By Luigi Prouse, Giovanni, Amerio

ISBN-10: 1475712545

ISBN-13: 9781475712544

ISBN-10: 1475712561

ISBN-13: 9781475712568

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Hence, hjftj can be expressed as a finite combination, with integer coefficients, of the A,,'s, which is absurd. 20) This system has, by Kronecker's theorem, a solution T when we set In this case, in fact, each relation of the form N L" dj"iI" = ljfLj (dj", lj integers) 1 implies lj 21T l' = m j 27T = 0 (mod 21T). J Let us now consider the function g(t) ~ 2:l1) l n sup Ilg(t+T)-g(t)11 = J bnei~nt. 20), we obtain 21T 21T 1'-8:::; fLjT:::; 1'+8 (mod21T) J J and, consequently, taking 8:::;1T(l;, we have 1T f'.

4). The continuity of f(t) is therefore proved. , and, to do this, we shall utilize Bochner's criterion. Let s={sn} be an arbitrary sequence. 5) lim f(7]k+ Sn) = gk. n~oo Let us show that the convergence of the sequence {f(t+s n )} is uniform; the thesis will then follow from Bochner's criterion. Assume that the convergence is not uniform. P. 9) By Hahn-Banach's theorem, there exists a functional z* E X* such that

11_00 T_oo Let P(t) be a trigonometric polynomial such that sup Ilf(t) - P(t) I ~ E. J We have then P(t) and n = Lk bkeivkt + b' eill't 1 WII = IIA't(P(t)e-ill't) I ~ + 11A'{(j(t)e -ill't) I 11A'{(P(t)-f(t))e-ill'tll :0; e. Thereforc, 8ctting Q(t) = P(t) _b'eill't, we obtain 8Up Ilf(t) - Q(t) I J :0; 2E. 24 HARMONIC ANALYSIS OF ALMOST-PERIODIC FUNCTIONS It is then possible to define a sequence of trigonometric polynomials such that I sup IIPn(t)-f(t)11 ~ - n J and such that A' is not a characteristic exponent for P n(t), Vn.

### Almost-Periodic Functions and Functional Equations by Luigi Prouse, Giovanni, Amerio

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