We prove that the conjugate convolution operators can be used to calculate jumps for functions. Our results generalize the theorems established by He and Shi. Furthermore, by using Lukács and Móricz's idea, we solve an open question posed by Shi and Hu.
[1] Chui, C. K. 1992 An Introduction to Wavelets Academic Press Boston.
[2] Daubechies, I. 1992 Ten Lectures on Wavelets CBMS-NSF Series in Applied Math 61 SIAM Philadephia .
[3] Fejér, L. 1913 Über die Bestimmung des Sprunges einer Funktion aus Ihrer Fourierreihe J. Reine Angew. Math. 142 165–188 .
[4] Gelb, A. Tadmor, E. 1999 Detection of edges in spectral data Appl. Comput. Harmon. Anal. 7 101–135 .
[5] Golubov, B. I. 1975 Determination of jump of function of bounded variation by its Fourier series Math. Notes 12 444–449.
[6] He, Z. T. and Shi, X. L., Determination of jumps for functions via conjugate convolution operators, Acta Mat. Sci., to appear.
[7] Hu, L. Shi, X. L. 2007 Determination of jumps for functions with generalized bounded variation Acta Math. Hungar. 116 89–103 .
[8] Kvernadge, G. 1998 Determination of jumps of a bounded function by its Fourier series J. Approx. Theory 92 167–190 .
[9] Lukács, F. 1920 Über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierrieihe J. Reine Angew. Math. 150 107–112 .
[10] Móricz, F. 2003 Determination of jumps in terms of Abel–Poisson means Acta Math. Hungar. 98 259–262 .
[11] Móricz, F. 2003 Ferenc Lukács type theorems in terms of the Abel–Poisson mean of conjugate series Proc. Amer. Math. Soc. 131 1243–1250 .
[12] Shi, Q. L. Shi, X. L. 2006 Determination of jumps in terms of spectral data Acta Math. Hungar. 110 193–206 .
[13] Shi, X. L. Hu, L. 2009 Determination of jumps for functions based on Malvar–Coifman–Meyer conjugate wavelets Science in China 52 443–456 .
[14] Shi, X. L. Zhang, H. Y. 2009 Determination of jumps via advanced concentration factors Appl. Comput. Harmon. Anal. 26 1–13 .
[15] Shi, X. L. Zhang, H. Y. 2010 Improvement of convergence rate for Móricz process Acta Sci. Math. (Szeged) 76 471–486.
[16] Zhou, P. Zhou, S. P. 2008 More on determination of jumps Acta Math. Hungar. 118 41–52 .
[17] Zhou, Y. Y. Shi, X. L. 2009 Determination of jumps for functions via derivative Gabor series Appl. Math. J. Chinese Univ. 24 191–199 .