(1.2) Nim {t; AN ax.f (nxt) C w}=e212 2 du,

Transcription

1 A GAP SEQUENCE WITH GAPS BIGGER THAN THE HADAMARD'S SHIGERU TAKAHASHI (Received July 3, 1960) 1. Introduction. In the present note let f(t) be a measurable function satisfying the conditions; (1.1) f(t+1)=f(t), rlf(t)dt=0 and f1f2(t)dt=1. is a function of Lip c or of bounded varia- In [2] M. Kac noticed that if f(t) tion, then it is seen that (1.2) Nim {t; AN ax.f (nxt) C w}=e212 2 du, where -x =1 ti 2 nk j is a sequence of integers such that (1. 3) limn nk+i/nk=+00 and ax is any sequence of real numbers satisfying the following conditions (1.4) AN=LamMa-+00 and max aki=o(an), as N00 Also in [4] G. Morgenthaler proved that if f(t) is bounded and {ak} satisfies (1.4), then there exists a sequence {f(nkt)} independent of {ak} and (1.2) holds. On the other hand P. Erdos [2] showed that if f(t)=cos 27rt + cos 4 7rt, then 11T 1 f1w iz cos1tx 1 euz mt; f((2-1)t) C G= dx du lim NN x=17t From above facts we see that if (1. 2) holds, the properties of nk+l/nk and the smoothness of f(t) become subjects of considerations (cf. [3]). The purpose of this note is to prove the following THEOREM. Let {nk} and {ak} satisfy (1.3) and (1.4) respectively and for some e>0, (1.5) 1t-S(t)2dtj 1112=O1 1 as n-00

2 106 S. TAKAHASHI where S(t) denotes the n-th partial sum of the Fourier series of f(t). Then, for any measurable set E, E C [0, 1], of positive measure, 1 N 1 wli m1 E t; t E E, A akf(nkt) <W=e2 du. N-co I I N k 1 2 7T From the above theorem it is seen that under the conditions (1.3) and (1.5), a=+cc implies the almost everywhere divergence of the series akf(nkt). On the other hand in [1] S. Izumi proved that under the conditions (1.5) and the Hadamard's gap condition nk+1/nk > q> 1,; a <0O implies the almost everywhere convergence of the sequence lira m-8j c=1 ak f(nkt). 2. Proof of the theorem. By (1.3) and (1.4) we can take a sequence of positive integers {qa} such that and (2.1) nk+l/nk>4 qk for k=1. 2, 3,..., *) (2.1') max gk'2ak =o(an) and qn-+oo, as N -+goo. 1<k5N We put (2.2) f(t)>2 ci cos 2 7rlt and, for k= 1, 2,... Where (2.2') gk(t)cl cos 27lt and Rk=1 ci. z>qtt>k LEMMA. 1. We have 1 1 N 2 11m {-akgk(nkt)} A N dt=0. N-oon PROOF. We have, by Parseval's relation for k>j, gk(nkt)g(n,t)dt = 2 cldi<2 ci (-2 d), j l>qk l>qk l>qk *) The condition (2.1) need not hold for small k, but without loss of generality we may assume that (2.1) holds for all k.

3 By (2. 1), (2.2'), A GAP SEQUENCE WITH GAPS BIGGER THAN THE HADAMARD'S 107 dt= cnkz, if n, nkl, 0, if otherwise. (1.5) and the definition of d1, 1J2 1/2 1 di <1 c i o(1), as (k-j)-+oo, lqk lz41c- Hence, by above relations 1N 2 1 N 1 J 1 AN+akgk(nkt)dt=AN a+gk(t)dt+2 a,ak gk(nkt)g,(n,t)dt 0 k m1 k=t 0 1+1<k+N 0 =1JaxRq + O 2 aka. R1 J 2 1 AN+AN+qk =1 1<7<k<N (k-j) k-1+ek 2 arq 1 1 t Rqbaa2+r ANk 1q N r 2 k=1 r=1 i=1 N A arq+o a2 112 k=1 A Since R2-0 as i -o, by (1.4) we can prove the lemma. and LEMMA 2. We have (2. 3) max aksgk(nkt)=o(an) 1k<N (2. 3') f 1+N A N alsgk(nkt) dt o(1), as N + 0 PROOF. By (2.1'), it follows that max aksgk(nk t) <0< max ak I cl max 2 ak qk=o(an), 1kcN+1k<N Further, by (1. 4) and (2.2'), N 15k<<-N as N-+oo. N 2 2 AN alsgk(nkt) -1 k=1 A N N 2qk q+1 f ak> cos 2 urnkzt =1 akrk NANk k=1 t=1 t-j=1

4 1 S. TAKAHASHI By (2.1) if k=k', then for any 1, l'(1<1<2q. and 1<1'<2q), cos 2 7rnkl t cos 2 7tflk,l'tdt=0, and AN a R2k 0, k=i as N-+o. Hence for the proof of (2.3'), it is sufficient to show that IN AN ak cc= o(l)'as N-p + oo, k=1 l=1 2-j On the other hand, by (2.1') and (1.4), By Lemma 2, we know that if we put IN=O max akgk =o(1), as N+ co. 1SkSN<T (2.4) EN={t; N aksgk(nkt)-1c1 k=1 then (2.4') lim EN=1. For the proof of the theorem it is sufficient, by Lemma 1 and the theorem of Glivenko, for any fixed X and any interval I, to show that cjn(x, I)1j-jexp I r1aksqk(nkt)dt-ei, 1 N A2 as N-+oo. By (2.3), (2.4), (2.4') and that exp the z=(1+z) fact exp + O(Iz3), as z-0,, as N e1 N Za+akSgkCnkt2 N 2L N(X, I)=1+A exp-a2>aksqk(nkt) dt.iii IfEN 1 N 2 N By Lemma 2, (2.4), and the fact that I ex-1 NZXa I<I x I ex 1, ks3k(nkt)2 N L z21+ A exp-2azn aksgk(nkt)-exp-dtien 1 N ex X2 - z aksgk(nkt) + dt -0, as N -. I fl E N pc+2 AN k=1 2

5 A GAP SEQUENCE WITH GAPS BIGGER THAN THE HADAMARD'S 109 Hence for the proof of the theorem it is sufficient to show that (2.5) 11+ ZX aksgk (nkt) alt -1., as N>+oo. IlEN 1 AN LEMMA 3. We have, for all N, PROOF. We have fl 11(1+ 2XakSgknkt) 2 cit M. I N 2 N 2ak CL {- ia,aksgk(nkt) }- a'akt k(nkt) 01 AN 1 2A NX 12akL c22 =I (1 + j 1 +ly(t, X) < e+ 1If(t, x) A Pr(t, x) is the sum of terms of the following form (2.6) (constant) X JJ cos 2 7r-nkglit, where (2.6') 1 <k1<k2...<k, N and qkt (2.6) can be expressed as the sum of the following terms (2.7) (constant) X cos 27r(nkjS+...+nkil). On the other hand by (2.1) and (2.6), nksls-nks-1ls nk1l1>nks nky I-2/4 1-1/4) nks/3>0. Hence N(t, x)dt=0. This completes the proof. By Lemma 3, and (2.4')

6 110 S. TAKAHASHI (2.8) - ZXGlkSgkCnkt/< dt E N- C I/L [j 1 II N ( 1 Za,CLkSgkCZktJ 2 1/L fjen AN 0 1 AN dt0 LEMMA 4. We have Nia S (n t) lim 1+k gk k dt=ji. N-j 1 AN PROOF. If we put [f(1+t)) k gx k=1+on(t, x), then 9w(t, x) consists of the terms (2.9) s J ixa k,cj cos 2 7r nkfl t=s ixak, c'cos 3=1 AN jl 2 AN 2 r If we put f cos atdt<2 jjj(xak,c,cos2 7r nk l t dt C s I Xak jcj I nk). Ij=1 AN 1=1 fn1+ 21, C1SQk(nkt)dt=II+n k lin(x, I), then, by (2.1) and (2.1') for N>N0, C N i Xak I qk-i n A I I fji+ k=l k N l=1 s=i qa I Xasc1 I A+A l=1 N qf Xalc1 II N(X, I) f n Z=1 N 1 1/2 I N k-i f g1/l I 2 I max 2 Xakqk 1 1-2, A as S A,CL1g112 1<ksN ANk=2 nk s=1 N Nn1

7 A GAP SEQUENCE WITH GAPS BIGGER THAN THE HADAMARD'S 111 This completes the proof. I(2N)Zk <max 1SkN (Xakgk=o(1), AN 4 as N -+-oo. By (2.5), (2.8) and Lemma 4, we can prove the theorem. REFERENCES [1] S. IZUMI, Notes on Fourier Analysis (XLI); On the strong law of large numbers and gap series, Tohoku Math. J., 3(1951), [2] M. KAC, Probability method in analysis 55 (1949), and number theory, Bull. Amer. Math. Soc., [3] M. KAC, On the distribution of values of sums of the type f (2t), Ann. of Math., 47 (1946), [4] G. W. MORGENTHALER, A central limit theorem for uniformly bounded orthonormal systems, Trans. Amer. Math. Soc., 79(1955), DEPARTMENT OF MATHEMATICS, KANAZAWA UNIVERSITY.