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I nternat. J. Math. Math. Sci. (1978)47-56 Vol. 47 ON L I-CONVERGENCE OF WALSH-FOURIER SERIES C. W. ONNEWEER Department of Mathematics University of New Mexico Albuquerque, New Mexico 87131 (Received February 14, 1977 and in revised form October 25, 1977) ABSTRACT. Let G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. LI(G) Ll(G)-norm. As In this paper we formulate a condition for functions in which implies that their Walsh-Fourler series converges in a corollary we obtain a Dini-Lipschltz-type theorem for L I (G) convergence and we prove that the assumption on the LI(G) modulus Similar results also of continuity in this theorem cannot be weakened. hold for functions on the circle group T and their (trigonometric) Fourier series. Let G be the direct product of countably many groups of order 2. Thus G-- {x x,y E x G the sum x (xl) 0 + y with x i {0,i} for each i >_ 0}, and for is obtained by adding the i-th coordinates of C.W. ONNEWEER 48 >_ 0. x and y modulo 2 for each i The topology of G can be described by means of the (non-archimedean) norm if x 0 0 and x k Xk_ I G and for k {x e G llxll <_ 2 -k} lxll on G, where II011 i, and the subgroups Gk of G by G O Gk II’II 0. Also, if we define 0}, 0 --...=Xk_ I For k then the Gk form a basis for the neighborhoods of 0 in G. k, -k >_ I {x e G; x we define the cosets l(n,k), 0 < n < 2 2 >_ 0 k If 0 < n < 2 of Gk as follows then n can be represented uniquely as n with b i e{0,1} in G and let b02k-I + bl 2k-2 + ...+ bk_ I, Let e(n,k) for each i. e(n,k) l(n,k) Gk. (bo,bl,...,bk_l,0,0,...) So, in particular, l(0,k) Gk. Furthermore, in order to simplify the notation, we shall denote e(l,k) by e (k). Next, let Its elements are the denote the dual group of G. Walsh functions and Paley defined the following enumeration for them. For each k >_ 0 and x #k(X) e G define (xi) 0 by k(X) exp(ixk). If n > 0 is represented as n with a i e{O,l} a0 a12+ ...+ + a k 2k for all i, then the n-th Walsh function defined by a Xn(X) 0 0 is a (x)-..." kk(x). Let dx denote normalized Haar measure on G. . Xn For f e LI(G) we define its Walsh-Fourier series by f(x) k--O }(k) Xk(X), where (k) g f(t)Xk(t)dt. CONVERGENCE OF WALSH-FOURIER SERIES For the partial sums of this series we have n-1 7. Sn(f;x) f(k) f(x-t)Dn(t)dt-- Xk(X) k--0 where , Dn(X) G n-i k=O k (t) Dn (t) f 49 is called the Dirlchlet kernel of order n. The following properties hold. (D1) If n > 0 is expressed as n 2 k + n’ with 0 < n’ < 2k then D2k(X) + k(X)Dn’ (x). Dn(x) (D2) For each k > 0 we have k, 2 D2k(X) (D3) If f e LI(G) 1 < x e l(m,k) IS2k(f)-f Ii .I O. n. < 2 k and 0 < n < 2 m Gk, if x e G k G k [0 0 we have D (0) n (D4) for each n > (DS) If k > 0, lk_= then if x e: k, then for each e(m,k) + Gk we have ISn(x) [Dn (e(m k))l < m-12k+l A proof of these properties and additional information on Walsh-Fourier [2]. series can be found in y e G the function f Theorem I. as n / Proof. 0 < n’ . Y Finally, if f is a function on G and if is defined by f Let f be a function in Then ISn(f)-f Ii Y (x) LI(G) o(I) as n f(x-y). for which n / . Let n > 0 be given and assume that n < 2 k. Then lSn(f)-fll I <_ 2 k + n’ f-f e(n) with lSn(f)-S2k(f)ll I + lS2k(f)-f II I. o() C.W. ONNEWEER 50 Thus, according to (DI) and (D3) we have IISn(f) IIl Ik Dn’ < fl * I + o(i) A + o(i), as n / . In order to find the appropriate estimate for A we continue as follows. 2k_l I (p,k) G p=O k (t)Dn’ (t) f (x-t) dt Clearly, Dn,(t) l(p,k) l(2p,k+l)l(2p+l,k+l) l(2p+l,k+l) then whereas if t [ A < [ D p--0 G l(p,k)G. is constant on each set n’ (e(p k)) IDn’(e(0’k))I 2k_l and if t E l(2p,k+l) then k(t) k(t) (x-t)d (x-t)dt G I (2p+l, k+l) If(x-t)-f(x-t-e(l,k+l))Idt dx I G Gk+ I -f(x-t-e(2p+l,k+l))Idt =B+C. According to (D4) we have IGk+ Ilf(x-t)-f(x-t+e(k+l))Idx B <n’ I II n’ f-f dt G e (k+l) II I at o(i) as n / Gk+l Finally, if we use (D5) and apply Fubini’s Theorem we obtain 2k_l C <_ p=l e(k+l) p G I dt i, Therefore -i. I (2p,k+l) I IGk+ p--i Also dx CONVERGENCE OF WALSH-FOURIER SERIES 51 2k_l I p-I 2k+l 2-(k+l) ilf_f e (k+l) i p=l < cI log 2kl f-f e (k+l) o(i) as n i / according to the assumption of the Theorem. II ISn(f)-f Thus o(i) as n/ Before stating a corollary to Theorem i we first introduce some additional terminology. LI(G) For f Definition i. > 0 the integral modulus of continuity and is defined by sup{l ml(; f) 2-(n+l) lle(n) ll si== Ify-fl Ii; lYll for each n >__ <_ 6}. 0 we see immediately that the following holds. Corollary i. If f e ISn(f)-f Ii then Remark i. LI(G) and if o(i) as n o(llog ml(;f) 61-1 as / 0, / Corollary i can be considered as the L I analogue of the Dini- Lipschitz test for uniform convergence of Walsh-Fourier series, see [2, Theorem XIII]. We first show that Corollary i is weaker than Theorem i by giving an example of a function f e as / by fk(x) i >_ Xk; such that (i) If-f e(n) II I 0 and (ii) n For each k LI(G) 0 and x then f(x) fk (xi) 0 e L I) o(i) as n ml(6;f) + o(llog 61-1 / in G define the function and lfk (k+i)-3/2 fk(x)" k=O i I " fk Next let C.W. ONNEWEER 52 Clearly, the sum defining f(x) converges for each x e G and applying the Monotone Convergence Theorem to its partial sums we see that I(G). f e L Fix r >_ Then for all k 0. >__ if k k(X), 0 and x e G we have + r, fk(x-e(r)) -fk(x), if k r Thus, [ f(x-e(r)) k--O (k+l)-3/2fk(x) + (r+l)-3/2(l-2f r (x)). Hence, f(x) and since f(x-e(r)) 2fr(X) 1 (r+l) _r(X), -3/2 (2f r (x)-l) we obtain II f-f e(r) II i (r+I)-3/21 l*rl 11 Next, let d(r) k--r e(k). o(r Then for each k I k(x)’ I fk(x), fk(X_d(r)) -i >_ 0 and x e G we have if k < if k >_ Thus f(x) (k+l)-3/2(2fk(X)-l) [ f(x-d(r)) rl r k--r [ (k+i)-3/2 k(X)" k--r From a well-known inequality for Rademacher functions, see [4, Chapter V, Theorem (8.4) ], we obtain llf-fd(r) ll I >_AI( for some positive constants A I [ (k+i)-3)i/2>_ k;r and A 2. A2r-i Therefore, since lld(r) ll 2 -(r+l) CONVERGENCE OF WALSH-FOURIER SERIES i 2-r ;f) we see that + o(r-l), + o(IXog ml(6;f) that is, 53 as 6+0. We shall now show that Corollary I is the best possible in the following sense. Theorem 2. There exists a function f in L (i) converge in L I(G) Proof. 61-1 0(flog l(;f) properties: I(G) with the following two and (ii) {S (f)} does not n In [2, p. 386] it was shown that if n 2 n I + n 2n2 + + 2 with n I > n 2 > ...> n then -n -i llDnll [ n P( p=l 22s + 22(s-l) Thus, if n 2 [ 2 r). r=p+l + 22 + 20 for some s > 0 then s p-i p=l r=0 + [ 2- 2p [ --(s+l)- 2 mr) S >(s+) [ > p=l _f 2 Also, it follows immediately from (D2) that for each k > 0 we have llD2klll Next i. if k is odd and a k for each n > 0, let I if k is even. n [ n-i k= 0 n D 2n+l (x) and 2 k with a Furthermore, for each n let P (x) ak Qn(X) D 2n+ n (x) k >_ 0, 0 54 C.W. ONNEWEER {Ienlll Then polynomial 1 and if n is even then Qn "part" of is {]QnlII the polynomial P > n/4, and the (Walsh) In order to simplify n the notation we shall from here on write k’ for 2 k and k" for (k’)’. Consider the function f(x) k=l 2-k (k+l) (x) Pk, (x) Clearly, the series defining f(x) converges for all x 0 and f L I(G). For k > i we have Qk’l IS(k+l),,+ (f)-S (k+l)’ Thus, the sequence {Sn (f) with 0 < g < 1 and let Then lYll all k _< <_ does not converge in L <__ (x) X(s+2) . implies that n < 2 k:l 2-(1+1)< Ps (x) hence n(X+y) is of degree _> ’. < 2 $ -1. The last condition that (s+2)’ > k-l, so that 2 Xn(X) -s Consequently, we have G 2-k IX (k+l)" (x-y) D (k+l)" (x-y)-x (k+l)" (x) D (k+l)" (x) G + 2 -k k=s+l + k:s+l G 2-k (k+l),, (x-y)D (k+l),, (x_y) il(k+l) ’’(x)D(k+l) G 0(. -1 for all x in G and all n such Ilf(x-Y)-f(x) lfY-fl Ii < Next, take any G. Choose the natural number s so that for X(k+l),,(X)Pk, (x) is of degree < ’, whereas implies that 2(s+2)" > 1’ that 0 I(G). i implies y e $ the polynomial G l2-kx (k+l)" Qk’ II i >-- 2- k/4 ll i be the natural number for which s the polynomial Also y (k+l)" (f) ’’(x) ). . CONVERGENCE OF WALSH-FOURIER SERIES 0 + 2 2 -k 0(2 -s) 55 0(-i). k--s+l Therefore, if that is, lYll <_ 6 then I-i). 0(flog l(;f) lfy-f Ii 0(flog 2 -(+1) This completes the proof of Theorem 2. As was observed in the abstract, except for some minor Remark 2. modifications the theorems presented thus far also hold for functions defined on the circle group T and their (trigonometric) Fourier series. In this context we have Theorem i’ LI(T) If f e ISn(f)-f ll o(1) as n and if log n / . If-f /n II i o(I) as n / then Theorem l’ can be proved by modifying the proof of a test for uniform [i, Chapter 4, 5]. Also, convergence of Fourier series due to Salem, see in order to see more clearly the similarity between Theorem i and Theorem i’ If-f e(n) II i le(n) ll)-ll If-f e(n) II 1 we mention that the condition n is equivalent to Theorem 2’. as / log( There exists an f e L 0 and (ii) {Sn (f) I(T) o(i) as n + o(I) as n such that (i) does not converge in LI(T). In order to prove Theorem 2’ we use a result of F. Riesz, who showed that for each n > I there exists a trigonometric polynomial P 2n such that "part" of en llenlll i, and a polynomial and such that ....llQnll I in Theorem I Qn n of degree of degree n, which is > C log n, see [i, Chapter VIII, 22] or [3]. REFERENCES i. Bary N. K., A Treatise on Trigonometric Series, Vol. II, MacMillan, New York, 1964. 2. Fine, N. J., On the Walsh functions, Trans. Amer. Math. Soc. 65(1949), 372-414 56 C.W. ONNEER 3. Zygmund, A., A Remark on Conjugate Series, Proc. London Math. Soc., 34(1932) 4. 392-400. Zygmund, A., Trigonometric Series. Vol. I, 2nd. ed. Cambridge University Press, New York, 1959. KEY WORDS AND PHRASES. Walsh-Fourier Series, Convergence in norm, Dini-Lipschltz-type test. AMS(MOS) SUBJECT CLASSIFICATION (1970). 42A20, 42A56, 43A75.