SOME ABSOLUTELY EFFECTIVE PRODUCT METHODS

It is proved that the product method A(C,1), where (C,1) is the Cesàro arithmetic mean matrix, is totally effective under certain conditions concerning the matrix A. This general result is applied to study absolute Nörlund summability of Fourier series and other related series.

than the absolute summability defined by the product matrix A.B (if it exists).
It may observed that the condition (1.2') and a ]ortiori (1.2) are automatically satisfied whenever A is a lower triangular matrix (that is, a. 0 for k > n ).
(1.3) (a) for each fixed n, there is a positive integer r(n) such that {cnk} is nondecreasing for <_ k <_ r(n) and nonincreasing for k >_ r(n), or and then (1.1) defines the series-to-series (N, p)-transformation.The (N, p)-transformation with p.
(iii) IF-effective ir {-'e,()lv(0, )inplies that F'(x)is IAI ummable end absolutely total effective if it is effective in each of the sens defined almve.
For a general mtrix A. 'rripathy [11] (s als Kuttner and Tripathy [8]) and, Kuttner and Sahney [7] have obtained sufficient conditions s that A is F-effeetive.The restrictions itnposed on ,I it [11] arc qtit(' general.Ittt it is ,tsually (lillicult to v(,rifv them for special 1" iterest.The ature of the corresponding conditions used in [7] is such that they can be easily v.ritied, and, theretbre the following result due to Kuttner and Sahney h the advantage o1" Iavig sone direct al)plications.THEOREM A. Let the matrix A be absolutely coeative and a 0 for all n, k.Suppose that either (b) for each fized n, there ts a posttive tnteger s(n) such that {cr,,,/k} is nondecveasin9 for < k < s(n) and nonmcreasng for k > s(n).
Suppose also that in case (a).for ko > r(n)+ko r(n))_2ko and tn case (b), for ko >_ o.t, 0(1 Then A is IFol-effective. Starting with an absolutely conservative method A. we obtain in the present paper a sub licient condition that connects the proof of absolutely total effectiveness of ..I(C', 1) with the proof of It"l-effectiveness of the A matrix.As we shall see.such a result has some interesting applications.We will first prove the following.
We shall obtain the following corollaries to 'rhrcm I.
2. SOME PRELIMINARY RESULTS.We shall nd the following lemm for the proof of Theorem and its corollies.LEMMA 1.The necessary and sufficient condition that A be absolutely conservative is that for all k 0, Xh result of mm is wn known.S, .g., [9] o [6].
LEMMA 2. Let {a,} be a given sequence; then for any number z we have -(1 .rakz az TM a,,z" + a,+ where m and n are integers such that n m 0. The proof of Lemma 2 is a straight forward calculation.LEMMA 3. If A satisfies the hypotheses of Threm A, then (1.7) holds.PROOF.Observing that (sin kt)/k is the imaginary part of exp(ikt)/k, we shMl prove (1.7) by following closely the proof of Threm A, which contorts the proof for ]b,(t)l O(1) with b,,(t) the A-transformation of (sinkt)/k.Thus.considering those values of n (if any) for which r(n) 5 20(or s(n) 5 20), we have that {a,/k} is nonnegative and nonincreing for k 20.Therefore, by an application of Abel's Lemma to the inner sum and using the fact that the partial sums exp(ik/) are O(l/t), we have cxp(ikt) ,(n)<O k=20 n=l (or (,)<O) th 0(1) timat i otincd fron Lmma and tim hypothss that A is solutly onsrvtiv.
Lemma 4 may be prov by following the prf of mma 10 in [4].

IP.I lPl
The first part of the lemma follows directly when we observe that k + and appeal to the hypotheses {}B and {S}B.The second part follows an carlier result (see [5], Lemma 3; see also [2]).
Lemma 6 may be proved by applying Abel's transformation and using the result that the partial sums exp(-ikt) are O(1/t).
Ve shall first prove (2.3). ,'Now observing that Pk P-t P.  g,lA&l IP, I (n + 1)IP.I by virtue of Lemma 5 and the hypothesis that { }eBV.Taking 0 if a > & breaking the range for k into w parts vi.. k < 0 and k 0 and observing that by Lemma , Pg O(IP), we here N I,lPol nlP.-,I + I, nlP._,l pexp(-ikt)l.
Since p P(R/(k + 1)} and the partial sum of exp(-ik) O(1/), an application of the (2.5) Writing r' for min (r, n 0), we see that by a change in the order of summation, we have for any by virtue of the condition (1.2).In view of (3.4) and (3.5), we have 2 [tR(n,t)-ot,,, ReS,(u)du]dda(t) (3.6) b,,(x) -rr where R(n, t) is the real part of Since by hyl)othcsis fo Id,(t)l <_ K, in order to prove that A(C, 1) is IF, l-,:frertive, it is sufficient to slmw that the following estimates hold for re(0, S' S .,1,c,.ReS(u)dul 0(1).
We lirst proceed to prove (3.7).Breaking the range of summation for the inner sum into < k < 20 and 20 < k, we use for the former range the fact that &,(t) O(1), while for the latter rlulge we replace b(t) by the following expression whicl is equal to it: (t exp(it))-'k+l {.1 -oxp(i,'t)-exp(i[k + ,]t)}.
Thus. if b(x) denotes the A(C.1)-trsformatioa of F'(x) then in view of (3.4), we have _4 " !lt)I(n,t) sin(t)R(n,t)IdO(t).But by hypothesis (t)eBY(O,r); therefore, in order to prove the [Fl-effectiveness of A(C, 1) it is sufficient to show that (3.7) holds.We thus complete the proof of the [Fl-effectiveness of (IV).IFl-effectiveness: For the series F'(x) Z u,(x), we have and.therefore, 2 [ d u,(z) --(t)tt Re{ (t)}dt. (3.9)Following the proof of the [F l-effectiveness part, we deduce IF]-effectiveness of A(C, 1) from (3.7)   and (3.8) by comparing (3.9) with (3.2) and then appealing to the hypothesis that ((t)/t)eBV(O, ).Since (t)eBV(O, ) implies that ,(t)eBV(O, r), the IFol-effectiveness of A(C, ) is included in its F-effectiveness.Thus combining (I) (IV), we complete the proof of Threm 1.
To prove Corollary we note that if A satisfies the hypotheses of Threm A, then Lcmma 3 ensures the condition (1.7) of Threm 1.This together with (1.2) implies the conclusion of Theorem l, which is also the conclusion of Corollary 1.Since for the (N, p) transformation, (1.2) holds automatically we use Lcmma 4 and Lemma 7 to s that the hypotheses of Threm are satisfied and in conclusion we obtaitt Corollary 2.
We shall also prove the following.