A Version of Lagrange's Theorem for Some Classes of Functions of Many Variables

Abstract

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. Actually, this result is a consequence of almost periodicity
in Weyl's sense of amplitude increments over segments of the
length 1. Here we consider the problem for some classes of almost
periodic functions of several variables.

Keywords

• exponential polynomial, Lagrange's conjecture, mean motion, Weyl's almost periodic function

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20. Weyl H., Mean Motion I, Amer.J.Math., 60 (1938), 889–896. Sergey Yu. Favorov, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : [e-mail] Natalya P. Girya, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : n [e-mail]