By Luigi Prouse, Giovanni, Amerio

ISBN-10: 1475712545

ISBN-13: 9781475712544

ISBN-10: 1475712561

ISBN-13: 9781475712568

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Y;. p. c. = k). 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

If, and only if, from every sequence {In} it is possible to select a subsequence {sn} such that the sequence {f(t+s n )} is uniformly weakly convergent. The sufficiency of this condition is obvious. Let us prove that the condition is necessary. For this purpose, we select, from {In}, a subsequence {sn} such that, Vk, lim eillksn = eiak. 10) lim Qm(f(t+s n)) n_oo = 1 uniformly. 10). p. OBSERVATION II. Let Z be the· topological vector space defined in observation I in § 2. Bochner's criterion can then be expressed in the following way.

3) when 8 -3> O. The condition is sufficient. p. function. 10). For this, we observe that, for certain integers q~, . " q~ and q~, . 12) o. Lk (q~-q~)Bk = 0 (mod 217) 1 and, consequently, (;J .. (~) expfi(t ~Ie q~Ak- ~k q~Bk) J + (~) ... (~) expfi(t ~k q~Ak- ~k q~Bk) J Observe now that the square of the sum of the absolute values of complex numbers having the same argument (mod 217) is not smaller than the sum of the squares of the absolute values themselves. Therefore, once the reductions are effected, the sum of the squares of the absolute values of the single terms of (cp(t))P is not smaller than the sum of the squares of the coefficients of (CP(tl" ..

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

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