  In this paper we construct a wavelet basis in weighted L^2 of Euclidean space
possessing vanishing moments of a fixed order for a general locally finite
positive Borel measure. The approach is based on a clever construction of
Alpert in the case of Lebesgue measure that is appropriately modified to handle
the general measures considered here. We then use this new wavelet basis to
study a two-weight inequality for a general Calder\'on-Zygmund operator on the
real line and show that under suitable natural conditions, including a weaker
energy condition, the operator is bounded from one weighted L^2 space to
another if certain stronger testing conditions hold on polynomials. An example
is provided showing that this result is logically different than existing
results in the literature.
