Mathematische Zeitschrift

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1 Math. Z. 83, (983) Mathematische Zeitschrift 9 Springer-Verlag 983 Generalized Hausdorff Matrices as Bounded Operators on! p* David Borwein Department of Mathematics, University of Western Ontario, London, Ontario, N6A 5B7 Canada. Introduction For p > let I p be the normed linear space of all complex sequences x = {x,} with norm i co \lip IIxlIp= ~ Z tx.f') \n= / Let B(l p) be the normed linear space of all bounded linear operators on I v into l', so that a matrix A=(a,k)eB(IP ) if and only if, for every x~l p, y,=(ax), = ~ a,kx k is defined for n=o,... and y={y.}e/p. The norm [[A[Ip of a ma trix AsB( p) is given by < oo IIAIIp= sup IIAxllp. [Ixllp= < Weighted mean matrices. Let a-- {a,} be a sequence of positive numbers and let A, = ~ a k. The weighted mean matrix M, = (C,k) is defined by k= C,k=~, for O<k<_n; C, for k>n. The following theorem is due to Cartlidge [3]. Theorem A. If p >, p > c > and A,+ I A <=c+ ~ for n=s,s+l,..., an + an then Ma~B(l; ) and, when s=, IlMallv<=p p-c" * Supported in part by the Natural Sciences and Engineering Research Council of Canada, Grant A-2983

2 484 D. Borwein The primary object of this paper is to extend Theorem A to generalized Hausdorff matrices. Generalized Hausdorff Matrices. Suppose in all that follows that 4= {4,} is a sequence of real numbers with 2 o >, 4, > for n>, and that e is a function of bounded variation on [, ]. For O<_k<n, let 2,k(t)=--2k+ ).....,2~i!(2k_Z).(2_Z),_ tzdz <t<l, 2.k( )=2.k(+), C being a positively sensed closed Jordan contour enclosing 2,,2k+~,...,2,. We observe the convention that products such as 2k+... 4,= when k=n. Let 2,k=~2,k(t) de(t ) for O<_k<_n, )~,k= for k>n, (2) and denote the triangular matrix (2,k) by H(2, e). This is called a generalized Hausdorff matrix (see [2]). We shall prove the following theorem. Theorem. If p >, c > and 2,+<e+2. and if ~ t "c/p ]de(t)] < % then where H(2, e)eb(i p) and for n=s,s+l,..., [IH(2, e)l[pn#l/pst-c/p [de(t)] when s= 2k+ 4, # = max ' when s >. o_~k_~,_~ (2k + c)... (4,_ +c) Hardy [4] established this theorem for ordinary Hausdorff matrices, i.e., 4, =n, and showed that in this case, if e is non-decreasing, then IlH(2, e)llp = ~t -/pde(t). Jakimovski, Rhoades and Tzimbalario [5] extended Hardy's re- sults to the case 2, = n + a, a >. A "generalized weighted Hausdorff" matrix W=(w,k ) is defined by WO ~--.~, Wnk=2nk(t].k/.)~n) lip for n>l; and W is defined to be the matrix ([w.kl). Borwein and Jakimovski [2] proved that /f p >, (3) ~<2<.. <An,.~n -"~ OO,

3 Generalized Hausdorff Matrices as Bounded Operators on I p 485 and if (2) holds with a normalized, i.e., ~()= and 2a(t)=~(t~-)+a(t-) <t<, then W, lyveb(io, Wll~<= llwllp and for Let [d~(t)l- I~( +)l _-< II WIt,_-< ~ W~(t)l. Do=(l+2o)do=l, D,=(l+~)...(l+~)=(l+2,,)d, for n>l. (4) Then 2o t- ~ d k for n>. (5) D"=2"+id"+l-l+2o It is known (see [2]) that O <2.i(t)=< ~ 2.k(t)N for O_<t_<l, O<=j<=n, (6) k= i dk S2.k(t)dt=-- for _<k_<n. (7) o Dn When a(t)=t and 2o=, H(2, a) reduces to the weighted mean matrix M d with d= {d.} given by (4). Conversely if d= {d.} is a sequence of positive numbers with d o =, then (4) yields a sequence 2= {2.} such that H(2, a) becomes Md when ~(t)=t. These observations together with (7) show that Theorem A is a special case of Theorem. 2. Preliminary Results * - 2k/2 n for n >_. Then, for m >_ n >_ O, Lemma. Let 2"=2oo, 2.k--2.k 2ink= ~ 2k*. (8) k ~ k~ ~ Proof It follows easily from () and (2) that, for m>_k>_o, 2,.+,k -2.,k = (2m+,k 2k -2,.+,k+ 2k+ /2,.+ - We proceed by induction on m. Clearly (8) holds for m=n. Assume (8) holds for some m_> n. Then m+l m+l m -- Z 2k *--- ~.(2m+~,k--2,~k)+2m+l,m+t--2,.+l,. * k=n k=n k=n m k2_n(2m+i,k2k 2m+l,k+2k+l)+2m+l,m+i 2m+l,n 2m+l = = 2,.+,. 2./2.,+ - 2m+,,.+ + 2m+ i,,.+ a - 2"+ i,. =. Thus (8) holds with m + in place of m. This completes the proof.

4 486 D. Borwein * - >. Then Lemma 2. Let 2~o(t ) =2(, 2.k(t )- 2,k(t ) 2ff2, for n o y' 2L(t)< = for _<t_<, n>_o. _ k=. Proof By Lemma and (6), we have that, for m>_ n_>, < t <, The desired result follows. 2k,(t )- ~ 2,.k(t ) <. k=ri k=n 3. Proof of Theorem Let < t <, and let w.=w.(t) = ~ 2.k(t) xk where x= {x.}el p. Then, by H6der's inequality and (6), (9) and so Iw.l'< ;~.k(t) lxkl p I Z 2.k(t) < 2.k(t) lxkl p k= \ i k n=o n=k () Let 2,=2.+c and define 2,k(t) by () with {'~n} in place of {2,}. Since 2k+... 2,<#2g... 2,- for O<k<n by (3), it follows from () that Hence, by Lemma 2, and so, by (), Now let Then, by (2) and (9), 2.k(t)t--#2.k(t)2k/2. n=o Y,,~.k (t) t ~ < ~, n=k for n>k. Iw.lP~#t -c ~ IXkl p. () Yn= ~ 2nkXk" Y. = S w.(t) dc~(t). (2)

5 Generalized Hausdorff Matrices as Bounded Operators on l p 487 It follows from () and (2), by a form of Minkowski's inequality, that ( oo \lip / o \l/p yol,.i / oo \l/p i.e., I[yll, < llxllp~l/pst -c/p Id~(t)l. This completes the proof of Theorem. 4. A Subsidiary Theorem Theorem 2. If p >, d,+ ~:-d, for n > s, and S t-lip Id~(t)l < o, then H(2, c~)eb(lp). o Proof By (4) and (5), we have that, for n >s, o.+ o. (;) --2n D n +<. 2"+a d,+ d. ~/,+ The desired result is now an immediate consequence of Theorem. Cartlidge [3] proved the special case ~(t)= t (i.e., H(2, e)=md) of Theorem 2. References. Borwein, D., Jakimovski, A.: Matrix operators on p, Rocky Mountain J. Math. 9, (979) 2. Borwein, D., Jakimovski, A. : Generalization of the Hausdorff moment problem. Canad. J. Math. 33, (98) 3. Cartlidge, J.M.: Weighted mean matrices as operators on ft. Ph.D. thesis, Indiana University Hardy, G.H.: An inequality for Hausdorff means. J. London Math. Soc. 8, 46-5 (943) 5. Jakimovski, A., Rhoades, B.E., Tzimbalario, J.: Hausdorff matrices as bounded operators over p. Math. Z. 38, 73-8 (974) Received November 3, 982