We consider the space hν∞ of harmonic functions in R+n+1 with finite norm {norm of matrix}u{norm of matrix}ν = sup {pipe}u(x, t){pipe}/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in hν∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces hν∞ ∼ l∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in hν∞ along vertical lines. © 2014 Hebrew University Magnes Press.