  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.

Classical Analysis and ODEs