AN ARITHMETICAL PROPERTY OF SOME SUMMABLE FUNCTIONS

Transcription

1 MATHEMATICS AN ARITHMETICAL PROPERTY OF SOME SUMMABLE FUNCTIONS BY J. F. KOKSMA (Cmmunicated by Prf. J. A. SCHOUTEN at the meeting f May 20, 950). Intrductin... If X dentes an irratinal number, it is weil knwn, that the sequence X, 2 X, 3 X,... is unifrmly distributed md ; frm this it fllws that () N Lim N 2: g(nx) = f g(t) dt, N-+ n=l 0 if g(t) dentes any bunded R-integrable functin with perid. It is clear that () generally becmes false, if in stead f suppsing that g(t) is Riemann-integrable, we nly assume th at g(t) is Lebesgueintegrable, since we can arbitrarily change the value f g(t) at au pints nx (md ), withut changing the integral. In 922 A. KHINTCHINE ), cnsidering the special case that the peridic functin g(t) fr 0 ~ t < dentes the characteristic functin f a measurable set & in (0, ), intrduced the questin whether in this case () is true fr almst all x. He prved this t be the fact under certain cnditins as t the nature f &. One may generalise KHINTCHINE'S. prblem by replacing the sequence (nx) by (À"x), where (2) À < À 2 <... dentes a sequence f increasing integers. In the special case I.,. = a,n, where a, IS a fixed integer > 2, RAIKOFF prved that the frmula (3) N Lim N 2: g(ànx) =: f g(t) dl N-+ == 0 hlds almst everywhere in x fr any (peridic) functin which is Lebesgueintegrable. 2) F. RIESS 3) pinted ut that RAIKOFF'S therem is actually an in stance ) A. KHINTCHINE, Ein Satz über Kettenbrüche mit arithmetischen Anwen dungen. Math. Zeitschr. 8, (923). 2) D. RAIKOFF, On sme arithmetical prperties f summabie functins. Rec. Math. Mscu, N.s., (936) (Russian with eng!. summary). 3) F. RIEss, 8ur la thérie ergdique. Cmment. Math. Helvet. 7, (945).

2 960 f the classical ergdic therem. This may be the cause that RAIKOFF'S special case Ä." = a n is easier t deal with than KHINTCHINE'S case J... = n. KAC, SALEM and ZYGMUND 4) prved a generalisatin f RAIKOFF'S result, cnsidering lacunary sequences (2), where Ä." need nt be an integer, but where Ä.,,+ > q Ä.", q denting an arbitrarily given cnstant >. They prved that (3) hlds almst everywhere in x, if g(t) dentes a functin f the class L2, the Furier cefficients f which satisfy a certain cnditin. As far as I knw, further results cneerning the abve prblem are unknwn, ó) even in the case Ä." = n.. 2. In this paper I shall cnsider a general class f sequences (f(n, x)) (n =, 2,... ; O~ x <) and a general class f peridic functins g(x) E L2 and I shall prve that the relatin ( 4) N Lim N L g(f(n, x)) = f g(t) dt N--> 0-0 hlds almst everywhere in x (0 < x ~ ) (Therem 2). The class f sequences (f(n, x)) includes e.g. a. the case t(n, x) = xn b. the case t(n, x) = Ä."x if (Ä.,,) is any sequence (2) f real numbers satisfying Ä.,,+ - Ä." ~!5 > 0, where!5 is a cnstant, c. the case t(n, x) = ( + x). Obviusly all these cases are included in the fllwing therem, which itself, as we shall prve in. 4, is a special case f the main therem 2. 4) M. RAc, R. SALEM and A. ZYGMUND, A gap therem. Trans. Amer. Math. Sc. 63, (948). 5) Af ter the finishing f the manuscript, I discvered the interesting recent paper f P. ERDÖS, On the strng law f large numbers. Trans Amer. Math. Sc (950). Mr ERDÖS deals with the special lacunary case (2) which als has been treatea by KAC SALEM ZYGMUND (l.c. 4)) and gives an interesting imprvement f their result. But still mre interesting is his Therem, which states that there exists a functin g(x) E L2, and a sequence f integers such that fr almst all x ~ f g(ä."x) -+. N n~ It fllws frm ERDÖS'S therem that in my abve Therema, 2 in any case sme cnditin n the Furier cefficients, like (3), is necessary.

3 96 Therem. Let b dente a psitive cnstant. Let (, x), (2, x),... be a sequence l real numbers delined lr every value l x (0 < x < ), such that, lr each n =, 2,..., the lunctin t(n, x) l x has a cntinuus derivative t~ > b which is either a nn-increasing r a nn-decreasing functin l x, whereas the expressin I~(nl' x) - ~(n2' x) fr each cuple f psitive integers n -== n2 is either a nn-increasing r a nn-decreasing lunctin l x lr 0 < x ::;;, the abslute value f which is > b. Let g(x) E L2 dente a peridic functin f perid, and putting let g (x) ~ C + L c k e 2 "ikz k ~ - k*o where 'YJ(N) dentes a psitive nn-increasing lunctin, such that L.J ~]N(N cnverges. Then (4) hlds almst everywhere in 0 < x <.. 3. We nw state the main result f the paper. Definitin. Let I(n, x) lr n =,2,..., be a real ditferentiable lunctin f x lr 0 < x <, the derivative l which is either a nn-increasing r a nn-decreasing, psitive functin f x. Then put (5) A" = Max (/, (~, 0), /' (~, )). Further, let lr each cuple l integers n >, n2 >, n -== n 2 derivate (j)~ l the lunctin the (6) be -== 0 and either a nn-increasing, r a nn-decreasing lunctin l x in O<x<. Then put (7) lr all integers M > 0 and N >. Definitin 2. Let g(x) dente a peridic functin l the class L2 in < x < with perid and mean value 0, i.e. 6) (8) and put (9) g(x) ~ L c k e 2nikz k~ - (m > 0). 8) As the assumptin C = f g(x) dx = 0 may be made withut 0ss f generality, we shall put C = 0 fr the rest f the paper.

4 962 Therem 2.. Let 0' O2, Oa dente 8uitably ch8en ab80lute cnstants. Let ](n} and ]2(n) dente tw p8itive nn-increasing lunctins l n = I, 2,... 8uch that ( IO) ~ TJl(n) < ',L.; n ' 2. Let f(n, x) (n =,2,... ) dente the functins f Definitin I and let (ll) M+N L A" < 0 N (N ~ I),,- M+l (2) A (M, N) < 02]~ (N) (M ~ 0, N ~ I). 3. Let g(x) dente the lunetin f Definitin 2 and let (3) (m ~ I). (4) Then N Lim N L g(f(n, x» = O. N->,,- I. 4. It is clear that the functins f(n, x) f therem I satisfy the cnditins f therem 2, fr if ne ranges the N numbers f~(m + I, x), f~(m + 2, x),..., f~(m + N, x) in increasing rder, the difference between each tw cnsecutive nes is at least {J, and we find and Hence M+N ~ ~,L.; ~:::;;,L.; ",5 < '" lg 3N.,, - M+l j ",(,) ' - ".- N ~ ",-M+l N ~ 2 :::;; 2 - < - lg 3 N.!",(n, x) -j",(nl' x) - ",5 5 '- M+N M+N M+N ~ A" < ~ I + ~ I :::;;; lg 3N.. -M+l,, - M+l j",(n, O).. - M+l j",(n, ) A(M,N) < : IOg;N. I. 5. With a different methd, which nly hlds fr the case f(n, x) = À,. x, where the Ä" are integers satisfying (2), I have prved a therem, which is smewhat sharper than Therem I, as the inequality fr L / c.. / 2 is replaced by a weaker ne. I shall publish thse.. -N+l results elsewhere.

5 2. Sme Lemma's Lemma. Let the functin ( 5) g (x) R:i L C k e 2 "ik" k~ - belang t the class L2 and put ( 6) (m > 0) and ( 7) a G(x,a) = 2aLg(x+t)dt (a> 0). Then (8) G (x, a) R:i L C k e 2nih ', k-- where (9) C _ sin2nka k - 2 nka Ck (k:s 0), (20) (2) f I C k I < --/ RI' (p > I), k=p+l - 2na p ~ IC I < VR;; k-l k - 4a ' whereas the relatin (22) G(x, a) = L C k e 2nikx hlds unifrmly in x. k- - Remark. The cnditin g(x) E L2 is nt required fr the prf f all the statements f the lemma, but a refinement in that directin wuld be f n use fr ur purpse. Prf. As is prved in textbks n Furier-series,7) any Furierseries, whether cnvergent r nt, may be integrated term by term between any limits; i.e. the sum f the integrals f the separate terms, is the integral f the functin f which the series is the Furier series. Nw applying this prcess t the functin g(x+ t) R:i L (c k e2 "ikx). e2 "ikt, k- - integrating with respect t t between the limits - a and a, we immediately find (22) and (9). We nw shall prve the frmula (2), frm which it is immediately clear that (22) hlds unifrmly in x. 7) E.g. cf. E. C. TITCHMARSH, The thery ffunctins, Oxfrd Univ. Press, (939).

6 964 Frm (9) we have fr p ~ 0, q ~ using the CAUCHy-SCHWARZ-RIEMANN-inequality; henee and, if p ~ : ~ IOkl ~ ~ VB < VR;; k-l 2na V6 4a I>f 0 I ~ RIl. (f du)i/. = ;R;, k=l>+l k - 2na I> ut 2na V p' I> whieh prves (2). Nw frm the faet that (22) hlds unifrmly in x we deduee immediately that the right hand side is the Furier series f its sum, whieh prves (8). Finally we nte that (20) is an immediate ensequenee f (6) and (9) because f I sin 2 nka I ~. 2nka - Lemma 2. f g(x) E L2 and G(x, a) (a> 0) dente tke functins f (5) and (7) and if Rm is defined by (6) (m ~ 0), we have (23) f Ig(x)-G(x, a) 2 dx ~ R m' a'+ 8R m, fr eack integer m ~ wkick satisfies 2!lma~. Prf. By Lemma we have (see (5) and (8)) henee g(x)-g(x, a) ~ L (c k -O k )e 2 "ik:, k - - () G( ) ~ ( sin2nka) 2nik: g X - x, a ~ L. c k - 2 k e, k- - n a by (9). Nw, sinee g(x) and G(x, a) belng t L2 (G(x, a) being a entinuus funetin f x), we have, by PARSEVAL'S therem, (24) Nw we nte that ( _ sin 2 nka)2 < 4 2nka -

7 965 and als that ( I_Sin2nka)2 < (2nka)4 (6,5)4m4 a nka = (3!)2 < 62 < ma, if < k < m, and 2 TC m a <. Therefre we have by (24) m Jlg(x)-G(x,a) 2 dx<m 4 a 4 L Ic k 2 +8 L Ic k l 2 < k~ k~m+l Q. e. d. Lemma 3. Let I(n, x) lr n =, 2,... dente a real cntinuus lunetin l x lr a < x < b, and let (j)(~, n2, x) = l(n I, x) - l(n2, x) fr n -::/::- n 2 have a cntinuus derivative (j)~, which is -::/::- 0 and either nn-decreasing r nn-increasing lr a < x < b. Finally, put Then we have lr N > 2, h > 0 (h nt depending n n and x) b N A SI L e 2ni h/(n.xl 2 dx ~ (b-a) N + 2 N2. a n~l h This lemma I have prved in a previus paper. 8) I de duce frm it the fllwing Lemma 4. Let I(n, x) lr n =, 2,... dente the lunctins l Delinitin, and let A (M, N) be delined by (7). Then lr each integer h-::/::-o which des nt depend n n and x we have f MiN e2nihf(n.zi2dx < N + A(M,N) N2. n~m+l h (M > 0, N > ). Prf. Fr M > 0 cnsider the sequence I(M +, x), I(M + 2, x),... These functins satisfy the cnditins f Lemma 3 with I(M + n, x) in stead f I(n, x) and with a = 0, b = l. The crrespnding number AN defined in Lemma 3 (a = 0, b = ) is identical with the nu mb er A (M, N) which was defined by (7). Therefre Lemma 4 is an immediate cnsequence f Lemma 3. Lemma 5. Let I(n, x) lr n =, 2,... dente the lunctins l 8) J. F. KOKSMA, Ein mengentheretischer Satz über die GIeichverteilung mdul Eins. Cmp. Math. 2, (935).

8 966 Delinitin, let g(x) dente the lunetin l Delinitin 2, let G(x, a) (a> 0) dente the lunctin which is delined by (7) and linally, let M+N (25) S(M,N,x;a) =.L G(f(n,x),a). n-m+l Then lr each integer p > j l,s(m, N, x; al2dx~ 4 RNp+ 8 RN2A (M, N) lg 3p+ (26) _ + N '/,,: R Rf) + ~ N2 / R Rp A (M N) N2 Rf) nal I p na r p, g P naz p, where A (M, N) and Rf) are delined by (7) and (9) (Delinitin and 2). Prf. Using Lemma, we deduce frm (22) and (25) M+N M+N S (M, N, x; a) =.L.L Ok e2nik/ln.z) =.L Ok.L e2nik/ln.z). n- M+l k- - k- - n-m+l Rence (p ~ ), as the inner sum is in abslute value ~ N. Therefre we have and applying the CAUCHY-SCHWARZ-RIEMANN-inequality f) f) M+N IS(M,N,x;a)2~4(.L IOkI 2 )( LIL e2nik/ln''')2)+ k-l k-l n- M+l f) M+N + 8N (.L IOkl) (.L IOkl.L e2nik/ln.z)i) + 4N2 (.L IOk!)2. k- f)+l k-l n- M+l k-v+l Integrating this and applying (20) and (2) we find: (27) Nw we have, by Lemma 4, j I MiN e2nik/ln.z) 2 dx ~ N + A (M, N) N2. n- M+l k

9 967 Mrever applying the CAUCHy-SCHWARZ-RIEMANN-inequality fr integrals, we find M+N M+N f I L e2nik f(n.,) I dx < (f 2 dx)i/. (f I L e2"ik/(n..,) 2 d~)'/ n-m+l 0 0 n-ji+l by MINKOWSKI'S inequality. Therefre it fllws frm (27) JIS(M,N,x;a) 2 dx<4r i (N+A(~,N)N2)+ k-l and hence, by (2), since we have + 4N I, Rp i IC I (ViV + NI" A (M,N)) + N2. R p na P k-l k k n 2 a 2 P PIP (P )/. V- ~ k~l ICkl. lik < (k~icki2)'/. k~ 7ë < 2 R V lg 3p, fis(m,n,x;a) 2 dx < 4RNp+ 8RN2A(M,N)lg3p+ + _N_'/'--,Vf-"-2R~ l,'r p + 8N2 VB '/ Rp VA (M, N) Vlg 3p + ~22 R p, na P na ' P nap which prves (26). Lemma 6. Let Q (x) > 0 dente a peridie lunctin with perid, which is Lebesgue-integrable. Let p(x) dente a dilferentiable lunetin in (0,), such that p' (x) is a psitive and either a nn-increasing, r a nndecreasing lunetin l x in (0, ). Put (28) A = Max (lp' ~O), lp' ~l), ). Then we have I (29) f Q (p(x» dx < 8 A.r Q(u) d'u. 0 (30) Prf. If lji(u) dentes the inverse functin f u = p(x), we have I f Q (p (x» dx = ''() f Q (u) IJl' (u) du. ''(0) We nw distinguish tw cases. A. If p()-p(0) < 8, then the right hand side f (30) is ''() < A f Q (u) du < 8 A f Q (u) du, ''(0) 0

10 968 because f (28) and the fact that Q >- 0 is peridic. In this case therefre (29) has been prved already. B. If tp(i)-tp(o) > 8, then we write (30) in the frm (3) j where we have put B v+ f Q (tp(x)) dx = ~ f Q (u) lji' (u) du + 7~A v A ''() + f Q(u) lji' (u) du + f Q(u) lji' (u) du, ''(0) B A = [tp(o)] + 2, B = [tp(i)]-3. Nw by the same argument as we have used in A, we have A (32) f Q(u) lji'(u) du < 3,A f Q(u) du, ''(0) 0 (33) y>(l) f Q(u) lji' ('u) dn < 4,A f Q (u) du B 0 We nw deal with the sum n the right hand side f (3) and write B v+ B ~ f Q (u) lji' (u) du = ~ f Q(v + ) lji' (v + ) dv = v - A v.-a 0 B B+ = f Q (v) { ~ lji' (v + I)} dv ~ { ~ lji' (I)} f Q (v) dv,.-a v - A 0 since lji'(u) is either nn-increasing f nn-decreasing. Nw and therefre B+ B+2 ''() ~ lji' () ~ f lji' (u) du < f lji' (u) du = [lji(um!m = I, v - A A-I ''(0) R 0+ (34) ~ f Q(u) lji'(u) du ~ f Q(u) du - A 0 Cmbining (3), (32), (33), (34), we find because f,a > l. f Q(tp(x)) dx < 8,A f Q(u) du, 0 Q. e. d. Lemma 7. f I(n, x) (n =,2,...) and g(x) dente the lunctins l Delinitin and Delinitin 2, and il we put M+N (35) S*(M,N,x)= ~ g(f(n,x)), n-m+

11 969 where M 2 0, N > are integers, we have fr every psitive a, and every psitive integer m, satisfying 2 nma < and fr every psitive integer p the inequality (36) + {4 R Np + SRN2A(M,N) 0g3p + --l- Na/ l/ RR p + 8 N2 / RRp A (M N) N2 Rp }'I. 'na2 p na r p, g P n 2a2 p, using the ntatins f Definitins, 2. Prf. If G(x, a) (a > 0) is defined by (7) and if 8 (M, N, x, a) dentes the sum (25), we find frm (35) by MINKOWSKI'S inequality l {f IS*(M,N,x) (37) j 2 dx}'/. < UI8*(M,N,x)-8(M,N,x; a) 2 dx}'/ {f IS(M,N,x;a) 2 dx}'/. (3S) Nw by (25) and (35) f 8* (M,N,x) -S(M,N,x; a) 2 dx = u M+N =f L (g(f(n,x))-g(f(n,x),a)) 2 dx < n~m+l M+N < N L f Ig(f(n,x)) - G (f (n,x), a) 2 dx n-m+n U using the CAUCHy-SCHWARZ-RIEMANN-inequality fr sums. Applying Lemma 6 with Q (x) = I g(x) - G(x, a} 2 'lp (x) =!(n, x), we find f Ig(f (n,x) )-G (f(n,x), a) 2 dx < S (An+ ) f Ig(u) -G (u, a) 2 du, 0 where A" is defined by (5). Therefre we have by Lemma 2 flg(f(n,x))-g(f(n,x),a) 2 dx < S(A n +l) (R m 4 a 4 + SRm) lr every integer m > which satisfies 2 nma <. Therefre we have by (3S) M+N J 8 *(M,N,x)-8(M,N,x; a) 2 dx < (Rm 4 a 4 +SRm)N L S(A" + ),,~M+l + ais (M,N, x; a) 2 dx r/.. u

12 970 Nw appying Lemma 5 (p ~ ), we find frm (39) and (26) immediately (36), which prves Lemma 7. Lemma 8. Be 'fj (x) > 0 and nn-increasing lr x ~, such that (40) }; TI~) <. N-l Then we have lr any psitive e ( 4) and als N-l. (42) 'fj (N) lg N -+ 0, as N -+. Prf. A. It is clear that TI~e) and therefre is a nn-increasing functin f N ~ TI (Ne) l'f I TI (tt e ) dt ~ ~ <, cnverges. dt du e u Nw putting t" = u, we find -t = - -, hen ce T I T' TI (te) dt =..!. I TI (u) du t e u ', which prves the first assertin, since.) TI ~) <. B. As 'fj(n)fn is nn-increasing and as.) TI~) <, we cnclude N-l N - l frm a weil knwn therem that als the series L 'fj (2") <,,- Since the general term f this series is a nn-increasing functin f n, we have by anther weil knwn therem n'fj (2") -+ 0, as n -+. Nw let N dente a psitive integer> 2 and be n the integer fr which Then we have 2":-::;:: N < 2"+. 'fj (N) lg N <~'fj (2") (n + ) lg as N -+. Q.e. d.

13 97 Lemma 9. such tt Let 'YJl(N) and 'YJ2(N) be psitive nn-increasing lunetins }; 7Jl~N) <, Then V'YJl(N) 'YJ2(N) is als a nn-increasing lunetin l N, such that }; V7Jl (N~ 7Jz (N) <. N~l Prf. It is trivial that V'YJl(N) 'YJ2(N) is a nn-increasing functin. Further by the CAUCHY-SCHWARZ-RIEMANN-inequality which prves the lemma. Finally we use a lemma, which is a special case f a therem due t I. S. GÁL and the authr 9), and which has als been prved in a jint paper by R. SALEM and the authr: 0) Lemma 0. l,functins such tt Let I,,(x) E ij (0, ) (n =, 2,... ) (p > ) be a sequence M+N f I,,~JI+l L In (X) > dx < C (M + N)I>-U NU 'YJ (N), where C> 0, a> are cnstants and where 'YJ(N) dentes a p08itive nnincreasing lunetin l N such that L 'YJ(N)JN <.. Then N N L In(x)~O. as N~(), n~l almst everywhere in (0, ). (43) 3. Pml l Therem 2. We put (N > ) a = 2nNI/.' m = [NII.], p = [NI.]. 9). S. GÁL and J. F. KOKSMA, Sur l'rdre de grandeur des fnctins smmables. These Prc. 53, (950) = Indag. Math. 2, Fase. 3 (950). 0) J. F. KOKSMA and R. SALEM, Unifrm distributin and Lebesgue integratin, Acta Sci. Math. Szeged. 2B, (950). The main therem f this jint paper is clsely related with the abve therems, as it deals with the prblem, whether frm the unifrm distributin mdul f a sequence UI' U 2,, may be cncluded that Lim N N L g(u n + x) = f g(x) dx N-)- n - l 0 fr a given g(x) E L2 and fr almst all x.

14 972 Then m >, p ~, 2:rt m a ~. Further we have by (3) (44) (45) Rm = 0 (J2 (m)) = 0 (J2 (N'/.)), Rf) = 0 (J2 (p)) = 0 (J2 (N'/.)), as J2(n) is nn-increasing. We nw apply Lemma 7. Substituting (43), (44), (45), (ll), (2) in (36) we find + 0 ({ N'I, + N2 JI (N) lg 3 N + NUl. V J2 (N'/.) + NUl. V J2 (N'/.) J~ (N) lg 3 N + N''J2 (N'/.)}'/.). Rence using MINKOWSKI'S inequality and applying Lemma 8, i.e. Jl (N) lg 3 N ~ 0 and J (N) ~ 0, J2 (N'/.) ~ 0, we find dis (M, N, x) 2 dx r/. = 0 (NUl.. + N VJ2 (N'/.) + N VJl (N)). Rence by Lemma 9 fis (M, N, x) 2 dx = 0 (N2J (N)), where J(N) is a psitive nn-increasing functin such that E TJ~) <. N-I Therefre by Lemma 0 and (35) S (0, N, x) = (N) almst everywhere in 0 < x <. Q.e.d.