An algorithm is presented for raising an approximation order of any given or thogonal multiscaling function with the dilation factor a. Let Ф(x) = [φ1(x), φ2 (x), … , φr (x)]Tbe an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function Фnew(x) = [ ФT(x),φr+1(x), φr+2(x),… ,φr+s(x)]T with the approximation order m + L(L ∈ Z+). In other words, we raise the approximation order of multiscaling function Ф(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function Ф(x) = [φ1(x), φ2(x), … , φr (x)]T is symmetric, then the new orthogo nal multiscaling function Фnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given.
YANG Shouzhi & PENG Lizhong Department of Mathematics, Shantou University, Shantou 515063, China
We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+ L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(p＞1) for Ω ∈ Llog+ L(Sn-1 × Sm-1) satisfying the cancellation condition.
WANG Meng, CHEN Jiecheng2 & FAN Dashan Department of Mathematics, Zhejiang University (at Yuquan campus), Hangzhou 310027, China
