on cantor's theorem concerning the coefficients of a convergent trigonometric series, with...

On Cantor's Theorem Concerning the Coefficients of a Convergent Trigonometric Series,with GeneralizationsAuthor(s): William F. OsgoodSource: Transactions of the American Mathematical Society, Vol. 10, No. 3 (Jul., 1909), pp.337-346Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1988624 .

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ON CANTOR'S THEOREM CONCERNING THE COEFFICIENTS OF A

CONVERGENT TRIGONOMETRIC SERIES, WITH

GENERALIZATIONS*

Y

WILLIAM F. OSGOOD

§1. C:ANTOR S THEOP{EM. CANTORT has shown that if the trigonometric series

00

E ( an COS nx + bX, sin nx ) n=O

converges for a]l values of Z in any interval a c SG c b, however short, then

lima - O and limb: O.

n

n= q7>=o More generally, the conclusion holds tf merely

liln ( an cos nz + bn sin nz ) O t&=O

for all points of the interval. His proof is not simple, being based on arithmetic considerations of an intricate

character. The following proof has the advantage of transparency as well as of rigor, and it admits of extension to cases to which CANTOR'S methods do not apply.

Let A and ey be so chosen that n n an cos nz + bn sin nz An sin ( nz + t/n ) n

where An-O. Then AX 1/'a2 + b2.

* Presented to the Society April 24, 1909. tG. CANTOR, Uetier einen die trigonometrischen ReShen betre.ffieqlden Lehrsatz, Journal fur

Mathematik, vol. 72 (1870), p. 130; Sur les series trigoulometriques, Acta Mathematica, vol. 2 (1883), p. 329. For other proofs cf. HARNACE, Mathematische Annalen, sol. 19 (1881), p. 251 and p. 524; LEBESGUE, Lefons sur les series tryonometriques, 19068 p. 110.

337



338 w. F. OSGOOD: ON CANTOR S THEOREM CONCERNING [July

Obviously a necessary and sufficient condition that

lim an ° and lim bn ° w=ao n= is that

lim A _ 0. n=r Hence it is enough for the proof of the theorem to show that fi ons the hypothesis

limAnsin(nS+n/n) ° n=r follows that

lim A 0 . n=s Suppose the conclusion were false. Then there wollld exist a positive constant

h such that n -h

for an infinite set of values of the index, n nl s n2, * s - . Thus

An h (i_1,2, ). The funetion

sin (nit + fY)

we will drop the index °f t/n n since the value of 8

the latter quantity is immaterial for our proof has 1 -a- A

the period 1 15 \ 1 Z 27r 0 a ct A1 | b

@. . \ S ni \ I

Choose i_ il so that W Q)il<2 (b a).

Then the above function has a complete arch above the axis of , the base of the arch being enclosed in the interval (a, b). Hence throughout a certain subinterval al-' Z-,81, where a c al < /21-c b,

. sin(nilw+¢Y)--

Thus we have An sin (nz + ey) -2lh

when n nil and a,-z-121

Proceeding next to the interval ( al, ,S, ) we repeat the foregoing reasoning, choosing i - i2 > il so that

a)i2 < - (a1-al),

and thus obtaining a second arch above the axis of , its base being included



19091 THE COEFFICIENTS OF A CONVERGENT TRIGONOMETRIC SERIES 339

in the interval (al, 81) Thus we get all interval a2-z c 132, where a C a < ,8 C ,8 in whicl

sin (nZ2x + fy)- Hence we have

A1,sin(nz+¢y) -2lh when

n nZ2 and a2-CzC2

A continued repetition of the process leads to an infinite sequence of intervals (ak, k) each lying in its predecessor, whose lengths approach O as their limit and which are such that

Ansin(nx+¢y)2lh when

n = nik and ak-c z c ,8M.

The extremities of these illtervals determine a point

lim aAG lim ,S, X k_m k=ao

pertaining to all of them. For this point

An sin (nX+ 7) 1vh (n=nzlX ni> * * * ) 2

This is contrary to the hypothesis that

lim An sin ( nz + fy) _ O n=r

for every point of the interval (a, b). Hence the theorem is true.

A more general form of C:ANTOR'S theorem, to which botll his proof and the one here given apply, is obtained by replacing the expression

an COS nac + bn sin nsc by an cos Pn ac + bn sin Pn z s

where p7& iS any function of n which merely becomes infillite with n:

lim IPnl °° n=n

§ 2. A GENERALIZATION. THEOREM. Let fn ( sc ) be a contintlous ,function of Z in the interval

a c z _ b; let I be a certain positive constant, andN let fn have the following property. To every subinterval of (a, b), however short, there corresponds a positive inteyer n such that, if n -m be chosen at lvleasure,

[frl(z) I > I at some point of the latter interval.



340 W. F. OSGOOD: ON CANTOR S THEOREM CONCERNING [July

If under these conditions the series

E AnEn(z) n=l

converyes for every vcllue of z in the interval (, b), then

lim An = ° n=

The foregoing proof applies with slight modifications to this more general case. Suppose the theorem false. Then there exists a positive constant h such that

IA | -h (i=1, 2, .

Deterrtline ml so that, for any given n -ml,

Ifn(fS8)| > l

at some point of the interval (a, b). Denote by i, a value of i for which ni -ml. Then there will be a subinterval al c z c ,81, where a c al < vB1 c b, in every point of which

Ifn(e) |-I (n=ntl ) Hence

IAnfn(z)l -hl when

wa - ni and al cz C,B1.

A continued repetition of the process leads to a set of intervals (a1; Sek) like those of §1, for which

1 24nAntS8) I -hl where

n n, and a _ z c ,B .

Their extremities determine a point

lim ak-lim ,8k - ff k=w k=w

pertaining to all of them. For this point

{Anfn(X)t _hl (n=ns1, nt2, ),

and thus the theorem is pI'OVen.

Series Of Functions of Severctl Variables. The theorem admits of immediate extension to functions of several variables. Thus, if Jn(c y) is continuous throughout a region S of the (w, y)-plane, we shall demand that to every sub region :5 of <C, no matter how small, there correspond a positive illteger q^ such



1909] THE COEFFICIENTS OF A CONVERGENT TRIGONOMETRIC SERIES 3 4 t

that, if n m be chosen at pleasure,

Ifn(e Y) | > I

at some point of , I denoting as before a positive constant. It is immateriaL whether S incltlde its boundary, just as it was unimportant that the interval. ( a, b ) include its extremities.

The proof follows precisely the same lines as in the earlier case, a succession of squares, for example, E;1 2 *- * Ek *'-each lying in its predecessor,. replacing the intervals (ak, Sek)@

§ 3. APPLICATIONS. ZONAL HARMONICS. BESSEL S FUNCTIONS.

Zonal EEclrmowbics. The zonal harnlonics Pn(Z) have precisely the property- of the functionsfn(Z) in § 2, and thus we have the

THEOREM. Ibf a series of zonal harmo7wics,

Aopo(Z) + AlPl(Z) + * * *s

converges at all points of an interval a c x c b, however short, Iying in the interval 1 < Z < 1, then the coefeients A>, pproach O as their limit.

Bessel's Functions. If we write

Jv(Z) - 4-cos( 7) + gv

where ey - ( Is + 21 ) qr/2, it is well known that

lim gv 0. x=

We are thus enabled to state the following theorem regarding the behavior of the coefficients in a convergent series of Bessel's functions.

THEOREM. If the series

AlJv (\1z) + A2Jv (X2z) + * * * s

where ;tn > O does not depend on ac and limn=<,o kn = x converges at all pointsX of an interval O < a c Z _ b, however short, then

lim [ n_ O or | An | < 61 kns n=m X n

where e is an arbitrarily small positive quantity and n rn.

For the general term can be written in the form

AnJV(\n)- 1/72S- 1 knJv(\n Sc)



342 W. F. OSGOOD: ON CANTOR S THEOREM CONCERNING

[July and if we set fn(Z) = 6 kn Jv(k Sfc)s

then

Ifn( z ) I 1/ l/\n/c | Jv(kn/JC) |- -b ] P/Xeac Jv( k sc) [ . Hence the series

1 ( | A. )<?t(z) satisfies the conditions of § 2, and this proves the theorem.

We have cited the asymptotic expansiotl of Jv(e); but all that we need for our proof is the fact that the continuous function l/cJv(8) takes on values numerically greater than a certain positive constant I in every interval whose length exceeds a certain fixed qualltity Z alld which lies beyond a definite point Z = w. By the aid of the asymptotic expansions the theorems of this paragraph can also be established by Cantor's methods.

§ 4. DEVELOPMENTS IN MULTIPLE SERIES. The theorem of § 2 can readily be extended to developments ill multiple series

of functions of several variables, like

E Am n sin mz cos ny. nt X qW

The mtlltiple series here considered are said to converge if every siml?le series formed out of the terms of the multiple series converges. Thus if such a multiple series converges at all, it converges absolutely.

Let us plot the points (m, n) in a plane, and let R1, 1?2, *? be a sequence of regions each lying in its predecessor and each containing an infinite number of points (m, n) . Exatnples: If ( t, X ) is an arbitrary point of the plane, we may take as Ry the points (t, ) for which

(a) t-ys v-y;

or, again, those for which

(b) t +r/_X, t O, X _O.

THEOREM. Let fg, n ( SC C y ) be a continuous function of the two independent variables sc, y in a region S of the (ac, y)-plc4ne; let I be a certain 7)ositivre eon- stant, and let f-hate the following property. I'o every subregion E; of S, no qnatter how small, there corresponds a region RN such that, if the point (m, n) be chosen at pleasure in RX,

(1) Ifww,8(wny)l>

at some jpoint of E .



1909] THE COEFFICIENTS OF A CONVERGENT TRIGONOMETRIC SERIES 343

If undew these condttions the serie.s

E AsnX nef;nXn(zs Y) sn, n conveqXges Clt every poirtt (c, y) of S, then to an clrDitrarily smclll positive number e there corresponds Cl positive qnteyer p such that, for every point (nz, n) in Rp,

lAtn,nl As

If, tn jpclrticulclr, the regions Rk clre those of Encamqple (b), then

lim AsnXn-° m=, n=

The proof is similar to that given in § 2. Suppose the theorem false. Then there must be a point (9n, n) in each R for which

IAgn nl -h

where h denotes a certain positive constant. Thus we are led to a sequence of points (mi, ni), for which

[Ami,ni|-h (i=1,2, ),

and such that, for any preassigned Rk, all those points for which i q lie ill

that R. Now choose RV1 so that, for every point (m, n) in RV1 relation (1) holds at

some interior point of sS, and let il be a value of i for which (m*l, ni,) lies in R^. Then there will be an interior point (Z19 Y1) of X, for which (1) holds when m= m*1 and n n*l. Surround this point by a square E;1 lying wholly in S and such that (c>) I fxnx n ( bJC, y ) I - I

when n m*l, n n*1 and the point (w, y) lies within or on the boundary of El.

The reasoning is now repeated, a smaller square 2 lying in E1 being obtained, throughout which relation (2) holds if m m*2, n n*2 and the point (x, y) lies in 2. And so on. Thus a point (X, Y) is found, pertaining to all the squares and such that

I ftn. n(ff } ) | - t

when (ns, n) denotes successively the points

( m*k X n*k ) ( k - 1, 2, * * * ) @

Hence for these values of ( m, n )

iAen nftn n(Xs Y)j _ hl,

and the theorem is proven. Trans. Am. Math. Soc. 23



344 W. F. OSGOOD: ON CANTOR S THEOREM CONCERNING [July

The extension of both theorem and proof to r-fold serie.s of functions of s

independent variables presents no difficulty. By combining the above theorem and that of § 2 we are enabled to state

a further theorem. THEOREM. If the sertes

,AS ff fS ff (aG n Y)

n

sclttsf es the condttions of the foreyoing theorern, the regtorbs Rk being those of Eacclmple (ct); and if, filrthermore, every simple series formed out of the double sertes by holding one of the indices m, X fclst scltisfes the conditions of the theorerrz of § 2, the I of this latter theorens, however, being Clt liberty to vclry from series to sertes, then

lim Am n-° sn=wl n=

We wish to show that, under the above hypotheses, a positive e being chosen

at pleasure, there exists a positive integer Msuch that

I Asn, n g < 6

for all values of m and n for which

n + n _ ]1.

To do this, choose k so that for all points (m, n) of Rk

|Am,&1 As

Next, consider the series -

2..: Am ntgrrw, nt (xs y) s m=1

where 1-n' c k 1. By the theorem of § 2 its coefficients approach the

limit 0, and hence lAxn,nlAE (m_8/).

Let ,u be the largest of the k 1 numbers ,u' corresponding to n' 1, 2, * . h 1.

Then g AzmX n I < 6 (rn_S 1 CnCk-1 ) .

In like manner consider the series

00

AmxXnf;nn(S/8 8) n=l

where 1 =c m' = k - 1. Its coefficients satisfy the condition

IAm,nl Ae ( }& -vo ) -



1909] T1IL COEFFlCIENTS OF A CONVERGENT TRIGONOMETRIC SERIES 345

Let I) be the largest of the k-1 numbers aw' corresponding to qn' - 1, 2, * * * k-t. Then

l Anz, n l < f ( 1 _ m _ k-1, n v ) .

Thus we see that IA?4,nl As

for all but a finite number of points (m i n), and hence, in particular, f()r all points (?n, n) of a suitably chosen P£M of Example (b):

m+ n M.

The conditions of the last part of the foregoing theorem are herewith seen to be fulfilled, and the present theorem is thus proven.

This theorem also admits of extension to multiple series of higher order than the second. In particular, for triple series of functions of three independent variables it can be formulated as follows.

THEOREM. Let E A^nsp/;wtnsp(>s y, 2:)

rrw, n, p

be a triple series satisfying the conditions of thefirst theorenz of this paragrawh, the regions RX being those of Eacample (a):

RA,: rn -:X, n :N, np -X.

Fqlrthermore, let every double series formed oqbt of the triple series by holding one of the three brtdices zrn, n, p fast satisfy the conditions of the foreyotng theorem. 7Chen

lim Aqr> ff p - ; O . rn>=w, n=, p=m

§ 5. APPLICATION TO MULTIPLE TRIGONOMETRIC SERIES.

Consider any one of the doul)le trigonometric series

EAm n sinqnzsinny, xn, n

where either or both of the sine factors may be replaced by the corresponding cosines. Let such a series converge at all points of a two dimensional region S of the (w, y)-plane. Then it can be shown exactly as in §1 that this series satisfies the conditions of the first theorem in § 4, the regions i? being those of Example (ct).

Next, hold n fast and give y a constant value for which sin ny (or cos ny) does not vanish. Thus the conditions of the theorem of § 2 are seen to be ful-



346 w. F. OSGOOD: ON CANTOR S THEOREM

filled. Hence all the conditions of the second theorem in § 4 are satisfied and we have

lim Am n °@

n=, n=

Similarly it can be shown that if any one of the triply infinite series

E Am n p sinmzsinnysinpz, rn, n, p

where the sines may be replaced by cosines at pleasure, converges at all points of a three dimensional region V of space, its coefficients must approach the limit zero.

HARVARD UNIVERSITY, April, 1909.



on cantor's theorem concerning the coefficients of a convergent trigonometric series, with...

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