Invariance of Almost-Orthogonal Systems Between Weighted Spaces: The Non-Compact Support Case

If Q subset of R-d is a cube with center x(Q) and sidelength l(Q), and f : R-d -> C, define f(zQ)(x) equivalent to f((x - x(Q))/l(Q)) ("f adapted to Q"). We show that if {phi((Q))}(Q is an element of D) is any family of functions indexed over the dyadic cubes, satisfying certain weak decay and smoothness conditions, then the set {phi((Q))(zQ)/v(Q)(1/2)}(Q is an element of D) is almost-orthogonal in L-2(v) for one A(infinity) weight v if and only if it is almost-orthogonal in L-2(v) for all A(infinity) weights v. In the special case where every phi((Q)) = psi, a fixed Schwartz function, this universal almost-orthogonality holds if and only if integral psi dx = 0.