Localization operators of the wavelet transform associated to the Riemann-Liouville operator. (English) Zbl 1345.44001
Summary: We study the continuous wavelet transform \(T_\psi\) associated with the Riemann-Liouville operator. Next, we investigate the localization operators for \(T_\psi\); in particular we prove that they are in the Schatten-von Neumann class.
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
Riemann-Liouville operator; Fourier transform; continuous wavelet transform; localization operator; Schatten class operatorsReferences:
| [1] | 1. B. Amri and L. T. Rachdi, Beckner logarithmic uncertainty principle for the Riemann-Liouville operator, Internat. J. Math.24(9) (2013), Article ID: 1350070, 29 pp. [Abstract] genRefLink(128, ’S0129167X16500361BIB001’, ’000329392100004’); · Zbl 1282.42007 |
| [2] | 2. L. E. Andersson, On the determination of a function from spherical averages, SIAM. J. Math. Anal.19(1) (1988) 214-232. genRefLink(16, ’S0129167X16500361BIB002’, ’10.1137 |
| [3] | 3. L. E. Andersson and H. Helesten, An inverse method for the processing of synthetic aperture radar data, Inverse Probl.3(1) (1987) 111-124. genRefLink(16, ’S0129167X16500361BIB003’, ’10.1088 |
| [4] | 4. C. Baccar, N. B. Hamadi and L. T. Rachdi, Inversion formulas for the Riemann-Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci.2006 (2006) 1-26. genRefLink(16, ’S0129167X16500361BIB004’, ’10.1155 · Zbl 1131.44002 |
| [5] | 5. C. Baccar, N. B. Hamadi and L. T. Rachdi, Best approximation for Weierstrass transform connected with Riemann-Liouville operator, Commun. Math. Anal.5(1) (2008) 65-83. · Zbl 1163.44002 |
| [6] | 6. C. Baccar and L. T. Rachdi, Spaces of DLp-type and a convolution product associated with the Riemann-Liouville operator, Bull. Math. Anal. Appl.1(3) (2009) 16-41. · Zbl 1312.42028 |
| [7] | 7. P. Balazs, Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inform. Process.6(2) (2008) 315-330. [Abstract] genRefLink(128, ’S0129167X16500361BIB007’, ’000254425300011’); · Zbl 1268.42052 |
| [8] | 8. P. Balazs, D. Bayer and A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A: Math. Theor.45(24) (2012), page 244023. genRefLink(16, ’S0129167X16500361BIB008’, ’10.1088 |
| [9] | 9. W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, Vol 20 (Walter de Gruyter, 1995). genRefLink(16, ’S0129167X16500361BIB009’, ’10.1515 · Zbl 0828.43005 |
| [10] | 10. J. K. Cohen and N. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics44(6) (1979) 1077-1087. genRefLink(16, ’S0129167X16500361BIB010’, ’10.1190 |
| [11] | 11. E. Cordero and K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal.205(1) (2003) 107-131. genRefLink(16, ’S0129167X16500361BIB011’, ’10.1016 |
| [12] | 12. I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory34(4) (1988) 605-612. genRefLink(16, ’S0129167X16500361BIB012’, ’10.1109 |
| [13] | 13. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inf. Theory36(5) (1990) 961-1005. genRefLink(16, ’S0129167X16500361BIB013’, ’10.1109 |
| [14] | 14. I. Daubechies and T. Paul, Time-frequency localisation operators – a geometric phase space approach: II. The use of dilations, Inverse Probl.4(3) (1988) 661-680. genRefLink(16, ’S0129167X16500361BIB014’, ’10.1088 |
| [15] | 15. F. De Mari, H. Feichtinger and K. Nowak, Uniform eigenvalue estimates for time-frequency localization operators, J. London Math. Soc.65(3) (2002) 720-732. genRefLink(16, ’S0129167X16500361BIB015’, ’10.1112 · Zbl 1033.47021 |
| [16] | 16. F. De Mari and K. Nowak, Localization type Berezin-Toeplitz operators on bounded symmetric domains, J. Geom. Anal.12(1) (2002) 9-27. genRefLink(16, ’S0129167X16500361BIB016’, ’10.1007 · Zbl 1040.47017 |
| [17] | 17. J. A. Fawcett, Inversion of N-dimensional spherical averages, SIAM. J. Appl. Math.45(2) (1985) 336-341. genRefLink(16, ’S0129167X16500361BIB017’, ’10.1137 |
| [18] | 18. G. B. Folland, Real Analysis: Modern Techniques and Their Applications (John Wiley & Sons, 2013). · Zbl 0924.28001 |
| [19] | 19. K. Gröchenig, Foundations of Time-Frequency Analysis (Springer Science & Business Media, 2001). genRefLink(16, ’S0129167X16500361BIB019’, ’10.1007 |
| [20] | 20. N. B. Hamadi and L. T. Rachdi, Weyl transforms associated with the Riemann-Liouville operator, Int. J. Math. Math. Sci.2006 (2006) 1-19, Article ID: 94768. genRefLink(16, ’S0129167X16500361BIB020’, ’10.1155 · Zbl 1146.35426 |
| [21] | 21. M. Herberthson, A numerical implementation of an inverse formula for CARABAS raw data, Internal Report D 30430-3.2, National Defense Research Institute, FOA, Linköping, Sweden (1986). |
| [22] | 22. T. H. Koornwinder, The continuous wavelet transform, in Wavelets: An Elementary Treatment of Theory and Applications (World Scientific, Singapore, 1993), pp. 27-48. [Abstract] |
| [23] | 23. R. S. Laugesen, N. Weaver, G. L. Weiss and E. N. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal.12(1) (2002) 89-102. genRefLink(16, ’S0129167X16500361BIB023’, ’10.1007 · Zbl 1043.42032 |
| [24] | 24. N. N. Lebedev, Special Functions and Their Applications (Courier Dover Publications, 1972). · Zbl 0271.33001 |
| [25] | 25. Y. Meyer, Wavelets and Operators (Press Syndicate of University of Cambridge, 1995). |
| [26] | 26. S. Omri and L. Rachdi, Heisenberg-Pauli-Weyl uncertainty principle for the Riemann-Liouville operator, J. Inequal. Pure Appl. Math.9(88) (2008) 23. · Zbl 1159.42305 |
| [27] | 27. J. Ramanathan and P. Topiwala, Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal.24(5) (1993) 1378-1393. genRefLink(16, ’S0129167X16500361BIB027’, ’10.1137 |
| [28] | 28. E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc.83 (1956) 482-492. genRefLink(16, ’S0129167X16500361BIB028’, ’10.1090 · Zbl 0072.32402 |
| [29] | 29. K. Trimèche, Transformation intgrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur (0;+, J. Math. Pures Appl.60 (1981) 51-98. genRefLink(128, ’S0129167X16500361BIB029’, ’A1981LM86700002’); |
| [30] | 30. K. Trimèche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harm. Anal.4 (1997) 97-112. genRefLink(16, ’S0129167X16500361BIB030’, ’10.1006 |
| [31] | 31. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1995). · Zbl 0849.33001 |
| [32] | 32. M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, 2002). genRefLink(16, ’S0129167X16500361BIB032’, ’10.1007 · Zbl 1016.42017 |
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