A consideration on scaling functions of wavelets

A scaling function phi is defined by a functional equation with a sequence (h_n). We obtained a theorem concerning necessary and sufficient conditions for a finite sequence (h_n) to give a scaling function with compact support, and presented it in ISITA2006. The result was a variation of the theorem established in 1992, but our subject was a single sequence. In this paper, we propose a theorem easily derived from the former. The new one is convenient to select an appropriate wavelet. Along some examples, we discuss mainly about the continuity of scaling functions.