The best approximation[Figure not available: see fulltext.] [in the space L₂(Ω)] of a function f, satisfying a Lipschitz condition with exponent α, 0≤α≤1, with the aid of certain spaces of local functions, dependent on a parameter h, is discussed. We obtain the estimate {Mathematical expression}, where {Mathematical expression} and r = r(x) is the distance of the point x from the boundary of the domain Ω.