Translation, whose first traces date back at least to 3000 BC (Newmark, 1988), has always been considered time-consuming and labor-consuming. In view of this, experts have made numerous efforts to develop some mechanical systems which can reduce part of this time and labor. The advancement of computers in the second half of the twentieth century paved the ground for the invention of machine tra...

The purpose of this paper is to reveal the deep and rich structure of the set of Parseval frame wavelets. Two main directions are pursued. First, we study the reproducing properties of the translates of a Parseval frame wavelet within the closed linear span that the translates generate. In particular, we show that the translates need not have good reproducing properties, even though the transla...