Commun. Korean Math. Soc. 2000 Vol. 15, No. 2, 285-308
Hong Oh Kim, Jae Kun Lim Korea Advanced Institute of Science and Technology, Korea Advanced Institute of Science and Technology
Abstract : We generalize bi-orthogonal (non-orthogonal) MRA to frame MRA in which the family of integer translates of a scaling function forms a frame for the initial ladder space $V_0$. We investigate the internal structure of frame MRA and establish the existence of a dual scaling function, and show that, unlike bi-orthogonal MRA, there exists a frame MRA that has no (frame) `wavelet.' Then we prove the existence of a dual wavelet under the assumption of the existence of a wavelet and present easy sufficient conditions for the existence of a wavelet. Finally we give a new proof of an equivalent condition for the translates of a function in ${L^2({\mathbb R})}$ to be a frame of its closed linear span.
