Abstract: The new L^q approximation methods proposed by A.Pinkus and O.Shisha,which maintains the characteristic feature-alternation-of the classic Chebyshev uniform approximation, is extended to varisolvent approximation under the Young's condition.The theorems of existence, characterizations, unicity, de La Vallee Poussin and continuity of the best approximation are presented.

Key words: Varisolvency, Young's condition, Best approximation, Property Z