Multidimensional Version of Lagrange's Problem on Mean Motion

The famous mean motion problem which goes back to Lagrange as follows: to prove that any exponential polynomial with exponents on the imaginary axis has an average speed for the amplitude, whenever the variable moves along a horizontal line. It was completely proved by B. Jessen and H. Tornehave in Acta Math.77, 1945. Here we give its multidimensional version.