By Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov
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Extra info for Almost Periodic Functions and Differential Equations
A. This means that the system of inequalities lAkti < 8 (mod 2r) (k = 1, 2, ... , n) has solutions (and moreover arbitrarily large ones) for any real A1, A2,. An and an arbitrary 8> 0; (b) the numbers A 1 , A 2,. , At, are linearly independent (relative to the integers), that is, the equation 1 1 A 1 +12A 2 + + /„An = 0 only 1 It is not difficult to see this by using induction on n. For n =1 and for all p the number 17„,, of terms in the expansion (4) is (p +1), and consequently, the estimate holds.
The reader can find a detailed exposition in Bochner's book . § 2. The first application of the Bochner—Khinchin theorem to the proof of basic theorems in the theory of (numerical) almost periodic functions is due to Bochner , who used this approach to derive Parseval's relation, from which it is comparatively simple to derive the approximation theorem (see, for example, Levitan ). Bochflees proof carries over easily to abstract almost periodic functions with values in a Hilbert space.
TEl 2 It follows from results in Chapter 4, §§2 and 3, that if we presuppose that the almost periods of an almost periodic function are solutions of a system of inequalities of the form (1), then the proof of the approximation theorem is comparatively simple. See also the theorem of Bogolyubov in Chapter 4. 42 Arithmetic properties of almost periods Now we introduce a number of important new concepts which will be required later on. 2. Definition. A non-empty set of numbers is called a module if it is a group under the operation of addition.
Almost Periodic Functions and Differential Equations by Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov