it* VtO«+ M3 ■j** UC-NRLF SB 532 7MA 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<,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 < 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

(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+(/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 [— iW- We can now, by making use of the condition that/" (x) is integrable, show that

• • • > # "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; ,* (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 ?(r)smkr dr [0 3 n . We will now seek the limit of ( h *£?*■ (22) where ^ -~ = £ , ) H + x = J, + 2 , /„ -^ = A — o n . But since, in virtue of the condition of integrability,

«*(k-aJ \?V>*r [h-3.<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

m k =« Jo sin 7* 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) -

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

* = «> Jo sin y 2 6. If the function

(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 = i im f !]• 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

+ 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 , . . . 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 = + 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<*)-/(*-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. t 7Dec'4Q'Bfl 26Jan'* 3tAatfSQM SEP 81952LU 29Sep'52BR _ **C*D CD ^24195210^281^ **& 17195BL0 REC'l) LO JAN9 1957 MAR 15 SSI REC'D CD OCT 51960 LD 21-100m-9,*48(B399sl6)476 ■S %jr ~J t NW S0 19» ¥ /ft. t<* m~'.i ■^TL w. 8