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We establish the estimations of best approximations of elements of a Banach space by exponential type vectors associated with the positive operator. The corresponding estimations are expressed in terms of quasi-norms of the approximation spaces as Bernstein-Jackson-type inequalities. Such inequalities are applied to spectral approximations in the case of the positive operator with the point spectrum.doi:10.12732/ijpam.v112i4.10 fatcat:rmfyadfp4ze2vpespq4pdgytuy