On the Spherical Convergence of Multiple Fourier Series

On the Spherical Convergence of Multiple Fourier SeriesAuthor(s): Josephine MitchellSource: American Journal of Mathematics, Vol. 73, No. 1 (Jan., 1951), pp. 211-226Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372172 .

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ON THE SPHERICAL CONVERGENCE OF MULTIPLE FOURIER SERIES.*

By JOSEPHINE MITCHELL.

1. Introduction. In this paper we propose to use the methods and results of the lattice point problem for the hypersphere in number theory to obtain new results for the spherical convergence of multiple Fourier series. The set of functions { (2v)c1qe(n,x, + -+ nqxq) (nk - 0 + 1, ? 2, * * *; k = 1, , q) (where e (X) - eOX throughout this paper), forms an orthonormal systemii on the hypercube C(-ir < < r; IX 1, , q), that is,

7 r 7 r

( 1.1 ) (2ir )f fq . . . e (mixi + + Mqxq)) e (-nix, nqXq)

X dxl dxq q

TI 83jnk7 k=1

which is complete with respect to real Lebesgue integrable functions defined on C. We form the series

(1. 2) 'Y an,Le(nix,+*- * +nqXq) (nk=--s0 to 00; 7 1 , q),

where {a,,. .flq} is a sequence of complex numbers. Summing (1. 2) by

spherical, instead of rectangular, partial sums, we consider the limit of the sequence

(1. 3) SR-E an1...nqe(nlxl + * + nqXq) (v= n 12+* + nq2) v<-R

as R1 -- o, IR being the sequence of integers, which can be represented as the sum of q squares of integers, so that for q ? 4, R takes on all non-negative integral values [6].1 We ask ourselves what condition must be imposed on the

sequence {an,...n.q} in order that lim SR exist and show that if 2 R-*oo

(n 12 + -+ flq2):p Ia:.. 112 2 X

* Received June 5, 1950; presented to the American Mathematical Society, April 28, 1950.

'The numbers in brackets refer to the referen-ces at the end of the paper. 2 The summation for this series is always nk ?- c to + oo, k - 1, . . ., q, and

will be omitted. 211



212 JOSEPHIIANE MITCHELL.

lim SR exists almost everywhere on C, where p = 1/3 for q --2 and p = 4q R-->oo for q ? 4, while for q = 3 the sequence (n+2 + s22 + n32)P is replaced by (n12 + n22 + n32)2/3(log(n12 + n22 + n32)5/3. This theorem represents a con- siderable improvement over known results due to Chandrasekharan and Minakshisundaran, where p q [1, 2]. We prove it by finding a bound for the Lebesgue function

(1.4) LR(X) LR(a,, ,aq) f J R(a -X )I dxl dxq, where

(1. 5) KR(a X) -- KR(,l-xi, 7 , qx)

1/(27r) qe(n,(a, -x1) +- * nq(aq-Xq) P-lR

and then proceeding by classical miethods used in Fourier and orthogonal series to the conclusion concerning lim SR. Finer results are obtained for q = 2 and 3, namely, LR (a) 0 (R1/3) and 0 (R 2/3 (log R)5/3) respectivelv, by using new results on exponential sums due to L. K. Hua [3] in the lattice point problem for the circle and sphere.

2. The case q-2.

2. 1. We prove

THEOREM 2. 1. The Lebesgue function LR(a,,3) is O(R11/3).

Proof. From the periodicity of the exponential system

T p7r LRQ2, ,B)-,J ,JRI I (x, y) I dx dy.

7r

In the proof of this theorem we proceed as far as possible by means of the method used in the rectangular summation of Fourier series for the corresponding problem, but in order to complete the analysis some number- theoretic results are introduced (cf. Section 2. 3). Thus we subdivide the square (-7r < x <Sr, -7 < y < 7r) into smaller rectangles, oni each of which a suitable bound for KR (x, y) is utilized, by drawing lines parallel to the axes through the points x and y - X+ [E - -E 0, O, S ,r - b where e is an arbitrary small positive number. The square is divided into rectangles of 3 different types, of which (O - x < c, 0< y < E), (O < x E , e < y < r-E)

aind (e - x < Sr-c,E < y < 7r- E) are examples. (The fact that I x - y I



MULTIPLE FOURIER SERIES. 213

< 2- is used in the discussion of (2. 17 ).) Over squares of side E we use the bound I KR(x, y) ? E 1 0 O(R) [6], to obtain, for example,

m2+n2<R

(2.1 ) {e fi |KR(X, y) I dx dy O(RE2).

For rectangles of dimension E times ?r - 2E, if I y ? , we proceed as follows, [z] being the integral part of z,

[(f-M2) /2]

(2. 2) KR(x, y) _ e(mx) E e(ny) _R1 /2-<M:fRl /2 n=- [ ( R-M2) 1/2]

1(e (y) ) e(mx) {e( [(R q2a +-y -R1/2C_m-::R1 /2

e (-[ (R m2) ]

(O(RIcsc y).

Hence, for example,

(2. 3) f dy | KR(x, y) dx O(ER' logE1).

Similarly O r-e Y+jC

(2. 4) fe dy f K7R(X y) dx

zr-e ^ +jC (R csc2y dy dx) 0 (R1E log E-1)

7r-C X +e6 (Also, of course, dx f KR (x, y) I dy - 0 (R'E log E-1)) a result which

we shall need later. In the course of our argument we note that all bounids introduced except those involving 0( x- y 1-1) may be integrated over ( E y ?< - E, y - E< X y + 1 E) as well as over (E x _ 7 -

?yC ? y E), obtaining the same or even sharper results.

2. 2. In order to discuss the sum E over rectangles of the third )1b2+jb2<R

type we subdivide the circle m2 + y12 < 1R )y the lines n = m, n -

mr- 0 and n = 0. On the line it = n, for example, we have

[(IR)1/2]

(2. 5) KR(x, y) E e((x + y)m) R)Q. 1H1enc

{e( (x + y) ( (R ]+ )

-e(-(x + y) [('R)'-]))I{e(x + y)- 1}

Q.Hence



214 JOSEPHINE MITCHELL.

4 r-e > r-e ^ r-e ^ r- (2. 6) fIfQ, | dxdy O(f( f ?(x + y) -ldx dy)

>7-e -7r-'e

- ( log(x + y) dx),

and the second mean value theorem used on the latter term gives

O(log( + Y) dx + log(7r E+y) dx ) (E

which is 0( logE |). On the other hand, for example, for the rectangle (E? x <i E - 7r +- I- y < _ E), we use the procedure in (2. 4) over (c?x< X '7r c- X-jE ?< y x- + 1E) aiu d then, as is (2. 6)

7-r1r.5e -X-e2 P r-e -e (2. 7) + J f J f J dx dy O(log -l).

2 e -7r+e 1.5e X$+e/9

Analogous results are obtained when m - n, m = 0 or n 0. The remaining sums to be considered in E are all of the form

P1 2+112<R

(2. 8) , e( mnx ? ny) 0 < m-?a in < nc (KR-n2)1/2

(,E < x E r, E :< y -< 7 -E) (a ( 2@)l)p

or this sum with mn and n interchanged, of which, as we shall see, it is sufficient to study the cases mx +- ny and - mx + ny. Hence consider first

(2. 9) S , , e(mx + ny) 0 < 07--a mn < n? (R-m2) 1 /2

- e(y) ([R r2)1]y I- MX) - V e([(R-inlg 2 x e(y) - 1 o<mCa

e(y)-1 -, OKn?a

S1-+ S2. Now

S =(e( (x + y) [a])-) (e(y) l)'(e (x + y)-1)-le(x + 2y)

O (csc Y ycsc 2 (x + y)), so that

(2. 10) S2 dx dy = 0 2 y-1(x + y)-ldx dy

+ y-1 (2r x - y) ldx dy) > 7r-e >27r-2E

0( (x + y)-1{y-1 + (7r --- y)-'}dx dy)

o(f{ 1 (r1) -'log (x?+y)dy[ r 7r-




and using the second mean value theorem on log(x + y), we get (2. 10) to be O (log2 C-1).

Since the exponient [(R - i2)n')] in S1 is discontinuous, this sum cannot be dealt with by standard methods. Consequently we replace [(R-mn) ] by (R-M2)a. and consider

( 2. 1

)Sii M Y) e (mx + y(R _ 2 )1_l

and

(2. 12) (y) {e (mx + y[(R-_m2))

-e(mx + y (R-- in')2 _ = y)}

where S., + SI 2Si. The exponenit 2rf(u) =ux + y(R U2)2 in Si, satisfies the hypotheses of the following theorem which is due to vani der Corput [8]:

Oa Let D= ef(22rf(u))du E e(27rf(m)). If f'(u) is monotonic

0 < mt?a

and I f'(m)1 < then D-0(1). Hence

^a (2. 13) Si, (e (y) - ) e( "y) = e(u:x + y(R -- U2) I)du + 0(1) 2

11 + 0 (1).

Setting rv ==xu + y(R -u) in 11, where r- (x'+y')l and ru xv -Y( _)2)12 we get

>~ ~~ V22 P . 14 ) 11 = r-1 (x + yvl (R - 2)-)e (rv ) dv A: I,,ll+ 112

(v =yRf-rr-, v2= (x + y)ar-1). Now Iso=Q(X(X'-+y')') SO that as in (2. 10)

>7r 7r 0 o 2 - (2. 15) f csc Iy I 11, dx dy-0(log' E').

The function g (v) -= v (R- v 2) 1 in the integral '12 is montone and g(v1) = yx1', g(v,) = (x + y) X -y ' -'so that by the second mean value theorem 112 0(y2X-1(X2 + y2)1) -1+ 0(y(x + y) x X-y I -(X2 + y'2)-). As

for (2. 15), ff csc ly o(y'x'(x' + y2)')dx dy O 0(log c-1), and, using

(2. 4), with

( 7r-le rT rT r-7 (2. 16)

+ cscly y(X + y)l x-y I-1(x2+y2)-lddxy e 1 - 0 g.56 )

== 0(l?g2E-1),




7r 7 rT 7r

we see that both ff csc 2y I I12 I dx dy and ff SI I dx dy are

O(log'E-1) for all |x-y ? c.

Before colnsidering the sum S1, of (2. 12) we discuss that sum of (2. 8) whose exponent is - mx + ny. In particular we need onily consider the sum analogous to Sl,, namely,

(2. 17) T,1 e(y) = < e(-mx+y(R_M2)h-2 e (Y) -1 < om-?_a

since sums correspondin-g to S2 and Sl may be treated by the same methods as used for these sums. Here although the function 2wf(u) = _x -+ y(R - has a monotone derivative, it is not bounded by -1-, indeed f'(0) -x (2r) -1, f'(a)= (x + y) (27r) so that for (E - x -- 7r E, E < y < - E),

E(27r)' f '(u) I < 1 -E7-1, and hence the hypotheses of the vani der Corput theorem do nlot hold. However the same method of proof is applicable. By the Euler summation formula [5]

(2. 18) (e,(y) -1) e (- y) Tl e( u ( 2>du 2~~~~~ e(a(-x x-+ y))b(a)

- e(yR') + b(u)d(e (-ux + y(R-U2)1)),

where b (it) it [it] - 0(1) has the series representationi

00

--sin 2wu'7rv ([[U] < U < [U] + 1), V=1

which has uniformily bounded partial sums. The two middle terms of the right side of (2. 18) are 0 (1), so that the integral of their absolute value, multiplied by csc ly, over (E ? x ` 7r - E ? y < 7r-e) contributes 0(logEh1). Now consider the term

a (2.19) f {e (27rf (u) + vu) -e (27f (u) - vu)}f(u)dut

(27rwi)1 f(u) (f'(u) + v) -Id (e (27rf (it) + vit))

-a (2w7iY f'(u) (f'(it) - v)ld(e(2wrf (it) -vu)).

. o

The functions g, (it) = f(u) (f'(u) -+- v)'- and g9 (U) = f(Ut) (f'(U) - V)

are monotonie and do not change sign in (0, a). Also, I g (it) f v-1 for v ? 1 and g2(t)| < (v -1)-' for v - 2, while for v = 1, I g2(0)| I= x(27r- x)1 I ? 1, g2(a) I = (x + y) (27r - x - y) ? rE-. UIsing the secon-d mean value theorem on (2. 19), we thus get




a (2. 20) J J b (u) d (e (-- ux + y (R _m 2)-))

0

-0(1) + O( (x - y) ( -x y)- so that

(2. 21) fefe y y I J1 I dx dy O(log E-1)

+ O( log(2r - x y)y-ldy [) O(log2 E-1).

Similarly in the first integral of (2. 18), since f'(u) is monotone and different from zero in (0, a), (f'(u) ) -1 is monotone and boulided in (0, a) and using the second mean value theorem we get

a J2f e(27rf(u))dudt 27rx'O0(1)- 27r(x + y)10(1).

so that

ff sc ly I J I dx dy = 0 (log2 &-1) C 2

as before. Thus finally

f. .fw~~I T1,1 dx dy == O (log2 E-1),

2. 3. There now remains to consider the sum S12 of (2. 12). In this case we have been unsuccessful in using a direct analysis,-the difficulty apparentlv arising from the discontinuity of [ (R - i2)]; indeed the only property of the exponeiltial function that we use is that of boundedness. Since e (y [ (R- )] ) e (y (R 22 - 1y D(( ] 7t)1

(2. 22) S12 O(ysc Y b | 2((RYm2)_)_ )- 0< m <a

We now use the following theorem to be found in [3, Ch. XII]:

Let f(w) be a function with two continuous derivatives in 1 < u ?< p, p

and 0 < f'(U) < 1, f"(U) > Z-3. Then E {f(U)} lp + 0(z2), where

u=1

{f (u)} =min (f (u) [f()], [f (u)] + 1-f (u)). Taking f (u) 1 (R-m2)t, we find that both f'(u) and f"(it) are

monotonic non-decreasing in (0, a) and positive. Also,

f'(0) 0 < f'(u) < f'(a) 1, f"(i) > f"(0) =R--

and z =- R1/6. Thus all the hypotheses of the theorem are satisfied. Now




{t} = {d-1} and {J} {-t} so thlat {f(u)} ={(R-m2)'}. Also, O b(I)={ t}- for [ t]? [d ]+ -2, and 0 -?b(- )= {} for [t1 + - < t? [t] +]i1, that is b () - 1-{t}. IHence

(2. 23) S12 O(ycsc C y (2{(Rm 2)-})) O < mn <,a

0 (Ri/3),

and thus | Si I dx dy 0((R1/3). FuTthermore this result is inde-

pendent of the signs used in the term e(+ rmx + ny), and the proof of Theorem 2. 1 is complete upon choosing E = Rf.

2. 4. THEORE-M 2. 2. If (m2 + n2)1/3 | amn 12 < 00, then lim SR exists R-->oo

almost everywhere o0 C(- 7r x q< 7r -- 7r ? y ? 7r), where

SR (I, 3) E E amne,nma + n,3). v?R V=rn2+n2

Following a classical method of proof of Fourier and orthogoiial series [4J], we first show by means of Theorem 2. 1 that SR is 0(R1/6) a. e. (almosu everywhere) on C, then that the convergence of (1in2 + n2)1/3 1 amn 12 implies the convergence of a certain subsequence of {SI?} and finally that these two results imply the convergence of the sequence {SR} a. e. on C.

LEMMA 2. 1. If , I amn 12 <00, then SR(X, ) O(R1/6) a. e. on C.

Proof. Since Y I amn 12 < 00, by the Riesz-Fischer Theorem extended to more variables there exists a real Lebesgue square integrable function f, whose Fourier coefficients, (47r) -2f f f (x, y) e (- (mx + ny) ) dx dy, equal amn.

C

Now following a proof to be fouind in [4], we set vR=- max Sv(P)v1/6 O < Pv<R

t-1/6St(P) (R #-O), vo = aoo, where P = (a, 3) and t =- t(P). The sequence {vR} (R #/ 0) is monotone non-decreasing and we show that IR- = ff vRdx dy 0(1). The integral

C

I f f f(P') dx'dy'f f t-1/6Kt (P -P') dx dy, C C

where P -P' = (x - x', y - y'), and the Schwarz inequality, setting B = ff f2dx dy, we get 12R ?< Bf f f f f t-1/6Kt (P-P') dx dy 12 dx'dy'. Evaluating the latter integral [cf. 8, p. 253], we have

ff dx'dy' ff t-1/6Kt(P -P')dx dy f f to-1/6Ito(Po -P') dxodyo

= Sfff t-1/6to-1/6K(P -P')dx dy dxody, (T=-min(t, t,,)),




since by the orthonormal relations f f Kt (P - P) Kto (Po - P') dx'dy' KT (P - PO). Consequently

I213? B f ff f (tto)-1/6 I Kr(P- PO) I dx dy dxodyo ? B{f f t-1/3dx dy f f Kt(P -PO) I dxodyo

+ ff to-1/3dxodyo f f I Kto(P- PO) I dx dy,

which by Theorem 2. 1 is 0(1). Therefore lim vR(P) exists a. e. and the R-Aoo

sequence {SR(P)/R1/6} is bounded above a. e. Similarly the sequence is bounded below a. e. and the lemma is proved.

LEMMA 2. 2. If the series , av(av > 0) converges, then there exists a monotone sequence {pv}, pv - 0o0, such that E pvav <o.

This lemma is well-known.

LEMMA 2. 3. If E I a,,n 12( (M2 + n2)1/3 <oC, then {Svj approaches a limit a. e. on C, where {Vk} is a subsequence of the sequtence {R} of (1. 3) satisfying 7c3 ? vk < (k + 1)3.

Proof. By the orthonormal properties we have

ff 1f(P)-SVk 12dx dy E I |a,,, 12-rVk.

V<Vk Vz=7l 2+n2

The series rvk is convergent, for by Abel's transformation we get k=0

oo oo

'' r0V= E k (rv- rvk+1) + liM rVk+l k=O 0 k ---oo

oo

E 7c N I aM7n 12 + Jlimll k N a., 12. k=o vk < V<`Vk0 j V=m112+n2 k-oo V > Yk+j V=1112+12

But k ? Vk1/3 < V113 (M2 + n2)1/3. Hence the right hand side is less than or equal to

0E E (2 + n2)1/3 1 amn 12

k=o Vk < V<?k+l V=M2n2

+ lim E z (Mn2 + n2)1/3 1 amn 2 k->oo v > Vk+1 V=j)t2+n0

which equals E I am. 12 (n + n2)1/3, since the convergence is absolute. Thus 00 00 , f f I f(P) -SVk | dx dy < oo; consequently E I f (P) -S V 12 < oo a. e. k=V =

on C and lim Svk = f(P) a. e. on C. k--oo



220 JOSEPHINE MIITCHELL.

Proof of Theorem 2. 2 [4]. From Lemma 2. 3 it remains to prove that

lim (SR(P) Svk(P)) 0 a. e. for v < R < Vk,+, where from Lemma 2. 3 k->oo

k ? vkl/3 < k + 1. By Lemma 2. 2 there exists a monotone sequence {pv) with pv -oo, such that pt,2VI3 a inn 12 <00 (v0 =n2 + n2). Setting

binn== pvvc1/G61 a C p l,1/6 b = i nnb, (ma + nf) we have by Abel's transformation for k > 0 7f12+n2=J

v

SR - SVk P ( @ (CV- CV+1 CV+1 ( V + CR+1 4 cV. Vk < V?R j=O O<v<Vk o?v?R

Since | b,nn 12 <oo, bmn are the Fourier coefficients of a real Lebesgue square integrable function. Hence by Lemma 2. 1 4 V 0 (Vk 1/6) and therefore

SR - SVk 0 (R1/6) (CVk+l - CR+11) + 0 (pvk+1-) + 0 (PR+1i1).

Noxv Rvk~ ? C vk+lvA ?C (kc + 2) 3k03 0 (1), so that I SR-Svk O(pv 11), which --0oo as k ->oo. Hence the theorem is proved.

3. The case q - 3. The result corresponding to Theorem 2. 1 is

THEOREM 3. 1. The Lebesgue function LR (X, /, y) iS O (R2/3 (log R) 5/3).

Proof. As in the proof of Theorem 2. 1 we subdivide the cube

C(-?7r x < r v-?r < y -< ?r -r <Z < 7r) by drawinig planes parallel to

the y- and z-axes through the points - T + E, - c, O, c, 7r e, where E is an arbitrary small positive number. Using the bound KR (x, y, z) - 0 (R3/2), we get that the initegral of K K(x, y, z) I over parallelepipeds of dimension E X E X 2 I is 0 (7E2R3/2). For parallelepipeds of dimension E X 7r - 2E X 2r analogously to (2. 2), we use

(3. 2) KR (x, y,z) - 1 E e (mX + ny) e(;z) M2 +2 ny)P

X {e(([R-in2-n2)2] + 1)Z)- e(z[(R m2 n2)2])}

=O(R csc z)

if z E_ , so that, for example,

dx dz KR(X, y, z) I dy = (ER log cl).

Similarly as in (2. 4),

r 7r J Kr-( y z+ca (3. 3) dx, dzJ I KWR (x,y,z) dy O 0(,ERlog E-1) (a a constant).

-7 e Z-?:e




In order to discuss the sum E for parallelepipeds of dirneision M2+n2+p2?R

-r 2E X 7r - 2E X 27r, following Hua [3], we subdivide the sphere m2 + n2

+ p2 < R by planes m 0, n = , p 0, inmn, in p and n =p. On the plane n p, for example, we have

(3. 4) Q'== i e(mx + (y + z)nT) M2+92j2?R

1 [R1/2]

e(y+-I z) -1 m 2e(?){e([Q&(R-m2))!] +l)(y+z)

e- (_ [ ( I(_f M2) )'](y +Z))

O(R csc 2 (y + Z)) (y + Z 40 or1 4 27r),

s >7r-e 7r-C so that as in (2. 6) dx f Q', dydz =O(R logE-1). On the

other hand over (- 7r < x < 7r E - v,T + E z < E) we -use the procedure of (3. 3) with a = 1 and (2. 7) to obtaini the same bound. Similar bounds are obtained on the planes m = 0, n == 0, m n and m= p.

The remaining sums to be considered in : are all of the form m2+n2+p2?R

(3. 5) e( mx + ny + pz) n2+n2+p2?R

(7r < x < 7r c ?< y - Ez E c z ' < ,E) Pwh where 0 < m< n < p or this sum with m, n and p permuted. In considering (3. 5) the discontinuous bound O( y -amZ I-1), wherexm m(R -2m2)- (0 < m -(R13)`)p is encountered. This bound cannot be integrated over the domain

(3. 6) D1me= (-7r? ? c- X w< 7- E <Z Z <7 E E<<- ZmZ 1E )2

but as for the case q 2, where the bound 0 ( y - z I-') occurred, we replace O (I y - az 1-') over Dine by I KR (X, y, z) plus the sum of all other bounds used in considering (3. 5) and notice that all of these bounds may be integrated over D,, to give the same bound or an even sharper result thani the bound obtained over Pe. Of course, since a,n depends on mn it is necessary to integrate over Dine before summing with respect to m, which will be seen to be possible in all cases.

Defining G(R) as the set (0 < m < (R13)2, m < n < (-(R-m2) )), we consider as in (2. 9)

(3. 7) S= E e(mx + ny + pz) (m,n)eG(R) n<p< (R-m2-n2)1/2

- e(z) {e(mx + ny ' z[(R M2 n2)j] e(z) 1 (mn,n)cG(R)

-e(mnx + (y + z)n)}




= S. + S2. The sum S2 may be treated the same way as in (3. 4) arid hence we get fff I S dx dy dz = O(R' log2 c1) (cf. (2. 10)). We can use the

Pe same methods on the sum S1 as were used on the corresponding sum for q 2, that is, we consider

(3. 8) Sii e(z) E e(mx + ny + z(R-2- n2)- 2Z) e (z) 1 (m,n)eG(R) and

(3. 9) S12 = e(z) {e(mx + ny + z[(R m2- n2)1]) e (z) -1 (m,n)eG(R)

e(mx + Ty + z(R m2-n2)a- tz)},

where S1 = SlI + S12.

In S,, the function 27rf(u) =yu + z(R m2 2) (m < u < R, Rm= ( (R - 2))1) satisfies the hypotheses of the van der Corput theorem for (0? y?w , 0 z < 7r; 0 z< O m ? Ro (R/3)'). Consequently

^ Rm (3.10) e( 2Z) (e(zz) 1)Sjj e (mx + ity

O<m?RomR m

+ z(R-m2 '2)l)du + 0(1)}.

= 0<?R0 e(mx)Im + O(RI). O < ml-Ro

Using the same procedure as for q = 2, we show that

(3. 11) f f f csc z Im dx dy dz 0(R log2-E1), o<mRo Pe-Dme

where Pe - Dme is the set of points belonging to Pe but not to Dme and

a= m(R 2m2)-2.

Proof. Setting rv-uiy + z (R U where r (y2+z2)l and r yv-z(R _ V2) , we get

- ZV Im (y (Rm2 2) )e(rv)dv Im1 + Im2,

where v1 = r-1(my + z(R 2m2)l) and v2 = r1(y + z)Rm. Now Iml = O(yr2) and as in (2. 15)

E JbJ Cse 2z I Iml I dxdydz? < fff csclz Iml I dxdydz

o < m,--Ro Pe-Dme o < m-<Ro Pl

O- (RI log2 E-1) .




The function g(v) Ov(R m- v')a of 1m2 is monotone and hence by the second mean value theorem

Ims zr2g (V) O(l) + zr-2g (V2) 0(l),

where g (v1): (my + z(R- 2m2)) (mz -y(R -

and g(v2) =- (y +?z) I y-z j

which equals g(v1) for m= R. As a function of y and z, g(v1) is discontinuous at y =- az where RBa < a ? 1. Moreover for all such a. the line y == a,1mZ crosses the square (E ? y ? 7r-- E, E < z < r -E)if E is of the order o(Rd-) (cf. later). To avoid the discontinuity at y -CmZ we use (3. 3) with a = c,, and notice that all the other bounds are either 0 (RX log2 e') or Q1(R5/6E) (cf. (3. 20)) over the domain E Dme. Over PDm D, evaluating as in (2. 16) we get o <m-Ro

f fS zr-2 csc 2Zz(amy + z) I ammZ y I-' dx dy dz = 0 (log2 E-1), Pe-Dme

which proves (3. 11) upon summing with respect to m. As in the case q-- 2 and since our proofs are independent, of x, in

addition to S of (3. 7) it is sufficient to discuss that sum whose exponent is mx + ny - pz. In particular we need only consider

~~~~(3. 1z) T e(Z e(mx + ny-_z (R 92n))

e (- z)- 1 (m, O)eG(R) (x, y, Z) ? Pe

(cf. (2. 17)). As in the case q = 2 the van der Corput theorem fails for this sum but the method of proof is applicable. Hence by Euler's formula

3.e13) T ( = 2) ? e (mx + Uy _ z (R _ M2 _ U2')) du e,Q- z) - 1 o < 11?:RoIV.

- e (mx + Rmn(y-z)) b (Rmrn)-je (m (x + y)-z(R m_2) 1)

+ b(u6)d (e (mx+v (-2 q22)

e =z)) 1(Til + T12 + T13 + T14). e(- z)-

In this case the funiction 2Az1(u) =yn- Z(R- m2_U')t has a bounded monotone derivative which does not change sign in (m, Rm); indeed

(2r) -1E ? f'(m) = (27r) 1(y + amZ)? f'(U)? f'(RM) = (27r)-1(y + Z),




so that j f'(u) ? 1 - E(7r)-. Thus 1/f'(m) is bounded and monotone in the interval (m, Rm) and by the second mean value theorem

Rrn JimfRm (27rf'(u) )1ld(e (27rf (u))) 0 (y-1) + 0 ( (y +z),

since f'(m) ? (27r)1ly, so that

fJf csc 2Z j Y e(mx)Jim I dx dy dz = O(R`- log2 E-1). Pe ? < m<Ro

Also, T12 and T13 in (3. 13) are 0 (R':). On T14 we use the same procedure as for (2. 19) since the corresponding functions g91(u) = f'(u) (f'(u) + v) - and g2(U) = f'(u) (f'(u) - v)- are also moinotone and bounded and do not change sign in (m, Rm). The only difference is that

g2(m) = (y + mZ) (y + mZ 2V7rV)k1,

but | g2(m)j ? (y+ Z)(27rV- y -Z)-1,

which is the value of g2(Rm) and similar to the case q = 2. Thus

jSS T1 | dxdydz=O(R7 log2-1). Pe

Now consider S12 of (3. 9). Using the same argument as for the case q = 2,

e(z[(R ~m2_n,2)a]) - ((R n2) 2))

so that

(3.14) S12 O(zcsc C 2Z I b((R- m2 n2)1)) (m,n) e G (R)

O(Z 2 csc z {(R n2) (m,n)eG(R)

But by a theorem due to llua[3]

(3. 19) E {(R-m2( n2)L} = P + O(R 2/3(loglR)5/3), (m,n) e G (R)

where P = E 1. Consequently (m,n) e G (R)

7r 7

fwf7rf| S121 dx dy dz = O(R 2/3 (logR) 5/3).

In connection with (3. 11) we must show that

SS I b((R m jt2)2)| dxdydz (m,n) e G (R) Dme



AIULTIPLE FOURIER SERIES. 225

is sufficiently small. For fixed m the function f(u) 1 (R M2 2) satisfies the hypotheses of the theorem quoted on p. 8 where the z of the theorem may be taken to be R1/6 again. Thus by subtraction we get

E (R _m2 _n2)1} 2 0 1+ (R1/3) m?n < Rn rn?n < Rm

or

b(R-nM2 _n92):L O(R1/3), ni < n < R?n

so that

(3. 20) f f f b((R -n2-n2)')I dxdyddzO(R5/6E). O < mn<Ro Dme mn < n?Rin

Since these proofs are independent of the signs in front of the x, y and z terms in e (+? mx + ny ? pz), the proof of Theorem 3. 1 is complete if we take e = R-. Since the sequence { (in2 + nt2 + p2) 2/3 (log(Mn2 + n 2 + p2)) 5/3}

is monotonic increasing,, the proof of Theorem 2. 2 gelneralizes without difficulty so that we have

TIE0oREiAi 3. 2. If the series

E (M 2 + 1'2 + p2) 2/3 (log(jn24 + n2 + p2) ) 5/3 1 2inp I

is convergent, then lim SR exists a. e. on (-7r C x 7r, - 7r ? y ? 7r, 7r- - Z 7r

R>0

4. The general case. The following theorem is readily proved:

THEOREMu 4. 1. The Lebesgzie functio0n LR(al, .

q) is O(Rq)4

-Proof. The proof depends only on the boundedness and orthogonality properties of the exponelntial system. By the Schwarz inequality

(4. 1) L2 1, ,Xq) 4 a, KR(P Q)I dA}2 -71

=O4. .J?I R }(P Q) 2dA), 7r -7r

where P =--- , (, 1), Q (X1, , Xq) aiid dA dx1l .dq. By the orthonormal properties (1. 1)

7r 7 r

fK (P Q) 12dA E1, where v =M21 + + m2 .

Bult the right hanid side of this equation is the number of lattice points in the lhypersphere in21 + + m2(7 ? F, which is 0 (R1Q) [6]. Hence the coniclusion of the theorem follows.

ITsing Theorem 4. 1, it is easy to prove by the method of paragraph 2. 4

15



226 JOSEPHINE MITCHTELL.

THEGAiEM 4. 2. If E (m21 + * + n2 q) iq am....mq 12 < 00, then lim SR exists a. e. on (--?S X k?7r, k-1, * ,q). R-*oo

We remark that it would be impossible to improve this theorem by the methods used in Theorems 2. 1 and 3. 1 inasmuch as present results give the number of lattice points in the hypersphere

in21 +* + m2 q ?R as 7rhq(r ( l q + 1))-lRlq + Q (Rlq-1)

for q ?5 and as 17r2R2 + O(Rl+e) for q = 4 [7].

UNIVERSITY OF ILLINOIS..

REFERENCES.

[1] K. Chandrasekharan, "On the summation of multiple Fourier series III," Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 474-477.

[2] - and S. Minakshisundaran, " Some results on double Fourier series," Duke Mathematical Journal, vol. 14 (1947), pp. 731-753.

[3] L. K. Hua and L. Schoenfeld, Topics in Analytic Number Theory, New York (to be published).

[4] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monografle Mate- matyczne, vol. VI, Warsaw (1935).

[5] K. Knopp, Theory and Application of Inftnite kSeries, London and Glasgow (1928). [6] E. Landau, Vorlesungen iuber Zahlentheorie, vol. 1 and 2, Leipzig (1927). [7] A. Walfisz, " tber Gitterpunkte in mehrdimensionalen Ellipsoiden," Mathematische

Zeitschrift, vol. 19 (1924), pp. 300-307. [8] A. Zygmund, Trigonometrical Series, Monografie Matematyczne, vol. V, Warsaw

(1935).



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On the Spherical Convergence of Multiple Fourier Series