Uniform and Tangential Approximation on a Strip by Entire Functions Having Optimal Growth

For a functions f, continuous on a closed strip S and holomorphic on its interior, we discuss the problem of uniform and tangential approximation on S by entire functions with simultaneous estimation of their growth. For the optimization of this growth (depending on the growth of f on S and on the differential properties of f on ∂S) we develop an adequate approximation technique. In this way we obtain sharp new results on uniform and tangential approximation by entire functions of finite order. These sharpen results of H. Kober [Ko1], [Ko2], M. V. Keldysh [Ke], and other authors.