A Kolmogorov-Szego-Krein type condition for weighted Sobolev spaces Jose RodriguezDmitry Yakubovich An analogue of the Szeg\"o condition for density of analytic polynomials in weighted Sobolev spaces $W^{k,p}$ of the circle with general weights is given. This condition is always sufficient and is close to necessary. In particular, we prove that it is necessary and sufficient if all $k+1$ weights are absolutely continuous and their densities are piecewise monotone. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2507 10.1512/iumj.2005.54.2507 en Indiana Univ. Math. J. 54 (2005) 575 - 598 state-of-the-art mathematics http://iumj.org/access/