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

By Luigi Prouse, Giovanni, Amerio

ISBN-10: 1475712545

ISBN-13: 9781475712544

ISBN-10: 1475712561

ISBN-13: 9781475712568

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Extra resources for Almost-Periodic Functions and Functional Equations

Sample text

A(llf(t)n=L~l)n Ilan l1 2. p. functions. 1) 2" b"e jilkt . 2) i" a"eiilkt , 1 where a" is the Fourier coefficient of f(t) corresponding to the exponent A". A(P(t),f(t)) = 2" (b",a,,). n 1 i" Ilb,,1I2. 7) n n 1 1 = A(llf(t)il2)- 2:k Ilak l1 2+ 2:k Ilak -bk I1 2. Therefore,

IX. 1) admits a solution To, '18> 0, if and only if every time that a relation of the type (ql,' .. 3) The condition is necessary. 2) be verified for certain integers ql,' .. 4) mn being integers (depending on 8). It follows that mv' .. 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.

It follows that mv' .. 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" ..

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


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