with two window functions \(g_1\) and \(g_2\), as well as the symbol \({\mathcal{S}}\). We establish the boundedness and compactness of these operators. Ultimately, we introduce the Schatten-von Neumann class \(S_p\), where \(p \in [1, +\infty]\), and show that the Toeplitz operators are members of the class \(S_p\). Additionally, we provide a proof for a trace formula.
MSC:
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42A45 | Multipliers in one variable harmonic analysis |
| 44A35 | Convolution as an integral transform |
| 47G30 | Pseudodifferential operators |
Keywords:
Fourier multiplier; pseudo-differential operator; Schatten-von Neumann classes; Plancherel’s formulaReferences:
| [1] | Balazs, P., Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolution Inf. Process., 6, 2, 315-330 (2008) · Zbl 1268.42052 · doi:10.1142/S0219691308002379 |
| [2] | Balazs, P.; Bayer, D.; Rahimi, A., Multipliers for continuous frames in Hilbert spaces, J. Phys. A Math. Theor., 45, 24 (2012) · Zbl 1252.46008 · doi:10.1088/1751-8113/45/24/244023 |
| [3] | Bonami, A.; Demange, B.; Jaming, P., Hermite functions and uncertainty principles for the Fourier and the Gabor transforms, Rev Mat lberoamericana., 19, 23-55 (2003) · Zbl 1037.42010 · doi:10.4171/RMI/337 |
| [4] | Cordero, E.; Grochenig, K., Time-frequency analysis of localization operators, J. Funct. Anal., 205, 1, 107-131 (2003) · Zbl 1047.47038 · doi:10.1016/S0022-1236(03)00166-6 |
| [5] | Daubechies, I., Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inf. Theory, 34, 605-612 (1988) · Zbl 0672.42007 · doi:10.1109/18.9761 |
| [6] | Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36, 961-1005 (1990) · Zbl 0738.94004 · doi:10.1109/18.57199 |
| [7] | Daubechies, I.; Paul, T., Time-frequency localization Operators—a geometric phase space approch:2, Inverse Prob., 4, 661-680 (1988) · Zbl 0701.42004 · doi:10.1088/0266-5611/4/3/009 |
| [8] | De Mari, F.; Nowak, K., Localization type Berezin-Toeplitz operators on bounded symmetric domains, J. Geom. Anal., 12, 1, 9-27 (2002) · Zbl 1040.47017 · doi:10.1007/BF02930858 |
| [9] | De Mari, F.; Feichtinger, H.; Nowak, K., Uniform eigenvalue estimates for timefrequency localization operators, J. Lond. Math. Soc., 65, 3, 720-732 (2002) · Zbl 1033.47021 · doi:10.1112/S0024610702003101 |
| [10] | Flandrin, P., Time-Frequency/Time-Scale Analysis (1999), London: Academic Press, London · Zbl 0954.94003 |
| [11] | Folland, GB, Harmonic Analysis in Phase Space (1989), Princeton: Princeton University Press, Princeton · Zbl 0682.43001 · doi:10.1515/9781400882427 |
| [12] | Gabor, D., Theory of communication, J. Inst. Electr. Eng. Part III Radio Commun. Eng., 93, 26, 429-457 (1946) |
| [13] | Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Berlin: Springer, Berlin · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1 |
| [14] | Gröchenig, K.; Zimmermann, G., Hardy’s theorem and the short-time Fourier transform of Schwartz functions, J. Lond. Math. Soc., 54, 1, 64-72 (1996) |
| [15] | Hammami, A., Rachdi, L.T.: Heisenberg type uncertainty principle for the Gabor spherical mean transform. Integral Transform. Special Funct. 1476-8291 (2020) · Zbl 1479.42029 |
| [16] | Malinnikova, E., Orthonormal sequences in \({\cal{L} }^2({\mathbb{R} }^d)\) and time frequency localization, J. Fourier Anal. Appl., 16, 983-1006 (2010) · Zbl 1210.42020 · doi:10.1007/s00041-009-9114-9 |
| [17] | Mallat, SG, A Wavelet Tour of Signal Processing (1999), London: Academic Press, London · Zbl 0998.94510 |
| [18] | Ramanathan, J.; Topiwala, P., Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal., 24, 1378-1393 (1993) · Zbl 0787.94003 · doi:10.1137/0524080 |
| [19] | Saitoh, S., Theory of Reproducing Kernels and its Applications (1988), Harlown: Longman Higher Education, Harlown · Zbl 0652.30003 |
| [20] | Shapiro, HS, A quantitative multivariate version of Shapiro’s theorem for generalized dispersion, J. Multivar. Anal., 4, 49-59 (1964) |
| [21] | Shapiro, H.S.: Uncertainty principles for bases in \({\cal{L}\mathit{}^2({\mathbb{R}}})\). In: Proceedings of the Conference on Harmonic Analysis and Number Theory, CIRM, Marseille-Luminy, October 16-21 (2005) |
| [22] | Simon, B.: Schatten-von Neumann class \(S_p\). In: Trace Ideals and Their Applications, pp. 33-78. American Mathematical Society (2011) |
| [23] | Stein, EM, Interpolation of linear operators, Trans. Am. Math. Soc., 83, 482-492 (1956) · Zbl 0072.32402 · doi:10.1090/S0002-9947-1956-0082586-0 |
| [24] | Wong, MW, Weyl Transform (1998), New York: Springer, New York · Zbl 0908.44002 |
| [25] | Wong, M.W.: Localization Operators. Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul (1999) · Zbl 0945.42025 |
| [26] | 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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