Two-direction refinable functions and two-direction wavelets with dilation factor m

Motivated by cardinal B-spline scaling functions, we introduce the concept of two-direction refinable functions and two-direction wavelets with dilation factor m. We investigate the following two-direction refinement equation:ϕ(x)=∑kpk+ϕ(mx-k)+∑kpk-ϕ(k-mx),where m ⩾ 2 is an integer. Here, two sequences {pk+}k and {pk-}k are called the positive mask and negative mask respectively. Based on the positive mask and negative mask, the conditions that the above equation has compactly distributional solutions are established. The conditions that the above equation has L2-stable solutions are also presented. The support of ϕ(x) is discussed amply. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Based on orthogonal two-direction scaling functions and two-direction wavelets, orthogonal and symmetric multiscaling functions and their multiwavelets can be constructed easily. In the end, a construction example is given.