Wavelets and wavelet transforms have been studied extensively since the 1980s. It has been shown that the Discrete Wavelet Transform (DWT) can be applied to an enormous number of applications in virtually every branch of science.

The DWT is designed for decomposing and reconstructing infinitely long signals. In practice, we can only deal with finitely long signals. This raises the very important question of handling boundaries. What should be done near the ends?

Several approaches have been proposed to deal with this problem. In this dissertation, we consider the boundary function approach. The idea is to alter the DWT by constructing appropriate boundary functions at each end so that finite length signals can be analyzed accurately. This dissertation contains two sets of new results. One is about smoothness and approximation order properties of boundary functions. The other is about finding boundary functions. The results have been elaborated upon for some specific wavelets.