Analytical and numerical approximation formulas on the Dunkl-type Fock spaces. (English) Zbl 1361.32004
Summary: In this work, we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space \(F_{k}(\mathbb {C}^{d})\). Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for operator \(L:F_{k}(\mathbb {C}^{d})\rightarrow H\), where \(H\) is a Hilbert space. Finally, we come up with some results regarding the Tikhonov regularization problem and the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform \(\mathcal {B}_{k}\). Some numerical applications are given.
MSC:
| 32A15 | Entire functions of several complex variables |
| 32A36 | Bergman spaces of functions in several complex variables |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
Dunkl-type Fock spaces; reproducing kernel; Tikhonov regularization; uncertainty principlesReferences:
| [1] | Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337-404 (1950) · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7 |
| [2] | Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm. Pure Appl. Math. 14, 187-214 (1961) · Zbl 0107.09102 · doi:10.1002/cpa.3160140303 |
| [3] | Ben Said, S., Ørsted, B.: Segal-Bargmann transforms associated with Coxeter groups. Math. Ann. 334, 281-323 (2006) · Zbl 1109.33015 · doi:10.1007/s00208-005-0718-3 |
| [4] | Berger, C.A., Coburn, L.A.: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301, 813-829 (1987) · Zbl 0625.47019 · doi:10.1090/S0002-9947-1987-0882716-4 |
| [5] | Cho, H., Zhu, K.: Fock-Sobolev spaces and their Carleson measures. J. Funct. Anal. 263, 2483-2506 (2012) · Zbl 1264.46017 · doi:10.1016/j.jfa.2012.08.003 |
| [6] | Cholewinski, F.M.: Generalized Fock spaces and associated operators. SIAM J. Math. Anal. 15, 177-202 (1984) · Zbl 0596.46017 · doi:10.1137/0515015 |
| [7] | Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167-183 (1989) · Zbl 0652.33004 · doi:10.1090/S0002-9947-1989-0951883-8 |
| [8] | Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213-1227 (1991) · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8 |
| [9] | Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123-138 (1992) · Zbl 0789.33008 · doi:10.1090/conm/138/1199124 |
| [10] | Folland, G.: Harmonic analysis on phase space. In: Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989) · Zbl 0682.43001 |
| [11] | Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser, Boston (2001) · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1 |
| [12] | de Jeu, M.F.E.: The Dunkl transform. Inv. Math. 113, 147-162 (1993) · Zbl 0789.33007 · doi:10.1007/BF01244305 |
| [13] | Kimeldorf, G.S., Wahba, G.: Some results on Tchebycheffian spline functions. J. Math. Anal. Appl. 33, 82-95 (1971) · Zbl 0201.39702 · doi:10.1016/0022-247X(71)90184-3 |
| [14] | Lapointe, L., Vinet, L.: Exact operator solution of the Calogero- Sutherland model. Comm. Math. Phys. 178, 425-452 (1996) · Zbl 0859.35103 · doi:10.1007/BF02099456 |
| [15] | Matsuura, T., Saitoh, S., Trong, D.D.: Inversion formulas in heat conduction multidimensional spaces. J. Inv. Ill-posed Problems 13, 479-493 (2005) · Zbl 1095.35072 · doi:10.1515/156939405775297452 |
| [16] | Matsuura, T., Saitoh, S.: Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces. Appl. Anal. 85, 901-915 (2006) · Zbl 1101.65116 · doi:10.1080/00036810600643662 |
| [17] | Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519-542 (1998) · Zbl 0908.33005 · doi:10.1007/s002200050307 |
| [18] | Rösler, M.: An uncertainty principle for the Dunkl transform. Bull. Austral. Math. Soc. 59, 353-360 (1999) · Zbl 0939.33012 · doi:10.1017/S0004972700033025 |
| [19] | Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445-463 (1999) · Zbl 0947.33013 · doi:10.1215/S0012-7094-99-09813-7 |
| [20] | Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compos. Math. 85(3), 333-373 (1993) · Zbl 0778.33009 |
| [21] | Saitoh, S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74-78 (1983) · Zbl 0595.46026 · doi:10.1090/S0002-9939-1983-0706514-9 |
| [22] | Saitoh, S.: The Weierstrass transform and an isometry in the heat equation. Appl. Anal. 16, 1-6 (1983) · Zbl 0526.44002 · doi:10.1080/00036818308839454 |
| [23] | Saitoh, S.: Approximate real inversion formulas of the Gaussian convolution. Appl. Anal. 83, 727-733 (2004) · Zbl 1070.44002 · doi:10.1080/00036810410001657198 |
| [24] | Saitoh, S.: Best approximation, Tikhonov regularization and reproducing kernels. Kodai Math. J. 28, 359-367 (2005) · Zbl 1087.65053 · doi:10.2996/kmj/1123767016 |
| [25] | Sifi, M., Soltani, F.: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line. J. Math. Anal. Appl. 270, 92-106 (2002) · Zbl 1012.46033 · doi:10.1016/S0022-247X(02)00052-5 |
| [26] | Soltani, F.: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. Pacific J. Math. 214, 379-397 (2004) · Zbl 1052.33014 · doi:10.2140/pjm.2004.214.379 |
| [27] | Soltani, F.: Inversion formulas in the Dunkl-type heat conduction on ℝd \(\mathbb{R}^d\). Appl. Anal. 84, 541-553 (2005) · Zbl 1084.46019 · doi:10.1080/00036810410001731492 |
| [28] | Soltani, F.: Best approximation formulas for the Dunkl L2-multiplier operators on ℝd \(\mathbb{R}^d\). Rocky Mountain J. Math. 42, 305-328 (2012) · Zbl 1247.42030 · doi:10.1216/RMJ-2012-42-1-305 |
| [29] | Soltani, F.: Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator. Acta Math. Sci. 33B(2), 430-442 (2013) · Zbl 1289.42032 · doi:10.1016/S0252-9602(13)60010-7 |
| [30] | Soltani, F.: Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform on ℝd \(\mathbb{R}^d\). Bull. Austral. Math. Soc. 87, 316-325 (2013) · Zbl 1321.42022 · doi:10.1017/S0004972712000780 |
| [31] | Soltani, F.: A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform. Int. Trans. Spec. Funct. 24(5), 401-409 (2013) · Zbl 1270.42011 · doi:10.1080/10652469.2012.699966 |
| [32] | Soltani, F.: An Lp Heisenberg-Pauli-Weyl uncertainty principle for the Dunkl transform. Konuralp J. Math. 2(1), 1-6 (2014) · Zbl 1306.42017 |
| [33] | Soltani, F.: Dunkl multiplier operators and applications. Int. Trans. Spec. Funct. 25(11), 898-908 (2014) · Zbl 1308.42008 · doi:10.1080/10652469.2014.938650 |
| [34] | Yamada, M., Matsuura, T., Saitoh, S.: Representations of inverse functions by the integral transform with the sign kernel. Frac. Calc. Appl. Anal. 2, 161-168 (2007) · Zbl 1136.30304 |
| [35] | Zhu, K.: Analysis on Fock Spaces. Springer-Verlag, New York (2012) · Zbl 1262.30003 · doi:10.1007/978-1-4419-8801-0 |
| [36] | Zhu, K.: Uncertainty principles for the Fock space. Preprint 2015, arXiv:1501.02754V1 · Zbl 1499.30333 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
