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1996 | 119 | 1 | 37-64
Tytuł artykułu

Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets

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EN
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EN
We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g ↦ P_{g,ϕ}$, where for a fixed function ϕ, $P_{g,ϕ}$ denotes the one-dimensional orthogonal projection on the function $U_gϕ$, U is a group representation and g is an element of the group. They are defined as integrals $ʃ_W P_{g,ϕ} dg$, where W is an open, relatively compact subset of a group. Our main result is a characterization of function spaces corresponding to local Toeplitz operators with pth power summable eigenvalues, 0 < p ≤ ∞.
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Twórcy
  • Institute of Mathematics, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, nowak@math.uni.wroc.pl
Bibliografia
  • [D] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 6, SIAM, Philadelphia, 1992.
  • [F1] H. G. Feichtinger, Wiener amalgams over Euclidean spaces and some of their applications, in: Lecture Notes in Pure and Appl. Math. 136, K. Jarosz (ed.), Dekker, New York, 1992, 123-137.
  • [F2] H. G. Feichtinger, Generalized amalgams with applications to Fourier transform, Canad. J. Math. 42 (1990), 395-409.
  • [FG] H. G. Feichtinger and K. Gröchenig, Gabor wavelets and the Heisenberg group, in: Wavelets - A Tutorial in Theory and Applications, C. K. Chui (ed.), Academic Press, Boston, 1992, 359-397.
  • [Fo] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, N.J., 1989.
  • [FJW] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conf. Ser. in Math. 79, Providence, R.I., 1991.
  • [JP] S. Janson and J. Peetre, Paracommutators - boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305 (1988), 467-504.
  • [N1] K. Nowak, On Calderón-Toeplitz operators, Monatsh. Math. 116 (1993), 49-72.
  • [N2] K. Nowak, Some eigenvalue estimates for wavelet related Toeplitz operators, Colloq. Math. 65 (1993), 149-156.
  • [N3] K. Nowak, Singular value estimates for certain convolution-product operators, J. Fourier Anal. Appl. 1 (3) (1995), 297-310.
  • [Pe] L. Peng, Paracommutators of Schatten-von Neumann class $S_p$, 0 < p < 1, Math. Scand. 61 (1987), 68-92.
  • [PRW] L. Peng, R. Rochberg and Z. Wu, Orthogonal polynomials and middle Hankel operators, Studia Math. 102 (1992), 57-75.
  • [R1] R. Rochberg, Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators, in: Proc. Sympos. Pure Math. 51, W. B. Arveson and R. G. Douglas (eds.), Amer. Math. Soc., Providence, 1990, 425-444.
  • [R2] R. Rochberg, A correspondence principle for Toeplitz and Calderón-Toeplitz operators, in: Israel Math. Conf. Proc. 5, M. Cwikel et al. (eds.), Bar-Ilan Univ., Ramat Gan, 1992, 229-243.
  • [R3] R. Rochberg, Eigenvalue estimates for Calderón-Toeplitz operators, in: Lecture Notes in Pure and Appl. Math. 136, K. Jarosz (ed.), Dekker, New York, 1992, 345-357.
  • [R4] R. Rochberg, The use of decomposition theorems in the study of operators, in: Wavelets: Mathematics and Applications, J. J. Benedetto and M. Frazier (eds.), CRC Press, Boca Raton, 1994, 547-570.
  • [S] B. Simon, Trace Ideals and Their Applications, Cambridge Univ. Press, London, 1979.
  • [Sl] D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), 379-393.
  • [Z] K. Zhu, Operator Theory in Function Spaces, Dekker, New York, 1990.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv119i1p37bwm
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