A GENESIS FOR CESÀRO METHODS

Transcription

1 90 R. P. AGNEW [February 5. R. E. A. C. Paley ad N. Wieer, Fourier trasforms i the complex domai, Amer. Math. Soc. Colloquium Publicatios, vol. 19,1934, New York, pp. 151 ad E. T. Whittaker ad G. N. Watso, A Course of moder aalysis, Cambridge, 4th éd., THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY AND SYRACUSE UNIVERSITY A GENESIS FOR CESÀRO METHODS RALPH PALMER AGNEW 1. Itroductio. The Cesàro methods C r, itroduced by Cesàro 1 because of their applicability to Cauchy products of series, costitute the most publicized class of methods of summability. The regular Nörlud methods 2 of summability costitute oe of the two most publicized geeral classes of cosistet methods of summability. The regular Hurwitz-Silverma-Hausdorff methods costitute the other. This ote proves the followig theorem. THEOREM. The Cesàro methods are the oly methods of summability, regular or ot, which are both Nörlud methods ad Hurwitz-Silverma- Hausdorff methods. Thus if the Cesàro methods had ot bee previously itroduced ito mathematical literature, they could be defied ad exploited as the uique class of methods of summability ejoyig all of the properties of Nörlud methods ad all of the properties of Hurwitz-Silverma-Hausdorff methods. I 4, it is show that the oly methods which are both Riesz methods ad Hurwitz-Silverma-Hausdorff methods are methods T r closely related to the methods C r. 2. Nörlud methods. Each sequece po, pu of real or complex costats for which P ^po"\-pi+ +p ^0 for each defies a Nörlud method of summability by meas of which a sequece So, su is summable to a if cr ~><r where Preseted to the Society, August 14, 1944; received by the editors Jue 19, E. Cesàro, Sur la multiplicatio des series, Bull. Sci. Math. (2) vol. 14 (1890) pp N. E. Nörlud, Sur ue applicatio des foctios permutables, Luds Uiversitets Arsskrift (2) vol. 16 (1919).

2 1945] A GENESIS FOR CESARO METHODS 91 (1) <T = (^0 + p~\sl + * + pls-1 + P<)S )/P. The class of Nörlud trasformatios (1) is idetical with the class of triagular matrix trasformatios (2) öfcsfe fc-0 for which (3) 0> 7* 0, = 0, 1, 2, (4) #fc ^ 1>» = 0, 1, 2, ad, for each 2 = 0,1, (5) 2,, there is a costat b q j such that Q", q === UqQ"y % *z q. 3. Hurwitz-Silverma-Hausdorff methods. These methods (hereafter the HSH methods) costitute the class of triagular matrix trasformatios which commute with the arithmetic mea trasformatio (6) <r = (5o + 5i + + s )/( + 1) ad hece also with each other. As Hurwitz ad Silverma 3 ad Hausdorff 4 have show, with each method HSH there is associated a geeratig sequece Xo, Xi, such that the trasformatio takes the form (7) * «(- ïyc.jkiiï (- iyc itk s k 7-0 A-0 or (8) AT * ï 1 & Œ0 L i**o J Assume that (8) is a Nörlud trasformatio, ad let the quatity i brackets i (8) be deoted by a k. The, a =X so, by (3) ad (4), X 5^0 for each ad Xo = l. Moreover (9) a,~i = w(x _i X ) 3 W. A. Hurwitz ad L. L. Silverma, O the cosistecy ad equivalece of certai defiitios of summability, Tras. Amer. Math. Soc. vol. 18 (1917) pp F. Hausdorff, Summatiosmethode ud Mometfolge. I ad II, Math. Zeit. vol. 9 (1921) pp ad

3 92 R. P. AGNEW [February ad hece (S) with 5 = 1 guaratees existece of a costat (which we call r istead of qi) such that (10) w(x-i X ) = rx, g 1. Here r caot be a egative iteger; otherwise oe could set = r i (10) ad cotradict X-r-i^O. Therefore (11) X w = (*/(» + r))x-i, - 1, 2,. Sice Xo = 1, (11) implies that (12) Xi = 1/(1 + r), X, «(2/(2 + f))xx - 1-2/(1 + r)(2 + r) ad i geeral (13) X = /(l + f)(2 + r) (w + r) = lr!/(» + r)!,» = 0, 1,, where r\ is the factorial fuctio, T(r+1), defied for all complex r except 1, 2, 3,. By use of the familiar idetity (14) 2Lt d-fc+r-l^r-l fc=0 == C +r,r, T 7* 1, 2, *, we obtai the familiar fact that for each r5* 1, 2, the Cesàro trasformatio C r, (15) <T = 2^, IC-fc+r-l.r-l/C+y.rJ^A;, fc«=0 is the oe ad oly Nörlud trasformatio for which a =X = \r\/(+r)\. This completes the proof of the fact that the oly HSH trasformatios which are Nörlud trasformatios are the Cesàro trasformatios. It is oly whe r = 0 or r is a complex umber with a positive real part that C r is regular ad ca be writte i the Hausdorff form (16) <r = f EC,***(l - t)«- k s k d X (t) J 0 fc«o where (17) x(0-1 - (1 - ') r - For each complex r ot a egative iteger, the more geeral Hurwitz- Silverma trasformatio (8) becomes C r whex = w!r!/(w+r)!.

4 19453 A GENESIS FOR CESÂRO METHODS Riesz methods. Each sequece po, pu for which P ^po+pi + +^7^0 for each determies a Riesz trasformatio (18) <T = (p0s0 + plsi + h ps)/p,»» 0, 1, 2,. If oe of the two trasformatios (19) <T = ]C«ö ^> ^l' = E fl»,ii-fcî)li fc«=0 fc«0 is a Nörlud trasformatio, the other is a Riesz trasformatio. It ca be show, by a suitable modificatio of our treatmet of HSH ad Nörlud methods, that if A is both a HSH method ad a Riesz method, the there is a costat r ot a egative iteger such that A has the form (20) <T = 2_, [Cfc+r-l,r-l/C w +r t r]sfc. Jfc«0 Thus tóe 0wZ;y methods which are simultaeously HSH methods ad Riesz methods are the methods T r defied by (20). For each r^ 1, 2,, the trasformatio T r is obtaied from C r by reversig the order i which the elemets a%l are applied to So, $i,. The HSH sequece X geeratig T r is Xo = aoo = 1 ad (21) X = a = r/(» + r),» > 0. The followig discussio applies oly to regular trasformatios. The elemets a & of the matrix of the Hausdorff trasformatio geerated by x(0 are give by (22) a, k «f C,***(l - t) ~ h dx(t). If (22) holds, the a,-k = f C,***-*(1 - t) k d x (t) ad hece = - f C w,*«*(l - W) W ~^WX (1 - U) (23) a ^h = f C..*/*(l ~ *)^V«[l ~ x(l - *)]. It follows that if x(0 geerates oe of the trasformatios i (19), the xi(0 s l""x(l""0 geerates the other. Suppose ow that x(0

5 94 R. P. AGNEW geerates a regular Riesz method. The 1 x(l~0 geerates a Nörlud method, which must be a Cesàro method C r. Hece 1 x(l~~/) = 1 (1 t) r ad xco ^ Sice t r geerates a regular HSH method oly whe c Rr>O i ad sice V geerates the regular Riesz method T r whe 9^>0, we have proved the followig result. The methods T r for which %r >0 are the oly regular methods of summahility which are simultaeously Riesz methods ad HSH methods. The idetity r\\ (r \)\\ r (24) =, r^o, -1,, ( + r)l (» + r- 1)! + r implies the idetities C r ~C r -.\T r T r Cr-i ad r r =»C r C^! 1 = C r "l 1 1Cr ivolvig the methods T r ad C r ; this is a cosequece of the fact (Hurwitz-Silverma, loc. cit. p. 7) that if X» geerates A 1 ad X ' ' geerates A", the X 'X" geerates A'A". From the fact that C r ad the Holder method H r are equivalet (C r ~H r ) whe î^r > 1 we obtai, whe c Bj>O t the familiar formulas (25) C r ~ H r = i^r-lhl ~ C-.i#i = CiC r _i ad CrC^J^Ci. This gives the fact, proved by Hausdorff, loc. cit., that T r is equivalet to G whe î^r>0. CORNELL UNIVERSITY