We discuss determination of jumps for functions with generalized bounded variation. The questions are motivated by A. Gelb
and E. Tadmor , F. M�ricz  and  and Q. L. Shi and X. L. Shi . Corollary 1 improves the results proved in B. I.
Golubov  and G. Kvernadze .
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.
A characterization formula of an orthonormal multiwavelet with di_erent real dilations and translations for LE2(R) is presented. The result includes the known result on the classical Hardy space H2(R).
By using the critical point theory, the existence of periodic solutions to second order nonlinear p-Laplacian difference equations is obtained. The main approach used is a variational technique and the saddle point theorem. The problem is to solve the existence of periodic solutions of second order nonlinear p-Laplacian difference equations.