Multiwavelets--theory and applications
Vasily Strela · 1996
A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \\Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \\Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\\Gamma1 V j = L 2 (R), T 1 j=\\Gamma1 V j = f0g, and translates OE(t \\Gamma k) constitute a basis of V 0 . Then a basis fw jk : w jk = w(2 j t \\Gamma k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t \\Gamma k) form a basis of W 0 , V 1 = V 0 \\Phi W 0 . A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions OE 0 (t); : : : ; OE r\\Gamma1 (t) to generate a basis of V 0 . The vector OE(t) = [OE 0 (t) : : : OE r\\Gamma1 (t)] T satisfies a dilation equation with matrix coefficients C k . Associated with OE(t) is a multiwavelet w(t) = [w 0 (t) : : : w r\\Gamma1 (t)] T . Unlike the scalar case, construction of a multiwave...