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Let  S h   be a stripe of the width 2 h , symmetric with respect to the real axes ℝ. The aim of this paper is to solve (for sufficiently smooth functions on ℝ) the problem of uniform approximation on ℝ by functions harmonic on  S h   and having optimal growth at infinity (see Theorems 1.3–1.4). These are direct results of Jackson type, where the mentioned growth is expressed in the terms of the growth of functions to be approximated and their structural properties. The analogous approximation problems on R by functions holomorphic on  S h   have been discussed earlier by M. V. Keldish and other authors (see [1]–[4]). The methods used here are modifications and adaptations of those developed in [3]. In addition, a Bernstein type converse result is also presented (see 1.6).

Uniform approximation on the real axes by functions harmonic in a stripe and having optimal growth