Abstract
Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression, signal processing and statistics, etc. For Calderón-Zygmund operators T, it is interesting to construct estimator of Tf, based on wavelet shrinkage estimator of f. With the help of a representation of operators on wavelets, due to Beylkin et al., an estimator of Tf is presented in this paper. The almost everywhere convergence and norm convergence of the proposed estimators are established.
Similar content being viewed by others
References
Abramovich F, Amato U, Angelini C. On optimality of Bayesian wavelet estimators. Scand J Statist, 2004, 31: 217–234
Beylkin G. On the representation of operators in bases of compactly supported wavelets. SIAM J Numer Anal, 1992, 29: 1716–1740
Beylkin G. Wavelets and fast numerical algorithms. Proc Sympos Appl Math, 1993, 47: 89–117
Beylkin G, Coifman R R, Rokhlin V. Fast wavelet transforms and numerical algorithms I. Comm Pure Appl Math, 1991, 44: 141–183
Chen D R, Meng H T. Convergence of wavelet thresholding estimators of differential operators. Appl Comput Harmon Anal, 2008, 25: 266–275
Chen D R, Zhao Y. Wavelet shrinkage estimators of Hilbert transform. J Approx Theory, 2011, 381: 947–951
Daubechies I. Orthonormal bases of compactly supported wavelets. Comm Pure Appl Math, 1988, 41: 909–996
Daubechies I. Ten Lectures on Wavelets. Philadelphia: SIAM, 1992
Donoho D L. Denoising via soft thresholding. IEEE Trans Inform Theory, 1995, 41: 613–627
Donoho D L, Johnstone I M. Ideal spatial adaption via wavelet shrinkage. Biometrika, 1994, 81: 425–455
Donoho D L, Johnstone I M. Minimax estimation via wavelet shrinkage. Ann Statist, 1998, 26: 879–921
Liu Y, Wang H. Convergence order of wavelet thresholding estimator for differential operators on Besov spaces. Appl Comput Harmon Anal, 2012, 32: 342–356
Meyer Y. Ondelettes et Opérateurs. Paris: Hermann, 1990
Stein E. Sigular integrals and differentiability properties of functions. Princeton: Princeton University Press, 1970
Tao T. On the almost everywhere convergence of wavelet summation methods. Appl Comput Harmon Anal, 1996, 3: 384–387
Tao T, Vidakovic B. Almost everywhere behavior of general wavelet shrinkage operators. Appl Comput Harmon Anal, 2000, 9: 72–82
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, H., Wu, J. Wavelet shrinkage estimators of Calderón-Zygmund operators with odd kernels. Sci. China Math. 57, 1983–1991 (2014). https://doi.org/10.1007/s11425-013-4724-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-013-4724-8