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# Almost Periodic Functions and Differential Equations by Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov

By Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov

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Hence, the function a(A ; f) = WM) exP ( — in )} is defined for every A E J; a(A. ; f) is called the Bohr transformation of f. The next property is fundamental to the theory of almost periodic functions. Property 2. The function a(A. ; f) is non-zero for at most a countable set of values of A. To prove this property let {Pk (t)} be a sequence of approximating polynomials for which su? Ilf(t)—Pk(t)11--- 1/k (k =1, 2, Suppose that nk Pk(t) = ak, m=1 and {AO U Ak,m. k,m exp • • •)• 24 Harmonic analysis The set {gn} is not more than countable.

Tn))} it follows that the limit (15) is equivalent to (14) for T m ,n = tm — tn. Thus, Theorem 4 is reduced to Theorem 3. Now we make some observations regarding Theorem 4. For an almost periodic function f:J-4X we consider the family ge = k(f) (see Chapter 1, § 3). Each element f(s) E k is obtained as a limit f(s) =limm ,o, f(s + tm ). With each J(s) we associate the function 0(A) = m-,00 lim exp (Wm ) (A E nf), (16) A (s) there corresponds the function 0(A ) exp (iAt), Here, to the shift P that is, a shift on Ye goes into the multiplication of 0(A) by exp (Mt).

We consider the system of inequalities lAkTi< 8 (mod 27r) (k = 1, 2, . . 4k0 7- – 27r/V kol < 8 (mod 27r), where 8 > 0 and N are arbitrary. By Kronecker's theorem, for the system (20) to be solvable it is necessary and sufficient that from every equation Nk lko ktk o = E akiA ;, J=1 (21) where 1ko > 0 and the aki are integers, follows the congruence 2 lko — IT E 0 (mod 27r). / 1 k0 Next we are going to prove that this congruence holds. Every lko in the representation (21) is divisible by a minimal l' ko.