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**Additional resources for Almost-Periodic Functions and Functional Equations**

**Example text**

V. p. 2) T-+oo I T + s f(t)e-i~tdt = a(A), -T+s uniformly with respect to s EJ. Observe, first of all, that the function f(t)e-i~t is Riemann-integrable on every interval (- T +s)f--l(T +s), being weakly continuous. , V fixed x*. The mean value therefore exists. 3) followR thc existence of the weak limit a(A) = lim* - 1 T_oo 2T IT f(t)e-t~tdt -T = Jt(f(t)e- iAt ). 1) is therefore proved. Furthermore, Vx*, . 1 hm 2T T_oo IT+s -T+s .

The values {An} for which a(An;f(t))#O are called the characteristic exponents of f(t) and the sequence {An} constitutes the spectrum of f(t); obviously, {An}S{fLn} and setting an = a(An;f(t)) we can associate (for the time being in a purely formal way) to f(t) the Fourier series 00 f(t) '" Ln aneillnt, 1 thus obtaining, for f(t), a decomposition formula which constitutes its harmonic analysis. The an's are called the Fourier coefficients off(t). 9) = 1 2T fT -T f(t)e-illtdt, let us prove the following theorem.

P. P. , it is possible to select from s a subsequence s' ={s~} such that the sequence {llf(t+s~) II} converges uniformly. 25) n~ Vt EJ. 22) and the theorem is proved. The last criterion, which we shall now prove, holds for uniformly convex spaces (or Clarkson spaces). These spaces are defined in the following way. 27) Hence, Vx l , X2 :s; I-w(a). 29) Uniformly convex spaces satisfy condition (b); moreover they are reflexive (and consequently semicomplete). The spaces lv, LV with 1