Sum rules and analyticity in QCD

QCD sum rules of Shifman, Vainstein and Zakharov are investigated from the point of view of analytical extrapolations: in the weighted sum rule the analyticity connects the low energy hadron parameters with the high energy QCD parameters (coupling constant, quark masses, condensate parameters). It is shown that the optimal weight function in the sum rule is prescribed by the errors of the input information. The optimal tests of analyticity are proposed and applied (under some simplifying assumptions) to the vector and pseudoscalar charmonium channels.