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The generalized growth of entire transcendental functions in terms of polynomial approximation errors in some Banach spaces has been studied by various authors. The main purpose of this paper is to study the harmonic polynomial approximation of entire harmonic functions in space ℝn, n≥3, in certain Banach spaces. Moreover, the generalized type of harmonic functions of slow growth has been characterized in terms of best harmonic polynomial approximation errors. Our results add new aspects for the case of order zero.