Wavelet transform, Toeplitz type operators and decomposition of functions on the upper half-plane


  • Ding-Xuan Zhou Zhejiang University, China


continuous wavelet transform, function spaces, Toeplitz type operators, Schatten-Von Neumann class, Besov spaces


In this paper we consider the decomposition of functions on the upper half-plane into orthogonal subspaces which are isometric to \(L^{2}(\mathbb{R})\) by continuous wavelet transforms. A necessary and sufficient condition for such a decomposition is given. From such a decomposition by general Laguerre polynomials, we define a series of Toeplitz type operators and study the Schatten-Von Neumann classes of these operators.


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I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inf. Theory, 36 (1990), 961-1005, https://doi.org/10.1109/18.57199

I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure and Appl. Math., 41 (1983), 909-996, https://doi.org/10.1002/cpa.3160410705

A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representation II: examples, Ann. Inst. Henri Poincaré, 45 (1986), 293-309.

S. Janson and J. Peetre, Paracommutator-boundedness and Schatten-operties, Trans. Amer. Math. Soc., 305 (1988), 467-504. http://dx.doi.org/10.1090/S0002-9947-1988-0924766-6

Q. Jiang and L. Peng; Wavelet transform and Hankel-Toeplitz operators; Preprint. https://eudml.org/doc/167201

Q. Jiang and L. Peng, Toeplitz and Hankel type operators on the upper half-plane, Preprint, https://doi.org/10.1007/bf01200698

Y. Meyer, Ondelettes et opérateurs, Hermann (1990).

T. Paul, Functions analytic on the half-plane as quantum mechanical states, J. Math. Phys. 25 (1985), 3252-3263, http://dx.doi.org/10.1063/1.526072

L. Peng, Paracommutator of Schatten-Von Neumann class Sp, 0 < p < 1, Math. Scand., 61 (1987), 68-92, http://www.mscand.dk/article/view/12191/10207

G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publications, Vol. 23 (1939).




How to Cite

Zhou, D.-X. (1992). Wavelet transform, Toeplitz type operators and decomposition of functions on the upper half-plane. Rev. Anal. Numér. Théor. Approx., 21(1), 89–100. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1992-vol21-no1-art13