By Luigi Prouse, Giovanni, Amerio

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Bn ) takes on its minimum value when bn = an (and only in this case). 2), A(ilf(t) - F(t)il2) = A( Ilf(t) 112) - n Lk IIak l1 2 :2: b1 =a1 ,' . " 0, 1 that is, the series 2:m\ (Bessel's inequality) IIak l1 2 is convergent and satisfies the condition 00 2:k IIak l1 2 ~ A(llf(t)il2). 8) the equality sign holds; the theorem will then be proved. Let FE(t) be an approximation polynomial corresponding to an arbitrary e > 0, with characteristic exponents E {An} and such that sup Ilf(t) - FE(t) I ~ e.

P. for f{t). 24) has solutions for any N •. 23), Ilf{t+T)-f{t)11 ~ Ilf{t+T)-Pm{t+T)11 ~. > 0 and any + IIPm{t+T)-Pm{t)11 + IIPm{t)-f{t)11 Taking m. =n m£, let T, sup Ilf{t+T)-f{t)11 ~ e J and the theorem is proved. T be a solution CHAPTER 3 WEAKLY ALMOST-PERIODIC FUNCTIONS 1. P. FUNCTION Let X be a Banach space and X* the dual space. Let x denote the elements of X, x* the elements of X* (continuous linear functionals on X), Ilxll and Ilx*11 the respective norms. The function, with complex values, defined on X, obtained by applying the functional x* to the vector x, will be indicated by the symbol

P. As can be seen, the hypothesis of boundedness in the theorem of BohlBohr is substituted by the much stricter hypothesis of compactness. This hypothesis cannot, however, as we shall see in an example (Amerio [9]), be substituted in the general case by one of boundedness. }, with Ilxll = suP.. I~.. I. p. (t) = {cos t,' . " ~ cos ~, 0,0,· .. }, f .. (t)1I ::;; n+l' 53 54 THE INTEGRATION OF ALMOST-PERIODIC FUNCTIONS Moreover, F(t) = {Sin ~}. p. Since, in fact, ISin t : . d. }. p. This cannot occur if e < 2.