On the Represenation of a Function By a Trigonometric Series
On the Represenation of a Function By a Trigonometric Series
Edward Payson Manning
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, 1880, p. 104). Since this is rather condensed, I have expanded it a little in a few places. In particular, I have given a proof for the case where the function becomes infinite, du Bois-Reymond's treat- ment of which Sachse has omitted. tThis will be the case if they will have a form like that given in the theorem, as is shown in the footnote on page 8. X This requires of course that/(a) be integrable in the interval ( — tt, n) . § Shown in Riemann's Math. Werke, pp. 231-34 : Bull, des Sci. M...ath. , Vol. V, 1873, pp. 41-45 ; Picard, Traite d' Analyse, Vol. I, pp. 240-44. 7 3°. Lim F{x + e)+F(z-*)-2F(x) = Q for yalue of ^ e = £ Let us consider first a function f(x) which is finite and continuous and does not have an infinite number of maxima and minima. We must have necessarily for every value of x between — re and -f- rt, ^\ X dafe(P)dp==f(x). (9) Form the expression 0{x) = F(x)— Pda [/(/?) d/3. J IT * — It From the property 2° and from (9) it follows that lim 0(* + e)-20(*)+0(*-g) =lim 4*0 {x) _ Q « = o e 2 e=o s 2 and therefore 0(x)=ZC o -\-C 1 X.
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