Approximation of maps between real algebraic varieties

A nonsingular real algebraic variety Y is said to have the approximation property if for every real algebraic variety X the following holds: if f:X-->Y is a C^inf map that is homotopic to a regular map, then f can be approximated in the C^inf topology by regular maps. In this paper, we characterize the varieties Y with the approximation property. We also characterize the varieties Y with the approximation property combined with a suitable interpolation condition. Some of our results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties.