CONVERGENCE AND (, 1, 1 SUMMABILITY OF DOUBLEORTHOGONAL SERIES

B JOSEPHINE MITCHEIH

1. Introduction. Let E be the Cartesian product of two sets E1 and Ewhere E ( 1, 2) has finite measure and is embedded in Euclidean space.If Ibml (m 1, 2, ...) is a complete orthonormal system (CONS) of functionsof class L on E1 and Ibl a similar set on E then I bnl(m, n 1, 2, ...) forms a CONS on E; that is,

(1.1) f dA (ON relations),

where dA is the Euclidean volume element on E, and bff dA 0 (m, n1, 2, ...) for anf L impliesf 0 almost everywhere (a. e.) on E (completeness).The orthogonal development of any function $ L with respect to the ONS

{bm} is given by

(1.2) ] a,where

(1.3) a f 4f dA

The mn-th partial sum of series (1.2) will be denoted by S and the mn-th(C, 1, 1) sum by. The mn-th kernel function is

(.4) g.(P, Q) ,(P);(Q) (P, Q ),’,k-1

and the corresponding Lebesgue function is

L.(P) f Igor(P, Q) IdA.(1.5)

For (C, 1, 1) summability the analogous functions are

(1.6)K()(P Q)- K,(P, Q)mn

( )(1 J ! 1- 4,(P)4,(Q),n

(1.7) Q) dA.

Similar notation is used in referring to the simple orthogonal series.

Received February 16, 19’49; in revised form, January 2, 1951; presented to the AmericanMathematical Society at Chicago, February 26 1949.

211

212 JOSEPHINE MITCHELL

In 2, two well-known O, o properties of the Lebesgue functions of simpleONS are generalized for L..(P), where here f, O(g,) means that f, <Ag,, (g, > 0 and A an absolute constant) for all m and n. In 3 a theoremdue to Kaczmarz on simple orthogonal series [4], namely, if L(P) O(u2(n)),where u(n) <_ u(n 1), then the sequence {S,,(P)/u(n) approaches a limit a. e.,is generalized to double orthogonal series for CONS whose kernel functions arethe product of non-negative factors. A similar result is obtained for (C, 1, 1)summability in 4. In 5 these theorems are applied to a consideration of theconvergence of the double orthogonal series formed from the trigonometric andHaar CONS.

In order to simplify the notation the symbols -1 :.-1 and lim.(R)are usually replaced by and lim., respectively.

2. Properties of the Lebesgue functions [5].

(i) /f the ONS {b-} is bounded on E, then L,(P) O((mn)) a. e. on E.Proof. By the Schwarz inequality and the ON properties

L=( 0 K(P, Q)dA o(gm.(P, P)) O(mn)

(ii) For any ONS {] and arbitrary positive , L,.(P) o ([ran (log mlog n)l/’]) a. e. on E.

Proof. From the ON properties and the absolute convergence of the resultingdouble series

’ (mn)-l(log m log n)-1-’ f. .2 dA ’ n-(log n)

(" " means that meaningless tes are omitted). Hence the series-1-.(2.2) ’ (mn)-l(log m log n) .(

is convergent a. e. in E. Since the tes of series (2.2) are positive and thesequence {m (log m)+’ is monotonic increasing to , it is not difficult to prove,using the Kroneeker theorem for sple series, that the convergence of (2.2)plies 7.’- k-(log k)--’,(P) o(m(log re)a+’). From this result and thepositiveness of the tes of series (2.2), it follows easily that K.(P, P),. (P) o(mn (log m log n) +’) a. e. in E, from which (ii) results bythe reasoning of (2.1).

3. Convergence of the double orthogonal series. The proofs in 3 and 4depend upon a generalization to higher dimensional spaces and double sequencesof the results in 2 of a paper on the convergence of sequences of linear func-tionals by Banach [2]. Although the functions considered in [2] are defined onthe closed interval [0, 1], all the definitions and theorems in 2 hold withoutchange for higher dimensional measurable sets. However in order to carryover the results to double sequences it is necessary to assume the existence a. e.

213

of the functional M(X) defined at each point P of E as equal to lim(R)max. IU.(X)I, where {U.(X)} is a double s___equence of linear func-tonals and X J’(P) in our case (see [2; 30]). Since lim.. U(X) oo a. e.does not necessarily imply that M(X) a. e., we replace the hypothesis onthe limit superior assumed by Banach by that of boundedness a. e. on E of thesequence {U(X)} with respect to m, , which property is satisfied by thesequences under consideration. Thus the generalization of the Banach theoremwhich is sufficient for our purposes is:

LEMMA 3.1. Let {Un(X)} be a sequence of functionals, linear and continuousin measure, such that (i) for each X defined on E the functions U,n(X) are 0(1) a. e.on E, and (ii) lim., U,,(X) exists a. e. for each X belonging to a set B everywheredense in E. Then for each X the sequence {U(X)} approaches a. e. a functionalU(X), linear and continuous in measure.

In the proof of the following theorem we assume for simplicity that the setsEl and E2 are one-dimensional. Also, if K.,(x, s), Ll,(x) are the kernel func-tions and Lebesgue functions respectively of the ONS {(x)} and K2(y, t),L2n(y) similarly for {(y)}, then by (1.4) and (1.5) K,(x, y; s, t)K,,(x, s)K2.(y, t) and L.(x, y) LI(x)L,.(y).

THEORE 3.1. If (i) KI,(x, s) >_ O, K.(y, t) >_ 0 and (ii) L(x) O(u(m)),L2.(y) O(u2( )), where 1 <_ u(m) <_ u(m + 1) (k 1 2), then the sequence{S.,(x, y)/u(m)u2(n) is bounded a. e. and approaches a limit a. e. on E.

Proof. Set v(P) max,s.sn Si(P)/u,(2u2(k) S,(P)/u,(p)u2(q),where P (x, y) and p p(P, n), q q(P, n). We prove that/ f= v.dA. 0(1) from which it follows that the monotonic sequence {v} approachesa limit a. e. on E. By Fubini’s theorem and the Schwarz inequality

II f(Q) dAo ui’(p)u;’(q)K,,(P, Q) dA.(3.1)

<_ B u:’(p)ul(q)K,,(P, Q) dA, dA

where B = 12 dA Evaluating the latter integral in (3.1), the ON rela-tions give ] K,(P, Q)K,(P, Q) dA Ka(P, P), where p(P, n),, q(P, n), a rain (p, ) and f min (q, ). Thus using the symmetry ofthe kernel function

(3.2)

JOSEPHINE MITCHELL

By hypothesis (ii) the integral I1 is bounded. In order to separate the factorsof the kernel function in I, we repeat the argument used on In. By the Schwarzinequality (p being independent of P (s, t) and of (x, y)) we get

(3.3)

"(f. u(v)K2,(Y, t) dt) PIP.

Considering P1, we have from the ON relations, since p and p p(x, y, n)are independent of s

P --f.l ..f.. ..f.. K(x, Xl)U72(O)U72l)dr dr dy ( rain (p, p))

(3.)

<_2 , f. dx dy uT’(p) f,, K,(x, x) dx,

Hence by hypothesis (ii), P1 0(1). Similarly P is 0(1) so that h is bounded.Thus/ 0(1).

Since S,.(P)/u,(m)u..(n) <_ v.,.(P) <_ v..(P) if m <_ n, and similarly if n <_ m,the sequence {S,.(P)/u,(m)u2(n)} is bounded above a. e. Similarly by con-sidering u.(P) min,,nS(P)/u,(3)u.(k), the sequence may be shown tobe bounded below a. e.Now consider Lemma 3.1 with X $(P) and U,.(X) S,.(P)/u(m)u(n).

Since {. is complete, the set B may be defined to be the set of all finite linearcombinations of orthogonal functions which set is dense in E. Conse-quently he hypotheses of the lemma are satisfied and the theorem is proved.Remark. This proof can be generalized to multiple orthogonal series with q

subscripts (q > 2). No new points arise in the proof, although the number ofnecessary steps increases with each increase in q.

CORO.a 3.1. If (i) is satisfied and L,.(x, y) 0(1), then the double orthog-onal series (1.2) is bounded a. e. and converges a. e. on E.

For example a double orthogonal series formed from the double Haar system,defined on the unit square, satisfies the hypotheses of this corollary.

4. (C, 1, 1)summability. Let /.’’(1)’lm(, S), L((x) be the (C, 1, 1) kernelfunctions and Lebesgue functions respectively of the ONS {} and (1)... (y, t),L( (y) similarly for {.}. Then

THEOREM 4.1 If (i) K() r- s) > 0, ’(1).x, .. (y, t) >_ 0 and (ii) L((x) 0(u(m)),(1).. (y) ---O(u.(n)), where 1 <_ u(m) <_ u(m - 1) (k-- 1, 2), then the sequence

{a.,.(x, y)/u(m)u.(n) is bounded a. e. and approaches a limit a. e. on E [4].

Proof. Corresponding to (3.2) we have

215

, h,(x)4,(s) 1- 1- (y)(t)

(4.1) (l-b- 1)(1k- 1)q

where

u’;(p)uT()u(q)u(v) dAe dA.,,

J. f v (),, dA, v maxl,k<t

ai(P)/u,(j)u.(k)

and a, are obtained as in the proof of Theorem 3.1. Applying Abel’s trans-formation twice to the sum -1 in (4.1) and abbreviating further, we get

(4.2)

pp

[5; 192], and similarly for. Thus replacing the sum in (4.1) by Sa, and$,a where S, is the sum on the right of (4.2) with .a;r()(x, s) replaced by, (y, t), etc., and using the symmetry of the kernel functions, we get

(4.3)J <_ 2B f f u, (O&,} dA, dA,,

2B(J, -b J).

The boundedness of J1 follows from the methods used in [5; 192].decomposed into four sums of which only

i-11i

(4.4)

(f,. u;*()K(.)(y, t) dr)

J may be

need be considered since the proof that J.l is 0(1) is typical of the proof re-quired for the other three sums. Separating the two kernels by the Schwarzinequality as in (3.3), we get

(4.5)

ds dy u;’(v)K(’,’(y, t)dt PP,.

216 JOSEPHINE MITCHELL

The boundedness of P follows as in the case of the integral J but the evaluationof P involves some new points. From the ON relations (p and p p(x, y)being independent of s) and (4.2), we get

P, _< dx dx’ dy u-[’(p)uT’(p’)(pp’)-" 2 pn.,,"(’)’tx, x’)".. k.-1 I"1

(4.6)(1) kakKl (x, x’)

Also, we may prove that

(4.7)

Thus

(1 min (j, k)).

(4.8)

$ (a min (p, p’)).

f, fs, f=. dx dx’ dy u[2(p)uT=(p’)(pp’)-’$

f f ’ k f K’l’(x, x’) dx’ 0(1)<_ 4 dx dy u’[(p)p- ., p i,,

Finally we have that the second te on the right side of (4.6) is less than orequal to

f.f.f dx dx’ dy uT(p)uT=(p’)(’)- jk[K, (,

(4.9)

r"’(x,lk

k-’l

< 2 d. dy u-;’(p)p- j &’ .,,j..l

Consequently J. of (4.3) is 0(1) and the boundedness of J, ia (4.3) follows,from which the rest of the proof results by the method of Theorem 3.1.

5. An application of Theorem 4.1. Let the ONS {b(x) be the trigonometricsystem 1/(2), (1/r) cos rex, (1/) sin mx (m 1, 2, ...), which is completewith respect to class L on [0, 2]. With respect to y we choose the Haar CONS

CONVERGENCE AND (, 1, 1) SUMMABILITY 217

{C,’ (y)}, where 9Co) (y) 1 on [0, 1], ,1)(0) 2t", 9C,2") (1) -2t", 9C.) (y)2tin ((2k 2)/T/ < y < (2k 1)/2"/t), -2t" in ((2k 1)/T/x < y _<2k/2"/1), 0 elsewhere in [0, 1] for (k 1, 2"; n 0, 1, 2, ...) [5]. Thekernel functions,

are non-negative and take on values in such a way that the correspondingLebesgue functions r. (0, 1) [3].-2. (y) are all 1 onThe Fourier-Haar series is given by

(5.1) amo U.o (y) + Y] a’::gC.’(y) re(x),m--O n--O

where

$(s, t)(s)a:2’ (t) ds dr.

Let S()., $()=. be the partial sum and partial (C, 1, 0) sum respectively corre-sponding to the kernel function K,.(x, -’K()2. (Y, t).A method of proof used in Fourier series [7] enables us to prove the following

convergence theorem for this series. In this theorem it would be desirable toreplace the factor log (m + 2) in (i) by log (m + 2), but the generMiation ofthe corresponding theorem for simple series seems to break down at one ira-portant point.

THEOREM 5.1. If

a,.o + log2(m + 2)k-I

then the Fourier-Haar series (5.1) converges a. e. on E(O <_ x <_ 2r, 0 <_ y <_ 1).

The proof uses the following lemmas:(i) If the series u,.. is (C, 1, O) summable with bounded partial (C, 1, O) sums,

$., 7.’"- (1 (j 1)/m)ui and the mn-th partial sum S,.. of the seriesis o(tt:a), where t, is convex and ---,0, then the series t.,u., is convergent. IfAbel’s transformation is applied twice to the subscript j in the summu, (i) then follows by a proof similar to that in [7; 58].

(ii) The partial sums S(. of the Fourier-Haar series are o(log m) a. e. on E.Proof. In order to prove (ii) we need the following well-known results on

the differentiability of multiple integrals, namely, that for a function f of classL(p> 1)

(5.2) ,limo (hk)-a f’/, f/ /(s, t) y(x, y) Ids dt 0 n.e. [6!

218 JOSEPHINE MITCHELI

and

If(s, t) f(x, y) ds dt < D(x, y) < a. e.

for (0 < !!_, 0 < I1

_1).

Let y be an interior point of (0, 1).function ().. (y, t) assumes that

It is seen from the values that the kernel

K()(y, Of(s, t)dt k. f if(s, t)dr,

where i. is a sub-interval of [0, 1] of length k:, and k --,0 as n --,o [3]. Conse-quently setting f(x -t- s, t) -I- f(x s, t) 2f(x, y) F(s, t) (where outside of(0, 2)f(x, y) is defined by f(x, y) f(x :1: 2r, y)), ,k. F(s, t) dt F(s), andproceeding by well-known methods [7], we have

K,(x, s) ll/(, t)- f(, y) ids dt

(5.4)

sin2 sin 1/2s F..(s) ds

sin (m q- 1/2)s F(s) ds

where (1/m <_ <_ r) is an arbitrary fixed positive number.

k.r F(s, t)ds dt 0(1)(5.5) I < - -Similarly

(5.6)

By (5.3)

for fixed

ds(m + 1/2)s

F(s, t)ds dt 0(1).< 2r-mk

Finally by integration by parts with respect to s

L. <_ k,r- -s F,,(s) ds

(5.7) - -k,,o(n, i) mk,,O(m-, i) + k

219

where (s, i) j’ F(s) ds f; , F(s, t) ds dt. Now for sufficientlysmall and n sufficiently large by (5.2), ks-l(s, i) o(1) for 0 < s .Consequently I o(log m) and (ii) is proved.

Proof of the Theorem: Sce the series (a) of Theorem 5.1 is convergent by theRiesz-Fischer Theorem [1], there exits a function g, L whose Fourier coeffi-cients b equal a- log (m + 2).Obously the corresponding hypotheses assumed for the (C, 1, 0) kernel and

Lebese functions as Theorems 3.1 and 4.1 will lead to silar results forthe paial (C, 1, 0) ss8 of an orthogonal series. Consequently since eachfactor of the (C, 1, 0) kernel functions of the Fourier-Haar series is non-negativeand the corresponding Lebese function is bounded, the partial sums 8 arebounded a. e. and approach a lit a. e. in E. Also from (ii),o(log(m 2)) (k 1, 2"). Hence the hypotheses of (i) are satisfied forthe Fourier-Haar orthogonal development of g and the series (5.1) converges a. e.on E.

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1. RALPH PALMERAGNEW Onbvar$goa8, Proceedings of the London MathematicalSociety (2), vol. 83(1932), pp. 4.

2. STEFAN BANACH, r la verv pree par$o dfcionelle lieaire8 Bulletin desSciences Mathematiqu (2), vol. 50(1926), pp. 27-32, 36.

3. A. HAR Zur Theorie der or$glen Funk$ionysteme, Mathematische Annalen, vol. 69(1910) pp. 331-371.

4. S. KACZMARZ, r vr a 8obie des &veIoppe orhogoux, StuaMathematica, vol. 1(19), pp. 87-121.

5. S. CZMARZ AND H. STEINUST&r Or$gonalreihen Monografje Matematyczne,vol. VI, Warsaw, 1935.

6. A. ZY6MUN On differabili$y of multiple negrals Fundamenta Mathematicae, vol.23(1934), pp. 143-149.. A. ZYGMUND, Tro$r# Monografje Matematyczne, voh V, Warsaw, 1935

UNIVERSITY OF ILLINOIS