We consider the space h_v~∞ of harmonic functions in R_+~(n+1) with finite norm||u||_v = sup |u(x, t)|/v(t), where the weight v satisfies the doubling condition. Boundary values of functions in h_v~∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h_v~∞ ～ l∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h_v~∞ along vertical lines.
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