We analyze the internal structure of the multiresolution analyses of L2(Rd) defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define V0 to be the closed linear span of the shifts of the scaling function and W0 that of the shifts of the wavelets. Finally, define V1 to be the dyadic dilation of V0. We characterize the condi...

Surveys have shown radio-loud (RL) quasars constitute $10 \%$-$15 \%$ of the total quasar population. However, it is unknown if fraction (RL quasars/Total quasars) remains consistent among different parameter spaces. This study shows that increases for increasing full width half maximum (FWHM) velocity H$\beta$ and MgII broad emission line. Our data has been obtained from Shen et al. (2011) cat...

In [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556], a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in [B. Han and Z. Shen, J. Fourier Anal. Appl., 11 (2005), 615–637]. Motivated by these two papers, we develop in this article a general theory and a construction meth...