A duality proof of Tchakaloff's theorem
  Tchakaloff's Theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive, compactly supported measure that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^{d}$. Subsequent extensions were obtained by Mysovskikh and by Putinar. We provide new proofs and partial extensions of these results, based on duality techniques utilized by Stochel. We also obtain new uniqueness criteria in the Truncated Complex Moment Problem. 
Raul E. Curto and Lawrence A. Fialkow