NOTES ON FOURIER ANALYSIS (XLVIII): UNIFORM CONVERGENCE OF FOURIER SERIES SHIN-ICHI IZUMI AND GEN-ICHIRO SUNOUCHI. (Received April 5, 1951)

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1 NOTES ON FOURIER ANALYSIS (XLVIII): UNIFORM CONVERGENCE OF FOURIER SERIES SHIN-ICHI IZUMI AND GEN-ICHIRO SUNOUCHI (Received April 5, 1951) 1. G. H. Hardy and J. E. Littlewood [1] proved that THEOREM A. If (1.1) f(x+t)-f(x)=0(1/log/ltl (t-0) and the n-th Fourier coefficients of f(t) are of order n-s (0<d<1), then the Fourier series of f(t) converges at t=x. Recently O. Szasz [4] proved that THEOREM B. If f(t) is even (or odd) and continuous, and if av being the n-th Fourier cosine (or sine) coefficient of f(t), then the Fourier series of f(t) converges uniformly at t=x Especially if an is of order n-1, then (1,2) is valid. In the assumption of these theorems, the first is the continuity condition and the second is the Tauberian condition. We shall prove that the assumption of Theorem A is not sufficient to the uniform convergence of the Fourier series of f(t) at t=x. Further, even if (1.1) is replaced by the condition (1.3) f(x+t)-1(x)=0(ltl) (t-0) in the assumption of Theorem A, the Fourier series of f(t) does not converge uniformly at t=x in general. But we can prove that, if, instead of (1.1) and the n-th Fourier coefficients of f(t) is of order (log n)a/n (a>0), then the Fourier series of 1(t) converges uniformly at t=x. The condition (1.4) is the type of Dini-Lipschitz test, and (1.5) links Theorem B and this test. To prove the negative theorem, we construct an example of the type used by one of the authors [2]. For the proof of the positive theorem we use the method due to H. Lebesgue and R. Salem [3]. On the other hand, we can prove that Young's convergence test implies the uniform convergence of Fourier series at a point. This is a dual of

2 UNIFORM CONVERGENCE 299 Theorem B. 2. THEOREM 1. If (2.1) f(t)-f(t')=0(1/log1/lt-t'l), (t-x, t'-x) then the Fourier series of f(t) converges uniformly at t=x. PROOF. We can suppose, without loss of generality, that x=0 and f(t) is even. Let (xn) be an arbitrary sequence of positive number tending to zero and s, (t) be the n-th partial sum of the Fourier series of (t). Supposing 1(0)=0, it is sufficient to prove that For a given E>0, we can take b>0 such that if (2.3) lxn+(t+h)l<d, and lxn+tl<d, then (2.4) If{xn+(t+h)}-f(x,+t)I<E/log (1/h). If we put (2.5) gn(t)=f(xn+t)+f(xn-t), then we have say. Since the point t=0 is the Lebesgue point, In=0(1) by the well known method and Kn=0(1) by the generalized Riemann-Lebesgue theorem (see Zygmund [5], p. 22). If we put n/n=h and using =Ln+Mn+o(1), say. On the first term Lr, by (2.4) we have

3 300 S. IZUVII-G. SUNGUCHI It is therefore sufficient to prove that say Here by the second mean value theorem Also =0(n).

4 UNIFORM CONVERGENCE 301 and since t=0 is the Lebesgue point, we have 1P41=o(n2). Thus we can completes the proof of the theorem. 3. THEOREM 2. If, (3.1) f(t)-f(t')=0(1/logzlt-t'l-1) (t-x, t'-x), and the n-th Fourier coefficients of f(t) is of order (log n)a/n, (a>0), then the Fourier series of f(t) converges uniformly at t=x. PROOF. We shall adopt the simplification in the proof of Theorem 1. Then we have say, where (3.2) Bn=(log n)b/n, (a+1<b). By the condition (3.1) we have easily In=0(1). After R. Salem, we write, putting m=[(log n)b], where 0<0<n/n. If we replace nb+2kd and nb+(2k+1)ƒî by (2k+1)ƒÎ in the last sum, the error is o(1). Thus

5 since B>a+1. Analogously we get K2=o(1), and this completes the proof. 1. THEOREM 3. For any x, there is an integrable function f(t) such that (4.1) f(x+t)-f(x)=0(t)(t-0), and the n-th Fourier coefficients of f(t) are of order n-d (0<d<1), but the Fourier series of f(t) does not converge uniformly at t=x. PROOF. Let (nk) be a sequence of integers such that (4.2) nk+1>e2nk- (k=1,2,...) and let (Ak) and (ak) be sequences of intervals such that Then they are disjoint systems. Let us define an even function f (t) by f(t)=(-1)kt sin nk(t-d/logn) in Az. =(-1)+itsinnk(t-n/lognk) in Ak and f(t)=0 in (0, n)-vk=1(akuak). Then we can easily see that f(t) satisfies

6 UNIFORM CONVERGENCE 303 the condition (4.1) and that the Fourier coefficients of f(t) are of order n-s. Let us now consider the n-th partial sum of the Fourier series of f(t) at t= n where {xn} will be determined later. Then, as in the proof of Theorem 1, Thus (Unk) does not converge to zero. Hence sn (n/logn) does not converge. Thus the theorem is proved. 5. THEOREM 4. If (5.1) Q(t)=f(x+t)+f(x-t)-2f(x)-0, as t-o, (5.2) the function B(t)=tP(t) is of bounded variation in an interval (0,n), and for small h, where A is a constant, then the Fourier series of f(x) converges uniformly at the point x to the value f(x). PROOF. We have

7 where k is a fixed number and tn-0. For any E>0, we shall take b and k such that if Itnl<d and Itl<k/n, them lq(tn+t)i<e. Then Pn<Efk/n0ndt=o(1), and Rn<fƒÎ0q(tn+t)/t sinntdt=o(1) uniformly for to by the generalized Riemann-Lebesgue theorem. Put =0, otherwise, and if we prove that the total variation of an(t) in [0, n] is fin, then we shall have Since Sn(k/n)=(n/k)Q(tn+k/n)=0(n), it is enough to prove the same thing for the variation of Sn(t) in [k/n, n]. Let us write the total variation of f(x) in [a, b] by Vab[f], then say. we have Since I&(t+tn)I=1b(tn+t)-b(o)I<V0i+t[O(t)I<A(tn+t), for large k. Now and

8 <4AFn. Thus we get the theorem. REFERENCES (1). G. H. HARDY AND J. E. LITTLEWOOD, Some new convergence criteria for Fourier series, Annali di Pisa, 3 (1934), (2). S. IZUMI, Notes on Fourier analysis (XVI), Tohoku Math. Journ., 1(1950), ; Notes on Fourier analysis (XXVI), ibid. 2 (1950), (3). R. SALEM, Sur un test general pour la convergence uniforme de series de Fourier, Comptes Rendus Paris, 207 (1938), (4). 0. SzAsz, On uniform convergence of Fourier series, Bull. Amer. Math. Soc., 50 (1944), (5). A. ZYGMUND, Trigonometrical series, Warszawa, ToKYo TORITSU UNIVERSITY AND TOHOKU UNIVERSITY.