ABSTRACT
B-splines axe defined as B 3m i(x) := ( * < + * » . 4+i> · · · > 4+m]<(i - a 0+ 1 > where [ · , . . . , ·]* is the m-th divided difference of (t — x )™_1 with respect to the variable t . Note that B 3m^{x) = Nm(2^x — i) (i = 0 , . . . , — m ) , where iVm(x) := m[0,1 , . . . , m\t(t — x )™"1 is the cardinal B-spline of order m, while for i = —m + 1 , . . . , — 1, B 3m t· contains a multiple knot at 0 and for z = 2J — m + 1 , . . . , 2J — 1, a multiple knot at 1. To separate the effects of the two endpoints, it is necessary to restrict to j > jo , where jo is the smallest integer such that 2J’° > 2m — 1. A basis for Vjg j > jo , is then given by
Similarly, a basis for WfQ j > jo? can be found as
fii) The functions xpJk E WfQ (k = — m + 1 , . . . , 2 — m) are called dual to ψϊ E WjJj for dual wavelets) if
Note that on each level the dual scaling functions span the same space as the scaling functions and the dual wavelets span the same space as the wavelets.
