ON STRONG GENERALIZED HAUSDORFF

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1 ON STRONG GENERALIZED HAUSDORFF SUMMABILITY Aeta Math. Hung. 53 (---2) (989), D. BORWEIN (Lndn, Ont.), F. P. CASS (Lndn, Ont.) and J. E. SAYRE (Halifax) Intrductin Fr a series Z ak, let sn=.~ ak. Let Q= {qn,k} (n, k=,... ) be a matrix k=o and let a n : Q(s,) = ~ q,,ksk. k= The series Z ak is said t be summable Q t s if a, exists fr n=,... and tends t s as n tends t infinity. In this case we write sn--,s(q). The symbl P is reserved fr matrices {P,,,k} with Pn,k= >, and I dentes the identity matrix. We nw recall the definitin f strng summability intrduced by Brwein []. Strng summability. A series.~ak is said t be summable [P, Q]p (fl>) c t s if ZP,,k [ak--sl a exists fr n=,,... and tends t zer as n tends t infinity. k= In this case we write sn-~s[p, Q]a. Fr summability methds V and W, the ntatin Vc=W means that any series summable V t s is als summable W t s. The ntatin V-~ W means that bth Vc=W and W~V. Generalized Hausdrffmatrices. Suppse thrughut that 2 = {2n} is a sequence f real numbers with ~=~, inf2 n>o and,~l/~n=~~ n--~l n= Let I2 be a simply cnnected regin that cntains every psitive 2,, and suppse, fr n =, I... that Fn is a psitively sensed Jrdan cntur lying in f2 and enclsing every 2kqQ with O<=k~_n. Suppse thatfis hlmrphic in O and that f(2) is defined even when 2~ f2. Define () ~n,k ~ -2k+... ;~, F (;tk-z)... (,~,- z) [ fr k>n +t5 k fr O<-k<-n This research was supprted in part by the Natural Sciences and Engineering Research Cuncil f Canada.

2 96 D. BORWEIN, F. P. CASS AND J. E. SAYRE where 6k=f(2) if k= and 2~f2, and 5k= therwise. Here and elsewhere we bserve the cnventin that prducts like 2k+...2,= when k=n. Dente the triangular matrix {2,,~} by (2;f). This is called a generalized Hausdrff matrix. The set f all such matrices is dented by 2/g~. Fr ~ any real number, the generalized Hausdrff matrix H, is defined t be the matrix (2;f) with f(z)=(z+ )-L Fr ~> -, the generalized Ces~r matrix C~ is defined t be the matrix (L;f) with f(z)-- r(~+~)r(z+~) r(a+z+l) These reduce t the standard H6der and Ces~tr matrices when 2.=n. (See [].) Preliminary results Fr <t=<l, let }%k(t) dente the value f 2,, k btained frm () with f(z)=t z, and let 2,,r,(O)=2n,k(O+ ). Let Then, (see [3), If D = (l+2)d = ; f ~,,~( dt = -~, (2) f(z) = f t ~ where BV is the space f functins f bunded variatin n the dsed interval [, ], then z.,~ = f &~( az(. It fllws that " s that =(l+a,)d, fr n ~. fr O<-k~=n. dz(t ) with 2CBV (3) s.-cl(s.) = C~(2. a.). If f satisfies (2), 2()-2()= and Z(+)=Z(), then X=(X;f) is regular, i.e. s,-~s(x) whenever s,,~s. (See [2; Therem ].) Lemma 2 f [2] shws that if g and h are hlmrphic in f2 and defined at 2, X= (A; g) and Y= (,~; h), then (4) xr = YX = (2; gh). Acta Mathematica Hungarica 53, 989

3 ON STRONG GENERALIZED HAUSDORFF SUMMABILITY 97 Lemma f[3] shws that if X=.;f) with f satisfying (2), a~=(2;f) with f(z)= = f t" Idz(t)l, and fl=>, then, fr any sequence {w.}, (5) IX(w.)l a ~_ MP-~JT(Iw.I a) where M-- f Idz(. Frm (4) it can be seen that H~H~=H~+~ fr all real e, & Therem 2 f [2] shws that (6) C~H~ fr ~>-. (See als [5] and [6].) Thus (7) C~C~-C~+~ fr ~>-, 6>-, ~+3>-. Sme therems n strng summability The first therem generalizes Therem 5 in []. TnEOm~M. Suppse Q is a matrix, P is a regular matrix in 2/fa, and X=(2;f) where f(z)= f f dz(t) with zebv, Z()-Z()=I and Z(+)=Z( ). Then, fr fl=>l, [P, Q]p~[P, XQ]p. PROOF. Let )?={~7.,k}=(A;f) where f(z)=f t~ldx(t)l. Since zebv and Z(+)=Z(), it fllws that lira 2. k= fr k=,... and sup ~ I~,~< ~. (See [2, Therem ].) Hence ~(u,)~ whenever u,~. (See [4, Therem 4].) Let {s.} be a sequence, ~r.=x(s.) and w.=s.-s. In view f the regularity f X we have cr.-s=x(w.)+~, where e.-~. Frm (4) and (5) it fllws that (8) P(IX(w.)I a) <= Ma-IP~(lw.I a) = Ma-~.gP(ls.-sl a) where M= f Idz(t)l. Next, by Minkwski's inequality, <9) <- Suppse nw that P(Is.-slP)~O. Then, by (8), P(IX(w.)Ia)~O s that, by (9), P(l~.-sl-~. Hence [P, I]p=c[p, X]p, frm which it fllws that [P, Q]p~ c=[p, XQ]a" [] The next tw therems generalize crllaries t Therem 7 in []. 7 Acta Mathematica Hungarica 53, 989

4 98 D. BORWEIN, F. P. CASS AND J. E. SAYRE THEOREM 2. If XC 3cFz and fl=>l, then necessary and sufficient cnditins fr a series ~ a. t be summable [C~, X]~ t s are that it be summabte C~X t s and that )..a.-*[c~, CIX]~. PROOF. It fllws frm Therem in [] that ~ a. is summable [C~, X]a t s if and nly if it is summable CaXt s and summable [C~, (I-COX]p t. Further, by (3) and (4), The result fllws. [] (x-cx(s.) = = a.). In cnfrmity with ntatin intrduced earlier (see [ ; p. 23]), the generalized strng Ces~tr methd [C,, C._~]p will be dented by [C, ~]p and the generalized strng H6der methd 9 [ H ~, H,_~] a by [H, ~]~. We require the fllwing knwn result (see [8]). L~t~. Let F(6 + )F(z+ )(z+ ) 6 g(z)= F(6+z+l), 6>-. Then bth g(z) and /g(z) can be expressed as MeIlin transfrms f the frm f t=dx(t) with EBV, X()--Z()=I and X(+)=Z( ). TrmOl~M 3. If ~>= and fi>=l, then necessary and sufficient cnditins fr a series.~ a. t be summable [C, alp t s are that it be summable C~ t s and that 2.a.~[C, a+ ]p. PROOi~. It fllws frm Therem 2 that ~ a. is summable [C, ~]p t s if and nly if it is summable CC~_~ t s and 2.a.~[C, CC~_~] p. Next, it fllws frm Lemma with 6=~- and Therem that 2.a.~[C~, C~C._~]a if and nly if 2.a.~[C, It.]a. Applying Lemma and Therem I again, we see that )..a.~o[c~, C~C._I]p if and nly if 2.a,,-~O[C~, Cja. This tgether with (7) yields the result. [] The abve therem suggests the fllwing extensin f the definitin f [C, ~]p t the case ~= : ~ a. is summable [C, ]a t s if the series is cnvergent with sum s and ~ dk'l kaklp=(dn). When 2.=n, this definitin reduces t the ne k= given by Hyslp [7]. The next therem is an analgue f the equivalence relatin (6) fr strng summability. TnEl~EM 4. Fr ~>=, fl>= l, [C, ~]p~-[h, a]a. Acts Mathematics H, ungarica

5 ON STRONG GENERALIZED HAUSDORFF SUMMABILITY 99 PROOF. The case e = fllws frm Therem 2 and the definitin f [C, ]p. c Suppse therefre that e>. By Therem 3, ~' a.=s[c, el if and nly if z~ a,= =s(c.) and 2.a.~O[C,C.] a. Further, by Therem2, ~a.=s[h,e]p if and nly if ~a.=s(h~) and 2.a.-~O[C~, H~]p. The result nw fllws frm (6), Lemma and Therem. ~ Generalized Hausdrff matrices assciated with L p functins Let L' dente the functin space L"(, ). In this sectin we deal with Haus- drff matrices (2;f)with f(z)=f t~(t)dt where ~EL ~' fr sme p>l. An rdinary Hausdrff matrix {X.,k} satisfies these cnditins if and nly if ~ Ix,,,~lP< k= <M(n+ ) I-p fr n=,... where M is independent f n. (See [4, Therem 25].) The fllwing lemma is needed fr the prf f Therem 5. L~MMA2. Let q~cl p with p>l. Let X=(2;f) and X(P)=(2;f("O where f(z)=f ~z~(t)dt and t~lq)(t)lpdt. If # fl=l and /p=l/#-l/fl, then fr any sequence {w.}, where M= f l~(t)l p dt. JX(w,)l' <= M ~<I-/B) (c (Iw, lp))'/a-:x <p) (lw, t p) PROOF. Let f.(t)= ~2..k(t)w k where =<tn. Then, by H6der's inequality, k=o (See [3, (8)].) Hence () and If.C a --< Z 2,,k(t)Iwkl a. k= n f lf,(t)[ tjdt<= ~[wk[ p f 2",k(t) dt'='"~nk~= dk[wklo=cl(lwnl #) k=o f [q~(t)] p [L(t)] ~ tit ~= [wj f 2.,k(t)Ig(t)lPdt = X(P)(]w.[a). k=o 7* Acta Mathematica ttungarica 53, 989

6 D. BORWEIN, F. P. CASS AND J. E. SAYRE It fllws, by H6der's inequality, that IX(w.)l = if ~(,)I.( dt I <= (f l~(t)l"dtll-"(f If.(t)l' dtll/'-l/"(f I<:(t)l'lf.(t)l dr) ~/" -~ Mt-Vp(C~(IwJ)Va-ll"X@)(Iw.lO))v ~. [] The fllwing therem generalizes Therem in []. T~EOREM5. Let i z fl=l, /p=l+l/#-l/fl. Let X=(Z;f) where f(z)= = f t'q)(t)dt with 9EL p and f q)(t)dt=l. Then, fr any matrix Q, [C, O]aC= c=[ca, XQ]~. The therem remains valid when #= c (with lip = - /fl if 2> and p-- if fl-- ) prvided [C~, XQ]= is interpreted t mean XQ. PROOF. We use the ntatin intrduced in Lemma 2, and nte that X is regular and X~ whenever v,~. Suppse that s,~s[c~, Q]p, and let a,=q(s,), w,=a,-s, and v.=ca(lw,[p). (i) Suppse tt is finite. By hypthesis, v,-~ and hence, by Lemma 2, Cl(IX(w.)l ~) <- KCaX@)(IwJP) = KX(P)(v,) -~ where K=M# (-~Ip) sup v~/p-. Als, by the regularity f X, we have X(a.)-s= =X(w,)+ e, where e,-~. Thus, by Minkwski's inequality, (Ca (IX(aD-sl")) v" ~- (ca (IX(w.)l")) /" + (c (~.")) v" --, i.e. s,~ s[c, XQ],. (ii) Suppse nw that #= c. By H6der's inequality, = If dtl" m f IJ.(t)l" at where m=m p- if fl>l and m=ess sup [q~(t)[ if fl=l. Since () hlds under <t<l the perative hyptheses, it fllws that and hence that s.-,-s(xq). [] IX(w,)[# ~ mcl(iw, I #) = my, ~ O, Tra~Ol~Za6, Let Q>l/fl-/# where #>-_fl>=l. Then, fr any matrix Q, [cl, Q]~ c=[cl, c~ QL. Acta Mathematica Hungarica 53, 989

7 ON STRONG GENERALIZED HAUSDORFF SUMMABILITY PROOF. When /z=/~, the result fllws frm Therem. Suppse that /t>/~ and let /p=l+l/it-/~. Then CQ=(2;f), where f(z)=f tzp(t)dt with p(t)= = ~ ( - ~ Since pele, Therem 5 nw yields the result. [] THEOREM7. Let?>c~+ llfl--/# where #>=B>-_I and ~ is any real number. Then [H, ~]a~[h, 7]u. PROOF. Applying first Therem 6 and then Therem tgether with Lemma, we get [H, ~]a = [//, H~-lJp ~ [H, C~-~H~-I]~ ~ [/-/, Hr-,H~-a]v = = [u~, u~-d~ = [u, ~],,. [] R~MARI(. It is knwn that, in the special case 2,=n, Therem 6 als hlds when =l/fl-/# and Therem7 when?=~+l/fl-/p. (See [] and the references there given.) Whether the same is true fr mre general 2~ is an pen questin. References [] D. Brwein, On strng and abslute summability, Prc. Glasgw Math. Assc., 4 (96), [2] D. Brwein, F. P. A. Cass and J. E. Sayre, On generalized Hausdrffmatrices, YurnalfApprx. Thery, 48 (986), [3] D. Brwein, F. P. A. Cass and J. E. Sayre, On abslute generalized Hausdrff surmnability, Arch. Math., 46 (986), [4] G. H. Hardy, Divergent Series, Oxfrd University Press (949). [5] F. Hausdrff, Summatinsmethden und Mmentflgen I, Math. Z., 9 (92), [6] F. Hausdrff, Summatinsmethden und Mmentflgen II, Math. Z., 9 (92), [7] J. M. Hyslp, Nte n the strng summability f series, Prc. Glasgw Math. Assc., (95--3), [8] W. W. Rgsinski, On Hausdrff methds f summability, Prc. Cambridge Phil. Sc., 38 (942), (Received Nvember 4, 985) DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF WESTERN ONTARIO LONDON, ONTARIO N6A 5B7 CANADA DEPARTMENT OF MATHEMATICS MOUNT SAINT VINCENT UNIVERSITY HALIFAX, NOVA SCOTIA B3M 2J6 CANADA Acta Mathematica Hungartca 53, 989