Almost Periodic Functions and Differential Equations - download pdf or read online

By Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov

ISBN-10: 0521244072

ISBN-13: 9780521244077

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_ i We set n = Ne and 3 Ilbn,E11. Suppose that system of inequalities (1) with these n and 8. Then N. sup11PE(t +r) — Pe(t)11= E 11bn,e111exP (iAnT) — 1 1 tej T satisfies the n=1 N. <8 E n=1 11bn,e11---A-s. (t) — fit)11'= s, tEl that is, 2 T is an E-almost period of f(t). 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.

An cf and consider the function d = d(c i, c2 , . . , cn)= Mtlilf(t) — Enk=i ck exP (4041 2 1. We call NI d the deviation of the sum n s(t)= E ck exp (itikt) k=1 from the almost periodic function f(t). We are going to find min ciex d(ci, c2,. . , cn). kt) exp (—ittit)} k=1 1=1 n = MICRO, PO» - E (ck, a(t4k; D) k=1 n n E (ck, - E (a(ihk; f), ck)+ k=1 = n k1 Ck) n + E (a(ktk; A aCuk; D) — E (a(uk; f), a(Ak; A k =1 k1 = Harmonic analysis 32 n tk D) = Mtl(fit), PO» — kE 1 (a(lu k; f), a(k; n + E (ck—aCak; D, k=1 Ck — a(/uk; f)).

Gm ). It follows from (27) that the function x(g)= arg G(g)— R • g (28) Arithmetic properties of almost periods 50 is 27r-periodic in each variable. For, X(• • • + 2 /7", • • •) -- x(. ) = arg G(. )— arg G(. ) {gixi+ • • • + gi(x +210+ • • • + +{gix i + • • • + gixi + • • • + gm,xm }= O. 71) is an almost-periodic function with Fourier exponents 5 belonging to E. We can now prove the theorem in the general case by using the approximation theorem. Let e <7r/6 be any positive number and Qs (t) be a trigonometric polynomial with Fourier exponents chosen from those of f(t) (for example, a Bochner—Fejer polynomial) and satisfying the inequality f(t) — k sin E.

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


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