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LIBRARY 
 
 OF THE 
 
 University of California. 
 
 GIFT OF 
 
On the Representation of a Function by a 
 Trigonometric Series. 
 
 DISSERTATION 
 
 PRESENTED TO THE BOARD OF UNIVERSITY STUDIES OF THE 
 
 JOHNS HOPKINS UNIVERSITY FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY 
 
 EDWARD !P. MANNING. 
 
 ii 
 
 BALTIMORE, 1894. 
 
 nr 
 
 PRESS OF 
 
 THE FRIEDENWALD COMPANY, 
 
 BALTIMORE. 
 
-M3 
 
PREFACE. 
 
 In this thesis I have considered the representation, in trigonometric series, 
 of a function of a real variable only. 
 
 Since this subject has already been so fully treated I could hardly expect 
 to obtain many new results. Accordingly it has been my special aim, after 
 carefully examining the work of others, to investigate independently certain 
 phases of the question, and thereby to obtain if possible a simpler development 
 of the subject. To some extent my efforts have been rewarded, since, in several 
 cases, I have arrived at results by methods considerably shorter than those 
 which others have employed. 
 
 The first two sections of this paper are introductory. Section 1 is purely 
 historical, giving a brief account of the origin of the question and an outline of 
 the principal work done upon it up to the present time.* In section 2 is pre- 
 sented the theorem of du Bois-Eeymond which gives directly the form of the 
 coefficients in the trigonometric series and proves for all cases the uniqueness 
 of the development. 
 
 In the next two sections I have shown that the convergence of this develop- 
 ment of an integrable function / (x) to /(#„), for x = x , depends only upon the 
 behavior of the function in the vicinity of x . This is followed in sections 5 
 and 6 by a proof that the series thus converges tof(x ): 
 
 1°. When f(x), in the neighborhood of x 0> is finite, possesses only a finite 
 number of discontinuities and only a finite number of maxima and minima. 
 
 2°. When it satisfies the very general condition 
 
 lim r -|/(*o±2 r )-,/> t ±o) | 
 
 <=oJo y 
 
 This condition I have compared with other conditions and have shown its 
 application to an important class of functions in modern analysis, viz., to func- 
 tions having an infinite number of maxima and minima, or an infinite number 
 of discontinuities in a finite region. In section 7 I have given an illustration 
 of a function of this kind whose trigonometric development diverges for one 
 
 *For a more complete historical treatment see the papers of Riemann (Math. Werke, 
 p. 213 ; found also in Bulletin des Sci. Math., Vol. V, 1873, p. 20), Sachse (Bulletin des Sci. 
 Math., 1880, p. 43), and Gibson (Proceedings of the Edinburgh Math. Society, Vol. XI, p. 137.) 
 
 111.2 
 
value of the variable. Finally, in the last section, I have discussed the nature 
 of the convergence of the series, treating especially the question of uniform con- 
 vergence. 
 
 I desire here to express my acknowledgment to Professor Craig, who sug- 
 gested to me the subject of this thesis and who has had general supervision 
 over its preparation, and also to Professor Franklin, who, by his advice and 
 suggestions, has afforded me valuable assistance on several points. 
 
 Baltimore, April 3, 1894. 
 
CONTENTS. 
 
 Section Page 
 
 1. Historical account of the origin and development of the question, . 1 
 
 2. Du Bois-Reymond's theorem which gives the form of the coefficients 
 
 in the trigonometric series and shows that the development is 
 unique, ........... 6 
 
 3. The question of the convergence of the series, leading to Dirichlet's 
 
 integral, . . . . . . . . . . .12 
 
 4. Proof that the convergence of the series to the value of the function, 
 
 for any particular value of the variable, depends only upon the 
 nature of the function in the vicinity of that value, . . . 14 
 
 5. Proof that the development is applicable to functions satisfying Dirich- 
 
 let's conditions, 16 
 
 6. Derivation of the condition 
 
 lim f 'lpfr) — y(+Q)l dy _ o, 
 
 lo r 
 
 and illustrations of its applicability, . 
 
 7. Example of a function not developable at every point, 
 
 8 . Discussion of the nature of the convergence of the series, 
 Biographical Sketch, 
 
 18 
 21 
 24 
 27 
 
ON THE REPRESENTATION OF A FUNCTION BY A 
 TRIGONOMETRIC SERIES. 
 
 1. The question as to the possibility of representing an arbitrary function 
 of a real variable by means of a trigonometric series, first suggested nearly a 
 century and a half ago, has received considerable attention. Much has been 
 written upon it, for it is important, not only in pure analysis, but also in the 
 practical applications of mathematics to problems in physics and astronomy. 
 
 The question arose from a comparison of two different forms of solution 
 of the partial differential equation for a vibrating chord, viz : 
 
 a< 2 — c dx 2 [ } 
 
 Integrals of this equation were obtained in the form of a general solution 
 containing an arbitrary function and in the form of a trigonometric series. 
 The general solution, subjected to the condition that the extremities of the 
 chord are at rest, becomes 
 
 ?==/(« + C*)— f(ct — »), 
 
 where f is an arbitrary function having for a period the double length of the 
 chord. This general solution was first obtained by d'Alembert,* who, however, 
 supposed that,/ was of such a nature that it could always be represented by a 
 continuous curve, this being the idea conveyed to mathematicians of that time 
 by the expression "arbitrary function." But Euler, who in the following year 
 took up the question, f recognized the fact that /"could be perfectly arbitrary 
 and said that d'Alembert had imposed an unnecessary restriction .J 
 
 In the same article, as a special case, Euler gave, for the first time, a solu- 
 tion in the form of a trigonometric series. § He showed that if the initial 
 position was expressed by 
 
 . tzx , n . 2ttx , . 3nx , 
 y=asin— +0sm— + r sin — + . . . , 
 
 *Memoires de VAcad. de Berlin, 1747, p. 214. t Ibid., 1748, p. 69. 
 
 %Ibid., 1748, p. 70. (See also Mimoires, 1753, §§ III, V, IX, pp. 197-200.) In this 
 way the broader conception of an arbitrary function which we have at the present time 
 was first introduced. 
 
 §M6moire& de VAcad. de Berlin, 1748, p. 85. 
 
where a is the length of the chord and a, /9, y , . . . are any constants, the form 
 of the chord after a time t would be given by the equation 
 
 . 7tx itv , n , 2rcx 2tcv . . 3jtx Sttv , ,_. 
 
 v = «sm cos f- P Sln cos r T Sln cos !-•••> (2) 
 
 CI CL €L CL Cb CL 
 
 where v is a constant multiplied by t. (It is readily seen that if v = ct, (2) is a 
 solution of the differential equation (1).) This solution Bernoulli, who next 
 took up the subject, claimed was perfectly general,* and later Lagrange, from a 
 comparison of the two different forms of solution, was led to believet that a 
 given analytical function could always be represented by a trigonometric series. 
 
 Not much progress was made in the discussion of this question until the 
 time of Fourier, who in 1807 made to the French Academy of Sciences the 
 announcement : 
 
 Every arbitrary function of a variable, whether simple or composed of any 
 number of different parts defined by different laws, can be represented by a trigo- 
 nometric series. 
 
 The well known method by which Fourier obtained the coefficients a n , b n 
 of the trigonometric development 
 
 f(x) = X ( a n sin nx -f- b n cos nx) 
 
 n = 
 
 was by multiplying this equation by sin nx and cos nx respectively and inte- 
 grating from — rt to tt.% In this way is obtained readily 
 
 1 [ n i p If"' 
 
 6 =r— - \f[d)da; a n =. — /(«) sin na da; b n =z — /(«) cos na da. (3) 
 
 This method, however, is not due to Fourier. His merit lies principally in the 
 fact that he was the first to recognize the possibility of representing a completely 
 arbitrary function by a trigonometric series. § 
 
 Fourier did not prove generally that the series obtained converged to the 
 value of the function. He showed, however, in several examples that this was 
 so, and he considered that in the actual application of the development to any 
 particular case the proof of the convergence would be easy. 
 
 The first important contribution to the theory of trigonometric series after 
 Fourier's work upon this subject, was Dirichlet's memoir, published in Crelle's 
 
 * Memoirea de VAcad. de Berlin, 1753, p. 157, § XIII. 
 
 t Miscellanea Taurinensia, Vol. Ill, Pars Math., p. 221, Art. XXV. 
 
 % Fourier first obtained the coefficients by integrating from to * {Theory of Heat, 
 Arts. 219-221). Afterwards he showed that they might be obtained by integrating from 
 — 7T to n (Art. 231). 
 
 §See in regard to this Arnold Sachse's memoir, Bulletin des 8ci. Math., 1880, p. 47. 
 
Journal in 1829.* In this memoir he demonstrated rigorously for the first 
 time the possibility of representing by trigonometric series functions which, in 
 the interval under consideration, fulfil the three conditions, known as Dirich- 
 let's conditions, of being finite, possessing only a finite number of points of dis- 
 continuity and only a finite number of maxima and minima. His demonstra- 
 tion was based upon the proof that the following two equations 
 
 n*^^fto^*3P=* [0<,<W1] (4) 
 
 are satisfied whenever f fulfils the above conditions. His demonstration 
 showed that whenever / satisfies these two equations the trigonometric develop- 
 ment is applicable. A few years later (1837) he showedf that the function 
 might become infinite for isolated points between zero and h provided that the 
 
 integral ns 
 
 /(/?)#= JF09) 
 
 Jo 
 remain finite and continuous as /9 varies from zero to h. 
 
 The next important memoir published on this subject was that of Lip- 
 schitz which appeared in Crelle's Journal in 1864.J In this paper he made an 
 important extension to the class of functions which satisfy equations (4), and to 
 which therefore Fourier's development can be applied. He presented a theorem§ 
 which was, in effect, as follows : 
 
 If the function f (ft) is of such a nature that 
 
 l/(/5)l<-4 
 
 = 
 
 g<P<h 
 
 0<g<h< 
 
 Tl 
 
 tmi£i±J^mi <B [,<*<»], (5). 
 
 where A, B , a denote finite positive constants, then 
 
 fe=oo Jflr sin/9 r 
 equals -~-/(0) or zero according as g>0. 
 
 The theorem thus stated is not quite accurate, however, since simple con- 
 tinuity of the function at the lower limit of the integral is not sufficient, as is 
 shown by the illustration given on page 21 of this paper. In fact, Lipschitz's 
 
 * Vol. IV, p. 157 ; Werke, Vol. I, p. 117. 
 t Crelle's Journal, XVII, p. 54 ; Werke, Vol. I, p. 305. 
 
 % Vol. LXIII, p. 296. >^"\"« '* * *^S 
 
 §P. 301. jT *' *** 
 
 f I 
 
demonstration requires that the difference/ (0 -4- d) — /(0) when d tends to zero 
 approach zero with a certain degree of rapidity,* a point which he apparently 
 overlooked. This defect, however, is removed at once by making the condition 
 (5) include the limits. f The class of functions satisfying this condition includes 
 many functions having an infinite number of maxima and minima which would 
 be excluded by Dirichlet's conditions. 
 
 In 1867 Riemann's well known work upon trigonometric series was pub- 
 lished.;}: After an historical sketch he considered the question : What must be 
 the properties of a function which is supposed to be already represented by 
 Fourier's series ? From this point of view Riemann arrived at the important 
 conclusion§ that the convergence of the series for any particular value of the 
 variable depends only upon the behavior of the function in the vicinity of that 
 value. 
 
 Attention was now directed more particularly to the nature of the con- 
 vergence of Fourier's series, and to the question as to whether the development 
 was unique. Heine first demonstrated in 1870|| that a function satisfying 
 Dirichlet's conditions possesses a development uniformly convergent in each 
 interval comprised within the interval ( — tt, tt) and containing no point of 
 discontinuity for the function. He also showed that there can be only one such 
 development by showing that a uniformly convergent development representing 
 zero except for a finite number of points cannot exist, that each coefficient must 
 be identically zero. 
 
 Shortly afterwards Cantor proved** the more general theorem that any 
 trigonometric series of the form 
 
 n =co 
 
 2 (c n sin ruc-f- d n cos nx), (6) 
 
 n =0 
 
 * See the first two of his inequalities on the top of page 307. For his proof it is 
 necessary that lim ^i^ log 2m = . 
 
 a+u a+ /* 1 (2rmr\o. 
 
 fit can then be easily shown that lim — g- log 2m = 0. For — g — <-g B\ % 1 
 
 and, if we write */k = m + ff [0 < <r < 1] , we have 
 
 t ( m \ a i i- logm log m 
 
 hm Ufc) log * = lim pr + > zma + a y = ^-^T = ° • 
 
 \ m > 
 
 % Abhandlungen der Getelhchaft der WissenscJiaft zu Oottingen, 1867, Math. Glasse, p. 87. 
 It is also published in Riemann's Werke, p. 213, and Bulletin des Sci. Math., 1873, Vol. V, 
 p. 20. Although not published until 1867, it had been written for some time, having been 
 presented by Riemann in 1854 for admission to the Faculty of Philosophy at the Uni- 
 versity of Gottingen. 
 
 § Werke, p. 239 ; Bulletin des Sci. Math., 1873, Vol. V, p. 82. 
 
 | Crelle's Journal, LXXI, p. 353, §§ 7, 8, 9. 
 
 **Crelle's Journal, LXXII, p. 139; LXXIII, p. 294. 
 
5 (' 
 
 S 
 
 convergent and representing zero, except for a finite number of values of x , 
 cannot exist. This theorem Cantor soon extended* still further to the case 
 where the series represents zero except for values of x corresponding to the 
 points of a system of points P of the u th speciesf comprised within the interval. 
 Finally in 1875 du Bois-Reymond| completed this part of the theory very 
 satisfactorily by showing that whenever a function / can be represented by a 
 series of the form (6), the coefficients must always have the definite form (3), thus 
 showing that the development is unique. 
 
 Some other interesting results, which I will briefly mention, have been 
 obtained more recently in other directions. In the Comptes rendus for 1881, 
 p. 228, M. Camille Jordan has shown that functions having limited oscillation 
 satisfy Dirichlet's equations (4) and hence are developable in Fourier's series. 
 He also showed by an example that the class of functions possessing the property 
 of limited oscillation includes some functions having an infinite number of dis- 
 continuities scattered all along over a finite interval. 
 
 In the same volume of the Comptes rendus,% du Bois-Reymond gives an 
 account of the results of his researches upon integrals similar to Dirichlet's, but 
 more general. He obtained as a sufficient condition that equations similar to 
 (4) be satisfied, 
 
 limf e ^-mod[/(«)-/(0)] = 0. 
 
 e = 0J() a 
 
 Du Bois-Reymond has also investigated functions of the form 
 
 cos <p (x) 
 
 pW* 
 
 where p (x) and <p (x) tend to infinity as x tends to zero, /> (x) being always posi- 
 tive. He has shown that there are functions of this kind which cannot be 
 represented by Fourier's series at the point x = . 
 
 Holder, Kronecker, Weierstrass and some others have also written upon 
 the subject, but it is not necessary to dwell upon their work here. Kronecker || 
 
 * Malhematische Annalen, V, p. 123 ; Acta Mathematica, II, p. 336. 
 
 t If P comprises an infinite number of points, then there must be one or more points, 
 called point limits, in the neighborhood of which there are an infinite number of the 
 points of P. The aggregate of these point-limits is called the first derived system of P 
 and is denoted by P. The first derived system of P is called the second derived system 
 of P and is denoted by P", and so on. If P^ is the last derived system which P admits, 
 i. e., if P^ comprises only a finite number of points, P is said to be of the v th - species. 
 
 %Beweis class die Coefflcienten, etc., Abhandl. d. k. bayer Akad. d. W., 1875, Vol. XII, 
 pp. 117-167. 
 
 § Pages 915, 962; see also his memoir, TJntersuchungen uber die Convergenz und Diver- 
 gem der Fourierschen Darstellungsformeln, Abhandl. d. k. bayer Akad., Vol. XII, p. 1. 
 
 || Sitzungsberichte der Ak. der W. zu Berlin, 1885, p. 641. 
 
6 
 
 obtained in a number of different forms, conditions that equations (4) be satisfied, 
 and in the course of his paper showed that his forms included many of the con- 
 ditions which had already been given by others. 
 
 2. In considering the development of a function f(x) in trigonometric series 
 going according to the sines and cosines of increasing multiples of re, it is 
 natural to consider first what form, if f{x) is thus developable, the coefficients 
 of the sines and cosines will take. With this question naturally arises another, 
 viz : Can the coefficients take more than one form ? i. e. Is the development 
 unique ? As these questions are answered very satisfactorily by an important 
 theorem published by du Bois-Reymond in 1875,* I will give that theorem here. 
 
 In whatever manner a function f (x) can be developed in the series 
 
 f{x) = X («» sin nx -f- b n cos nx) , (7) 
 
 n = 
 
 holding within the interval ( — 7r, tt) } whose coefficients a n) b n become infinitely smallf 
 with — , the coefficients have always the form 
 
 If" 1 r" If" 
 
 b = ^- I f{a) da ; a n z=: — / (a) sin na da ; b n =z — / (a) cos na da , 
 
 &X) — n K J — jr 7T J— 7T 
 
 provided that the integrals have sense.% 
 
 The proof of this theorem is as follows: 
 
 Form the function 
 
 rpr \ i x 2 "^T a n sin nx 4- b n cos nx /Q \ 
 
 F(x) = b Q — — 2 22 J_» (8) 
 
 Li n = 1 IV 
 
 derived from (7) by two successive integrations. The function F(x) possesses 
 the following properties :§ 
 
 1°. It is uniformly convergent for every value of #. 
 
 2°. lim F ( x + £ ) + F & — £ ) — 21?0c) — f^ x ) except for values of x which 
 
 e = £~ 
 
 make the series (7) divergent or discontinuous. 
 
 * Abhandl. der k. bayer Akad. d. W., Vol. XII, 1875, pp. 117-167, Beweis dass die Co- 
 efficienten, etc. As I have not had access to du Bois-Reymond' s memoir, I follow here 
 Sachse's presentation of the proof {Bulletin des Sei. Math., 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. Math., 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. 
 
 Hence 
 
 F{x) = [*d« f>(/9) dp + c + ^r. (10) 
 
 » — ir « — it 
 
 If now, according to the method employed by Fourier, we multiply (8) by 
 sin nx and cos nx successively, and integrate from — rz to -\-tt ,we will get 
 
 IF* 1 
 
 flT . yjS Ctt 
 
 F(a) da =: -K- &o> -^(°0 s^ 11 wa ^a = 
 
 J — It " J_„- 
 
 J>(«) cos n« da = -^f +(-!)» 
 
 In these equations substitute for F(a) its value given in (10) and integrate 
 by parts. 
 
 [V (a;) <&> = [* {^da[f{P)dp+Co+<w\dx', f "da fda f/(/9)dj8 
 
 J T J — JT V J IT J IT J J — IT J — It J — It 
 
 = x [da |>(/3) d/9l- Hwte [>(/?) d^ 
 
 J — it J — 7r — ' — it J — it J — it 
 
 = ^j>^)dp]^-^|V(a)d«-^J>(/9)d/9]^+ \\^f{x) dx 
 = ~2){ 71 — a ff ( a ) da > I (°o + G &) dx = 2ttc . 
 
 \[{n-aff{a)da + 2r:c,= ^-b,. (12) 
 
 Hence 
 
 * Integrating by parts, it is readily seen that 
 
 [ n x* 2k f x H 
 
 -5- cos na; da: = (— l) n — r , — sin na; da; = , 
 
 J-1T Z n J — * 2 
 
j F(a) sin na da = I sin na d« \ da /(£) dp -f- c + cy/ > 
 
 = { i- cos na [| d« J> (/?) ^ + c + *«] } * 
 
 + -i-j ow na [ J/(0) d£ + Cl J da 
 = (:= ^T 1 & fc»^+ ( ~ 1)n r 27rCl -^f/Wrinnaci, 
 
 " J— IT -1— 7T " , J— IT 
 
 T" r / ( a ) sin wa da . 
 
 Hence 
 
 Therefore 
 
 (_^n + l ^ |y ( ^ ^ _ |*^ w ^ + 2 ^-| _ _^^ ^ nada = __W. (13) 
 
 .F(a) cos nadaz= cos na da *! da 1/ (/3) d,3 -f~ Co -f- cyz > 
 = { ~n~ Sin na U^" \l^ ^ + c o + c i«] } _ 
 
 — { W C ° S na L J/f ^ ^ + C l] } _ — ^2-j /(«) C 08 W « d «' 
 
 ^ j/j a) da - -w\h a) cos wa da = - h f + ( - i)n IF • ( 14 ) 
 
 From the three equations (12), (13), (14) we can readily obtain the values 
 of c , Ci, 6 , a n ,b n , by remembering that these equations are true for all integer 
 values of n greater than zero, and that a n) b n and the integrals 
 
 |V fir 
 
 /(a) sin na da, f(a) cos na da 
 
 J — It J — IT 
 
 tend to zero with * Thus from equation (14) by making n tend to infinity 
 
 *That these integrals tend to zero whenever/(a:) is integrable is proved by the same 
 
 method as the theorem given on page 14. In the integral f(a) sin na da, f{a) takes 
 
 the place of the function • in the proof of that theorem. For the integral 
 
 /(«) cos nada it is only necessary in that proof to make the subdivision of the interval 
 J-* 7T 
 
 ( — 7T, »r) in such a way that each small interval equals #~and the points of subdivision 
 are integer multiples of -g . 
 
we see first that 
 
 b °=itr\h a)da ' 
 
 and that then we obtain the value of b n for any value of n. The results which 
 we obtain from the three equations are 
 
 fc=4^j/ («)["§ ( 7r — a f~\ da > *=j£/W(*— *)«ki 
 
 If" i pf, if"' 
 
 6 =: -^-^/(ajda, a M = — I /'(«) sin nada, b n = — \ f (a) cos na da . 
 
 If now, however, supposing that f (x) is finite, we subject it to the single 
 additional condition of being integrable, the above proof that (P (x) = c -\- c x x 
 will not hold. We must make another investigation for this case. f{x) will 
 not have necessarily at each point a determinate limit, but its value will lie 
 between a superior and an inferior limit. Let us designate the half-sum of 
 these two limits by S (x) , the half-difference by D (x). We can evidently give 
 to the function the form S (x) -f jD (x) , where j denotes a real number lying 
 between — 1 and -j- 1 . Let us now, according to the method employed by 
 
 Biemann, find an expression for lim — -^-^ . Put 
 
 e = 
 
 X («n sin nx -f- b n cos nx) — S (x) -j- a \ • 
 »=o 
 
 Taking 3 an arbitrary small quantity, we can fiud a value of m such that 
 | a m | < D (x) -|- d . We can now write* 
 
 J 2 F{x) -,., 
 .2 =/» + 
 
 sin (n — 1) - ( 
 
 sin n -jr- 
 
 2^ 
 
 Taking e sufficiently small to have m -^ <[ tt , let us divide this series into 
 
 three parts. In the first let n increase from 1 torn; in the second from m -\- 1 
 
 to s, the greatest integer in -= — , and in the third from s -j- 1 to infinity. The 
 
 first part will tend to zero as e diminishes. The second part will be less in 
 absolute value than 
 
 * Riemann, Math. Werke, p. 223 ; Bui. des 
 d' Analyse, Vol. I, p. 241. 
 
 4. Math., 1873, p. 43; Picard, Traiti 
 
10 
 The third part will be less in absolute value than 
 
 Now passing to the limit we see that the first of these expressions will tend to 
 D (x) -f 8 , and the second to (* + — ) (D (x) -f 8) . Hence 
 
 ta^^^/sf(«)+i(i+-y + -i-)p>W+fl [— i<i<i]. 
 
 Putting 
 
 F 1 (x)z=\ X da\f^)d^ 
 
 • — IT J — It 
 
 we have that 
 
 lim -^-' = S(z) +j x D (x) [- 1 <j, < 1] . 
 
 But (x) = F (x) — F x (x) , and hence, neglecting the arbitrary small quantity 
 8 , the modulus of the greatest value which lim ^ can take is 
 
 e = £ 
 
 (a + ^r + T>W- 
 
 We can now, by making use of the condition that/" (x) is integrable, show 
 that <P (x) is a linear function of x . Let us divide the interval (x , x -f- a) into 
 smaller intervals limited by the points x , # -f- ^ , x + ^i + <^ > • • • > # "h ^i 
 -{-... -f<5 ft _i , a; -}- a and form the sum 
 
 d l D(z+ Pl dJ + d t D{z + 3 l + pA) + ... + 8J)(x + 8i + l+.--+pM, (15) 
 
 where the quantities p are positive fractions. This sum, in virtue of the con- 
 dition of integrability, must tend to zero with 8. Hence when the <5's become 
 infinitely small and e tends to zero, the limit of 
 
 J>[d 1 0(x + p 1 d 1 ) + d 9 0{x + 3 1 + pA)+> • .+8 n 4>{x + 8 1 + • - -+ftA)] 
 
 ? ' 
 
 which in absolute value cannot exceed the sum (15) multiplied by f 2 -f- —$- -\ J, 
 
 must be zero. That is, we must have 
 
 a 0(x + a)da = Q. 
 
 lim — y 
 
 e = £ 
 
 It follows therefore that 
 
 \ (x -\- a) da= c„ -\- CxX . ^V 
 
 Jo 
 
11 / 
 
 But since F(x) and F x (x) are continuous functions of a; ,* <P{x), = F(x%£~F x (x) 
 is continuous also. Hence, employing the theorem of means, we can write 
 
 <P(x + a)da = a0(x + aJ [0<« 1 <a]. 
 Jo 
 Consequently 
 
 Now let a tend to zero. The first member will tend to a definite limit, and 
 
 therefore the second member will also, •'. e., — and — will tend to fixed quan- 
 
 a a 
 
 tities c ' and c/'. We have then 
 
 4>(x) = c ' + Gl 'x, 
 
 and the demonstration can be completed as before. 
 
 Suppose now that /"(a) becomes infinite in the interval ( — tz , tz) , but still 
 satisfies the condition of integrability. Over each portion of the interval con- 
 taining no infinite point fovf{x) (x) is a linear function of x by the above 
 proof. Hence since (P (x) is continuous in the interval ( — tz , tz), the curve 
 y=0(x),&sx varies from — tz to tz , if it does not represent a straight line, at 
 least represents a continuous broken line, the corners corresponding to values 
 of x which makey(.r) infinite. We will show that the line is straight. 
 
 Writing \"A?W= r W , 
 
 it is easy to show that ? F (a) is continuous in the interval ( — tz , tz) . For 
 
 V(a + d)-¥{a) = ^f{P)dp 
 
 can evidently be made less than any assigned small quantity s by choosiug d 
 sufficiently small since, if,/ (x) becomes infinite for x=zx , the condition of 
 integrability requires that 
 
 lim [/•(/?) d^ = 0, lim f/t/3) W = ■ 
 
 Employing the theorem of means, and remembering that $ '(a) is con- 
 tinuous, we get at once 
 
 lim -ife) — lim -~|V (a) da =z lim i-[ [*¥*( a) da — fV (a) da] = . 
 
 *Fi {x) is continuous whenever / is integrable, since JFVis the integral from — - to x 
 of a finite function. 
 
12 
 But since by property 3°, p. 7, lim ^ ' = , we must have 
 
 e = * 
 
 That is, Hm 0(x + *)—0(x) = Hm 0(x)-0(x-e) 
 
 which tells us that the directions of the two straight lines meeting at any vertex 
 are the same. Hence, for values of x in the interval ( — tt , tt) , y z=z (x) repre- 
 sents a single straight line, and we can write 
 
 (x) = c + dx 
 and complete the demonstration as in the first case. 
 
 The theorem just proved shows that whenever /"(a;) is integrable and can 
 be represented by a development of the form (7), the coefficients a n and b n in 
 this development must always be of the form originally given by Fourier. But 
 the integrals giving these coefficients are perfectly determinate quantities. It 
 follows therefore that a given function can be developed in a trigonometric 
 series of the nature mentioned in only one way, or, in other words, that the 
 development (7) must be unique. 
 
 We have considered above a development available in the interval 
 ( — re t 1?) . If, however, we had desired to obtain the form of the coefficients in 
 a development holding for some other interval of 2~ , say [m- , (m -f- 2) ;:] 
 where m is an integer, the method of obtaining them would have been precisely 
 the same except that instead of integrating over the interval ( — -, -), as on 
 page 7, we would have integrated over the interval [mr: , (m -|- 2) tt] . If we 
 should do this, getting first equations similar to (11), and then after partial 
 integration equations similar to (12), (13) and (14), we would find that while 
 the expressions for c and c, would be different, 6 , a n and b n would be the same 
 as before (p. 9), except that the field of integration would be [m~ , (m -j- 2) ;r] .* 
 
 3. We have seen that if a function f (x) is capable of being represented by 
 a trigonometric series of the form (7), this development must be 
 
 f(x)=^f(a)da 
 
 sin x \f{a) sin a da-\- sin 2x \f{a) sin 2a da -\-. . . 
 + — { J -' r } ~ n } . (16) 
 
 i + cos x \f{a) cos a da-\- cos 2x !/"(«) cos 2ada-\- . . . J 
 
 ♦Evidently also a development of/ (x) t available in the neighborhood of any point 
 a , is at once obtained by developing f(a + x) , regarded as a new function of x , in the 
 ordinary manner. The two developments, holding for the interval [rmr , (r» + 2) t] , 
 obtained by the two methods are easily shown to be identical. 
 
13 
 
 We need then to consider under what conditions this series will converge to the 
 value of the function f(x). The sum of the first n terms, S n say, gives us 
 
 1 F 
 
 8 n =z — [| + cos (« — x) -f cos 2 (« — x) -\- . . . -f- cos n (a — x)\f (a) da 
 
 "■ J — ir 
 
 * f(a) sin (2ro + 1) %-? da 
 
 2 sin 
 
 a — x 
 
 Let us find to what limit 8 n tends when n increases indefinitely. By making 
 a change of variable S n can be written 
 
 s - 1 p*<— *> /(a? + 2 r ) sin (2n +1) 7 <fr 
 
 + 
 
 J_f*('-*)/(a; + 2/') sin(2n+l)rc?r 
 
 7T J sin p 
 
 jj^ rt (* + *)f( x _ 2 r ) sin (2n + 1 ) r <*r 
 
 7T Jo sin t' 
 
 (17) 
 
 where — tt<^x<^k . This form of expressing S n leads one very naturally 
 to consider the limit for k =. oo of the following integral, known as Dirichlet's 
 integral, 
 
 (> ?(r)smkr dr [0<i< , T ], 
 
 I" 
 
 JO 
 
 ;o sin y 
 
 where <p (y) has been put for/" (x ± 2y) . We will consider under what circum- 
 stances 
 
 (18) 
 
 Hm p yQQBinMr _ lim JL . M . 
 fc=ocJo sin y e =r0 2 
 
 For if (18) holds for all values of h lying between and iz , then when n 
 increases indefinitely, S n will tend to the limit 
 
 lim *[/(* + &) +/(*- 2*)], 
 
 6 = 
 
 i. e., to y(») unless /* has a point of discontinuity at the point x, and in this 
 case to a mean between the two values taken by f at the point. 
 
 It is only necessary, however, to consider 0</t< -^- , for if (18) holds for 
 
 such values of h, it will hold also when ~ <^h<^7r. In order to see this, 
 
 suppose (18) is true when < h < -& • This will evidently require that 
 
14 
 If now -jj- < h < z , then 
 
 f* eg (r) sin & r ^ __ f » <p (y) sin fr d<< , ( h <p( r ) s in k r d 
 Jo sin?- ' Jo sin-/- ' J, sin y 
 
 In the case where h is an odd integer, h = 2n -{- 1 , the second integral in the 
 right-hand member of this last equation will be zero, for, on changing y into 
 ~ — y, it will become 
 
 ir 
 
 p <p (tt — y) sin ky dy 
 iv-h sin?- 
 
 which is zero by (19), provided that, over the interval, tp (tc — y) fulfil the con- 
 ditions required of <p by equation (19). 
 
 4. We have shown in the last section that the question of the convergence 
 of the trigonometric series (16) to the value f{x) reduces to the consideration of 
 the circumstances under which 
 
 Km p y(r)*»*r<fr lim * () r < h «n . (20) 
 
 &=Jo sinr e=0 2 Yyj L ^ 2 J 
 
 We will now show that it is only necessary to consider what conditions <p must 
 fulfil in order that (20) be satisfied when A is a quantity greater than zero, but 
 as small as we please. This can be seen at once from the following theorem : 
 Whenever <p (y) is integrable over the interval (e, h) and (p (s) is finite, 
 
 lim [ V(r) sin hdy = r 0<e<A < *n 
 
 k= J e Biny L ^ ^ 2 J 
 
 To show this let us divide the interval (e , h) into n parts d 1} d 2) . . . d n , 
 
 making the points of division odd integer multiples of --^j- , distant from each 
 
 other by an amount equal to --.- . Let us also take s itself an odd integer mul- 
 
 tiple of-KT-, so that we shall have 
 
 flj = o 2 == d s = . . . = o n _! = -j- > 3 n . 
 
 We will now seek the limit of 
 
 ( h <p(y) sin ley dy ^ 
 
 } ( smy 
 
 as k tends to infinity. Let y p denote the value of y, in the interval 3 P) for 
 which <p (y) approaches nearest to zero, and in each interval d p let 
 
 sin 
 
 (r)_ y(r P ), jrf(r)i 
 
 u y sin y p ' Lsin y] ' 
 
15 
 We can evidently express the integral (21) as the following sum of integrals, 
 
 Tt^£**] + f *'i^** + f >*£?*■ (22) 
 
 where ^ -~ = £ , ) H + x = J, + 2 , /„ -^ = A — o n . But since, in virtue of the 
 
 condition of integrability, 
 
 <P Op) $p 
 
 
 **** 
 
 the first term of (22) is zero because 
 
 8111^^ = — np" J =»• 
 
 Now let fc increase indefinitely. The first term of (22) will continue to be zero. 
 As for the second term we have always 
 
 4my^ d H^m^A d <'n&^ 
 
 u 
 
 Lsm yj ' ' | \| e Lsm y. 
 
 which tends to zero from the condition of integrability. Finally, for the last 
 term of (22), since S1 . n J varies always in the same sense from h — d n to h, we 
 
 have, from Bonnet's theorem, that 
 
 [ h P(r)sinfer ,, 
 ]h — i n sin 7- 
 
 is not greater in absolute value than one of the two quantities 
 
 sin (A- K) fa* dr > «*(k-aJ \?V>*r [h-3.<e<h}. 
 
 But each of these approaches zero as h increases indefinitely, even though <p (y) 
 becomes infinite, because if y is a point at which <p (y) becomes infinite, the con- 
 dition of integrability requires that 
 
 fYo f Vo + « 
 
 lim <p (y) dy = , lim <p (y) dy =z . 
 
 a = 0-lyo — a n = 0Jyo 
 
 Hence the sum (22) tends to zero as k becomes infinite, and we have 
 
 lim 
 
 ' h <f{y)sinkydy _ 
 sin y 
 
 * *r 
 
 [ U ITV 
 
16 
 
 This conclusion is very important, for (see equation (17), p. 13) it shows 
 that if a function f{x) is to be developed in Fourier's series, it only needs 
 to satisfy the condition of integrability except for a region as small as we 
 please on each side of the point at which the development is to be made, or in 
 other words, that the convergence of the series for any particular value of x 
 depends only upon the behavior of the function in the vicinity of that value. 
 
 A particular case of the above theorem is when <p (y) is a constant, say 
 unity. The theorem will then become 
 
 lim \ h s™ k rdr = r <s<><4 
 
 * = «, J e sin y L 2 
 
 This can also be shown very readily by partial integration.* 
 
 (23) 
 
 5. The theorem proved in the preceding section reduces our problem to the 
 consideration of the conditions which <p (j) must satisfy in order that 
 
 Hm f- ?(r)sW r = ]im n ((J) = n { say> m 
 
 k =« Jo sin 7* <r=:0 ^ z 
 
 where £ denotes a quantity as small as we please but greater than zero. 
 Now, k being an odd integer, 
 
 sin y 
 and hence from (23) we must have 
 
 7. 
 
 P*" sin ky , n . 
 
 Jo isry^T^ 
 
 lim p™fed r = 7 
 
 Jo smr 
 
 fc=ooJo sin y 2 
 
 Accordingly we may write 
 
 lim P y fr) sin kr dr - lim [' <> fr) ~ ? ( + ° )] Sin hr dr + g r (+ 0) 
 *=oj sin?' fc=»J sin y 2 
 
 Equation (24) will therefore be satisfied if 
 
 lim [ e l> (r) - <P (+ 0)1 sin kydy _ n (26) 
 
 jfc=ooj sin y 
 
 * C h sin ky JL_ coaky~\ h l_ f* 
 
 J e sin y d y-~ k sin>J e k ] t 
 
 1 f a cos ky cos y dy 
 
 sin" y 
 
 which tends to zero with -j— 
 
 f This follows at once from the equation 
 
 sin(2?t + l)a ; : _ 1 + 2cos2a!+2co84a;+ _ + 2cos2tw; 
 
 sin x 
 
Let us first suppose that <p is a function possessing the property of limited 
 oscillation or limited variation, as Jordan has defined it,* in the interval (0, e). 
 It can then be written in the form 
 
 <p (y) = <p(+0) + P y -N y , 
 
 where P y represents the sum of the positive oscillations of <p from zero to y , 
 and N y represents the sum of the negative oscillations. In this case the left 
 hand member of (25) will take the form 
 
 lim H P v-^) sin i^ r , 
 fc=ooj sin y 
 
 But from Bonnet's theorem, since P y is a positive function, never decreasing, 
 
 rf T 8JnMr = f t W^ [0<*<«]. 
 
 Jo smy ]t sin y , 
 
 Suppose now that we divide the interval ( , ~ ) into the partial intervals 
 
 ( 0, ~f)' \T" "*/' \T' ~V)''''\T' ¥)' 
 
 r being the largest number of times that ~ is contained in -J- . We can write 
 
 k 2 
 
 7i [z sin ky j , , ,,._ 
 
 where 
 
 ^ sin ky dy \ \n ^2 1 r -, „ n 
 
 W)- sm IT 
 
 This gives at once 
 
 Px< 
 
 sm 
 
 ^ 2 . - . 2 
 
 (26) 
 
 and hence whatever value $ takes in the interval ( 
 
 0< ?*■£*<"+-* 
 
 J sm j' 2 - 
 
 But since 
 
 lim 
 
 
 k = oo 
 
 we have 
 
 lim 
 
 
 k = 00 
 
 lim psinMr = ^ 
 
 fc=ooJo smr 2 
 
 r 
 
 6 sin &;' cZ?- 
 * sin r 
 
 <- 
 
 (2' 
 
 *See Comptes rendus, 1881, p. 228; also Jordan's Cow« d' Analyse, Vol. II, p. 216. 
 
18 
 
 It follows therefore that 
 
 Similarly 
 
 Hence 
 
 lim 
 
 fc = oo 
 
 lim 
 
 k = oo 
 
 lim 
 
 £ 
 
 P y sin ky 
 siny 
 
 dy 
 
 tN y 8in k r dr 
 
 sin r 
 
 < 
 
 <\*.. 
 
 r 
 
 JO 
 
 {Py 
 
 -N y ) sin ky dy 
 sin y 
 
 <r:M, 
 
 where M is the largest of the quantities P ( and N € . It follows therefore that 
 (24) will be satisfied provided that P £ and N e tend to zero with e. This will 
 evidently be the case under the following circumstances : 
 
 1°. If the function is continuous and has only a finite number of maxima 
 and minima. 
 
 2°. If the function has a finite number of discontinuities,* but is finite, 
 and has only a finite number of maxima and minima. 
 
 These conditions are precisely those of Dirichlet, and hence we see that for 
 functions satisfying Dirichlet's conditions we must have 
 
 limf 6 ^)sm^ r _jr ( , Q)> 
 * = «> Jo sin y 2 
 
 6. If the function <p is continuous or has a finite number of discontinuities, 
 but has an infinite number of maxima and minima, it may still be true, when f 
 possesses the property of limited oscillation in the interval (0, e), that P e and 
 N e tend to zero with e. But for the general case of a function with an infinite 
 number of maxima and minima (with no discontinuities or a finite number of 
 them), and also for the case of an infinite number of discontinuities, a different 
 investigation is necessary. We can, however, derive a very general condition 
 which it is sufficient for functions of this kind to fulfil in order that the 
 equation just written be satisfied. 
 
 In equation (25) we can replace sin y by y since their ratio is very near 
 unity when e is very small. Now 
 
 Wr) 
 
 ^(-j-0)] sin kydy 
 T 
 
 Jo 
 
 f(r)-r(+o) sin kr 
 
 r <! 
 
 dy 
 
 y(r) — y(+Q) 
 r 
 
 dy. 
 
 * For if the function has a finite number of discontinuities, we can take e so small 
 that in the interval (0, c) the function will be continuous. 
 
19 
 But e can be taken as small as we please, and hence a sufficient condition that 
 
 i8 iimf I y(r)— y(+Q)i dr — ,* (28 ) 
 
 Let us compare this condition with one or two other conditions which 
 have been given. Lipschitz's condition was 
 
 l{m f±±^ZllSll < B, (29) 
 
 8 = ° 
 
 where B and a denote any positive finite quantities. His condition is less 
 general than (28). For if (29) is satisfied we can write 
 
 lim r- | pfr) -y( + o)| rfr<Hm f-Brdr 
 
 «=oJo « = oJo X 
 
 V « Jo 
 
 :0J0 « = 0^0 
 
 But 
 
 which tends to zero with e. Hence (28) is satisfied in this case. But on the 
 other hand (28) can be satisfied without Lipschitz's condition being fulfilled. 
 Thus if <p was of such a nature that 
 
 I!fo( log d l \?b + *)-H> (r) ] == X, 
 
 where Kis a finite constant, Lipschitz's condition would not be fulfilled, since 
 
 M K 
 
 a=o<* tt (logd) a 
 The condition (28), however, would be satisfied in this case, for we can write 
 
 l im f- k(r)-?(+0) l d UmK i- dr = Bm r_ *_T = . 
 
 «=oJo r e=o Jor( lo gr) .=<>l logr-io 
 
 Another condition which has been given is, denoting by D the difference 
 1P(r + *) — ?(r), " lim 2>log* = 0. (30) 
 
 8 = 
 
 The two conditions (28) and (30) differ from each other in comprehensiveness 
 very little, but the latter is slightly more general. Thus both can easily be 
 shown to be satisfied by a function for which 
 
 lim D (log dy—K [«>1], 
 
 8 = 
 
 *See page 5. This condition, for integrals similar to Dirichlet's but more general, 
 was obtained by du Bois-Reymond in another way. See his article, Comptes rendus, 1881, 
 p. 915. 
 
20 
 
 where K is a finite constant. But both cease to be satisfied for a < 1 . If, 
 however, we consider a function for which 
 
 lim D log d [log ( — log <*)]• = K [0<a<l], 
 «=o 
 
 the condition (30) will be satisfied while (28) will not. For 
 
 K 
 
 But 
 
 lim D log d = lim p^- 
 
 5 =o & 5 =o[log( — log^)> 
 
 = 
 
 i im f <k _ Iim r ^[log(-iogr)] 
 
 e =o Jo r log r [^g (— log r)] a £=0 Jo [log (— log r )Y 
 
 ~ lim \r~ -^[ lo g(— lo gr)] 1_a ]"=<». 
 
 e = o Li — a J 
 
 Let us now apply the condition (28) to one or two functions having an 
 infinite number of maxima and minima in a finite region. The function 
 
 f{x) = x sin — becomes zero for x = 0, having at that point an infinite num- 
 ber of maxima and minima. For this function the left-hand member of (28) 
 would become, since for # = <p becomes /"(i 2f)-=.f{2y), 
 
 But 
 
 limf 
 
 €=0jl 
 
 io sin iH<E 
 
 2sin-±-c7r. 
 2r 
 
 sin 
 
 2 r 
 
 d T <e. 
 
 Hence, since the condition (28) is satisfied, f(x) is capable of being represented 
 by Fourier's series at the point zero as well as elsewhere. 
 Again consider the function 
 
 /(*) = 
 
 (log 
 For this case we have to consider 
 
 x 
 JtF 
 
 y 
 
 sin 
 
 4 [*>!]• 
 
 sin 
 
 2r 
 
 lim i -, J . „ 
 
 r 
 
 Now 
 
 sin 
 
 2 T dy 
 
 ■(*£)' r 
 
 < 
 
 sin 
 
 2 r 
 
 ( 
 
 g l0° r 
 
 dr< 
 
 0Q- 
 
 1 — a 
 
21 
 
 which tends to zero with s if a y> 1 . Hence,/ (a;) is developable in Fourier's 
 series at the point zero. 
 
 7. Although the condition 
 
 imposes very little restriction upon the function <p , yet, as has already been 
 mentioned,* du Bois-Reymond has shown that there are continuous functions 
 which cannot be represented by Fourier's series at every point. The following 
 illustration given by Schwartzf is a special case of functions of this kind dis- 
 cussed by du Bois-Reymond. J 
 
 Recalling the value of the sum of the first n terms of Fourier's series, 
 
 8.= U"--/(«+2r)'m h dr+ J_f"> + y (*-2 r ) sinkr d [i=2n+1]> 
 Tt Jo sinr 7T Jo sin?- L J ' 
 
 and the theorem given on page 14, it is evident that we only need to find an 
 integrable function,/ such that for some value of x, 
 
 l im \< f(x + 2r)smk r d r =: ^ 
 
 *=°°Jo r 
 
 where e is a small quantity greater than zero, and of such a nature over the 
 interval (x — e, x) that the second integral of Sn is finite. 
 
 Divide the interval f — , OJ into intervals becoming smaller and smaller, 
 
 as follows : 
 
 IT' (I)]' L(T)' (|]'- , 1(T^'(|"]' , -'[(^^'(^)J , L(^' ]' (31) 
 
 where (X) = 1 . 3 . 5 . . . [21 -\- 1] , and consider a function which in the X th 
 interval is defined by the formula 
 
 /(/3)=c,sin(;)A, 
 
 where the constants c 1} c 3 , . . . <v +1 are positive and decrease indefinitely to 
 zero as ju becomes infinite. When //= oo this function, evidently continuous, 
 tends to zero with /3, presenting an infinite number of maxima and minima in 
 
 * Page 5. 
 
 \ Bulletin des Sci. Math., 1880, p. 109. I give here the function given by Schwartz, 
 but prove the divergence of the series differently. 
 
 \ Untersuchungen iiber die Convergenz und Divergenz der Fourierschen Darstellunga- 
 formeln. Abhandl. d. k. bayer Akad. d. W., II CI., XII Bd., II Abth., §§ 35-37. 
 
22 
 
 the region of the point zero. We will assume that for negative values of 
 fit f (ft) ^ 8 °f sucn a na ture that it fulfils the condition (28). This is allowable 
 since f is an arbitrary function. 
 
 In order to show that the Fourier series, which for all other values of ft 
 represents the function, diverges for ,3 = 0, we need to examine the integral 
 
 f' /(2r)smtr d 
 Jo r 
 
 We will show that this integral, with proper choice of the c's, becomes infinite 
 with h. To do this we will give to h only values of the form (/j.) , where fx 
 increases indefinitely.* Let us divide the integral into partial integrals cor- 
 responding to the intervals of (31), beginning with the right, and let / x be the 
 
 greatest integer for which ~ > e . Putting 
 
 ( x i) 
 
 J= nk f{2y) sin (fi)ydy ^ 
 
 Jo T 
 
 we have 
 
 J = c>x + i n^sin(// + l)rsin( / /) r d r+ Al | 1 t7A> 
 
 Jo T m 
 
 where 
 
 t _ f (A ~ 1) s * n M f s * n (aO y dy 
 
 w 
 
 Over the interval , j—.\ sin (fi) y is always positive, and hence applying the 
 theorem of means, we have 
 
 it I _ „ (V> sin(/<+l)r sin {fj)rdr -, f ^sin {^)ydy 
 Jo Jo 
 
 this last integral by equa 
 lim c^ + i = 0, we must have 
 
 TZ 2 
 
 But this last integral by equation (27), p. 17, is less than -^- -| , and since 
 
 lim J^ + i^O. 
 
 ft=O0 
 
 •This of course is allowable, because increasing k from («) to (,« + 1) amounts simply 
 to adding a large number of the terms of the series at one time instead of taking only 
 one term additional. Thus putting a equal to the number of terms of the series taken 
 when k is (/*) , and V the number taken when k is C" + 1) , we have 
 
 , _ 1.3.5.7 |>+1] — 1 ,,_ 1.3.5 [2.M + 1] [2.U + 3] — 1 
 
 a_ g ' A ~ 2 
 
 Hence a' = a + |> + 1] (u) 
 
23 
 
 To the integral J\ add and subtract 
 
 IT 
 J TT J 
 
 We get readily 
 
 (a) 
 
 J\ 
 
 = **[£ 
 
 - 1 ' coajY^Or—lM^: -- f (A ~ 1)c ° 8 C0») r + ^rM 
 
 (A) 
 
 r; 
 
 (A» 
 
 ]• 
 
 Integrating by parts we get 
 
 (A) 
 
 0«) + WJ_i 
 
 •(A-D sin[(//) r + (^) r ]c?n < (32) 
 
 7T / J 
 
 In particular, 
 
 (A) 
 
 
 Employing the theorem of means, the second integral of J^ is seen to be less 
 than 
 
 20") 
 
 [-±r i) =M^-^=^-^~)> 
 
 <*0 
 
 2^ + 
 
 which tends to zero when // increases indefinitely, since lim c^z^O. Conse- 
 quently lim «7 M = lim | c M log [2/j -f 1] . 
 
 /a =<x /a =00 
 
 Applying the theorem of means to (32) we have 
 
 (A) 
 
 Hence, 
 
 < 2 * C * [(«) - (*) + W + (/) J W < jgj ° A J0J ' 
 
 A, + 1 
 2 ^A 
 
 Aj + 1 
 < 2 
 
 Ca 
 
 W 0") 
 
24 
 
 But this series evidently has zero for its limit, as is seen at once by writing it 
 in the form 
 
 VrJ | V^2 
 
 2 <" + 1 ~ oArr [> + l]l>- 1] - 
 
 [2/i + l][2/* — 1] 
 
 J>) & + !) ' 
 
 (A+l) (a<) 
 
 But 
 
 *> +i 
 
 «^= «^M + 1 + «^» + 2 «A 
 ft — 1 
 
 and therefore 
 
 lim J= lim J c M log [2// + 1] . 
 
 Now since the c's are unrestricted, except that they are positive and lim c M = 0, 
 
 fl=0O 
 
 it is possible to choose them in such a way that the above product shall become 
 infinitely great with ft. This will be the case for example if we take 
 
 c 1 
 
 C * VlogO + 1]' 
 
 It follows, therefore, that with such a choice of the c's, the Fourier series, which 
 represents the function f (/3) for values of /9 different from zero, diverges for 
 0=0. 
 
 8. Let us determine as far as possible the nature of the convergence of the 
 trigonometric series 
 
 2 («« sin nx -f- b n cos nx) 
 
 n=0 
 
 which represents the function f(x), say, in the interval ( — n t 7c). Now 
 
 1 fV, \ . , 1 ["/(«) cos nay . 1 f* , , 
 
 a n = — / a sinwaaaz: k_ i_ i -J f (a) cos na da 
 
 7i )-„ ' tc L n _]_„. n>Tjl. w v ; 
 
 = tZ ^ 1 [/(?r) ~ /( - ^~ 4 fe (") Sin na da > 
 
 If* If" 
 6 n = — y (a) cos nada = f (a) sin na da 
 
 71 j — „. nTc J _ „• 
 
 = ^^ [/' (*) — /' (— *)] + i f /" (a) cos na da . 
 
 This shows that, if the first and second derivatives of/* are finite, the w th term 
 is of the order — , except when f ( — 7t)=f(7i), and hence that in general 
 
25 
 
 the series is only semi-convergent. But if,/"( — 7r)=.f(n), or if the develop- 
 ment contains only terms of the form b n cos nx , the series is absolutely con- 
 vergent. 
 
 The question also arises : Is the series uniformly convergent ? Manifestly, 
 it cannot be so in any interval containing a point of discontinuity. Let us then 
 consider the question for any interval (a, 6), comprised within ( — Tt, tz), con- 
 taining no point of discontinuity for the function, and over which the function 
 has not an infinite number of maxima and minima. We need to show that we 
 can take n so large that the sum of the first n terms of the series 
 
 #« = 
 
 1 f* (ir ~ a?) /(a?+2y) sin kydy 
 Tt Jo sin 7- 
 
 + 
 
 sin y 
 
 [Jfe=2n+1] 
 
 A = 
 
 shall, for any value of x within the interval (a, 6), differ from,/ (x) by a quan- 
 tity whose modulus is less than a, where a is an arbitrary small quantity 
 chosen in advance. Let us put 
 
 x Jo sin?" Tt J sin?- 
 
 f{x) under the conditions mentioned above evidently possesses the property of 
 limited oscillation in the interval (a, 6), and hence we can write 
 
 f(x + 2y) =/(*) + P; - JV;', /(*- 2y) =zf(x) + P>' - N< y >, 
 as long as x -f- 2y and x — 2y do not pass outside the interval (a, b). Or we 
 can write 
 
 f(x + 2y) +/(*- 2y) = 2f(x) + P - N y . 
 
 Consequently we have 
 
 A = 
 
 Xyf^dr. 
 
 smy 
 
 Since P y and N y are positive increasing functions, we get, employing Bonnet's 
 theorem, 
 
 = -§-/(*>! 
 
 rfnfy , P. 
 
 o sm y Tt 
 
 e sin ky 
 f sinf 
 
 dy 
 
 N e f e sin Jcy 
 
 N e psi 
 
 Tt L S 
 
 ? sm y 
 
 dy 
 
 ro<*<n 
 
 L0<£'<eJ- 
 
 But from (26) and (27), page 17, we must have 
 
 < 
 
 e smky 7 . n , 2 
 
 j smy 
 Hence ^L will differ from 
 
 -<f 
 
 sin ky 
 
 $ ' sin y 
 
 rfr<^ + 
 
 2 P 
 
 sin ky 
 
 d r =/{x) 
 
 f/W 
 
 jo sm r 
 
 by a quantity which is less in absolute value than 
 
 4 
 
 sin £7- 
 sin 7" 
 
 dy 
 
 M(JL + 
 
 ~\2^ 
 
 <M, 
 
26 
 
 where M is the largest of the quantities P e and N t . But since/ (x) satisfies 
 Dirichlet's conditions, we can choose £ sufficiently small to have M<^\a. 
 Having thus chosen £, it follows, from the theorem proved in §4, that we can 
 
 choose n =z — — so large that 
 
 JL fix) F ^fr dr JL f* (7r ~ z) /(^+2r)sin^^ r 
 n J J J t siuy " -Je sin r ' 
 
 l ■f*c+") /(g--2r)BiiiJb r d r (33) 
 
 x Je sin?- 
 
 shall, for any value of x in the interval (a, b), each be less in absolute value 
 than I a, and consequently we will have S n differing from /(a:) by a quantity 
 whose modulus is less than a. 
 
 Suppose now that f{x) has an infinite number of maxima and minima in 
 the interval (a, 6), but satisfies the condition 
 
 limp I /(»*»)-») dr = 0. (28) 
 
 We can write A , defined as above, in the form 
 
 SlD kr dr 4 1 P ^ + 2 ^ "^frfl sin ^ c?r 
 sin;- ' ~* 7i J siny 
 
 i f r./>)-/(*-2r)]sinfcr d 
 
 7T J sin y 
 
 Since the condition (28) is satisfied, it is easily seen, by employing the process 
 used at the beginning of §6, that we can choose £ so small that the last two 
 terms in this expression for A shall each be less than £ a in absolute value. 
 Having thus chosen e, we can, as above, now choose n sufficiently large to make 
 the modulus of each of the expressions in (33) less than \o., and hence we will 
 have S n differing from f{x) by a quantity whose modulus is less than a. 
 
 We have thus shown that the Fourier series representing a function/fa) 
 is uniformly convergent in each interval (a, 6), comprised within the interval 
 ( — 7r, tc) and containing no point of discontinuity for the function: 
 
 1°. When /*(») does not possess an infinite number of maxima and minima 
 in this interval. 
 
 2°. When it fulfils the condition 
 
 tM 
 
 lim 
 
 \f(x±2 T )-f(x) l dr ^ Q 
 
Biographical Sketch. 
 
 Edward Payson Manning was born in Antwerp, N. Y., March 21st, 1865. 
 In 1866 his parents removed to Raynham, Mass., where he lived until the fall 
 of 1882, when he entered the High School of Providence, R. I. 
 
 In 1885 he entered Brown University, from which he received the degree 
 of Bachelor of Arts in 1889. 
 
 After spending one year in private teaching in Baltimore, he entered the 
 Johns Hopkins University as a candidate for the degree of Doctor of Philo- 
 sophy, selecting Mathematics as his principal subject, with Physics and 
 Astronomy as subordinate subjects. During the last three years he has held 
 successively the positions of University Scholar, Fellow and Fellow by 
 Courtesy, this past year serving also as an assistant in the Mathematical 
 Department. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
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 8