Journal of Research of the National Burea u of Standards-B. Mathematics and Mathematical Physics Vol. 63B, No. 1, July- September 1959 Relations Between Summation Methods and Integral Transformations 1 Werner Greub 2 (M ay 12. 1959) R elations between the Lototsky method of summation and t ho se of Borel, Euler, a nd Knopp are obtained by associating an integral transformation wit h t he series transformation. 1. Introduction 2. General Relations Between Series and Integral Transforms In a r ecent paper , R . P . Agnew [1] 3 es tablishes re­ lations between the Lototsky method of summation for divergen t series and the classical summations of Let an, (n,v= 1,2, ... ) be an infinite matrLx whose Borel, Euler, and Knopp. clemen ts arc complex and subject to th e co ndition It is the purpose of t hi s paper to show that these relations can be ob tain ed in a very natural way if the lim ·~/ l anvl= O (n= 1, 2, . .. ) . (1) series tra nsformation is associated with an integral .->'" transformation . The following scheme describ cs the connections between the series and the integral We consider a llOt necessa rily convergen t infinite transforma tion : senes original series ~ transfo rmed seri es '" ~ U. (2) . =1 IL auren t expansion 1T aylor expansion wi th the proper ty analy tic fun ction ~ in tegral transform (3) of t be function In section 2 it is shown that und er certain con­ This cOllditioH assUl' C' s tha L Lite rad ill S 0 f convergence di tions a regular seri es tran sformati on can be of the power series ass igned to every integral transforma tion , and con­ versely. As an example, the integral transformation ± U. corresponding to the Loto tsky seri es transformation . =1 Z, (L-transformation) is constr ucted . This integral is positive. transfo rmation is used in section 3 for the di scussion Under cO:lldi tiolls (J ) and (3) all the series of the relation betwcen the Lo totsky and th e Borel integral methods. The ou tcome of this discussion is 00 that L-summability contains Borel integra l-summa­ Un= ~ anvuv bihty (BI-summability) but docs not contain a .=1 modified Borel sllmmability (BI*-summability as denoted in [1]). It is shown that the L-method can arc convergen t, as can readily be seen with th e help be ext ended to a summation method which is equiva­ of the root test. Ther efore, we can say that lInd er lent to the BI*-method. those conditions the matrix (an.) defines the series In section 4 it is proved that a power series is tl'ansfol'ma tion L-summable in all interior points of the Borel polygon '" U n---'7U n= ~ anvu v • (4) of t he corresponding analytic function but not v= 1 L-sllmmable in all the exteri or points. This proof is based on the relationship between the L-method We will show now th a t one can associa te witlt thi s 1 a nd the BI-methocl. series transforma tion a certain integral transforma­ In secti on 5 the Lototsky method is compared tion b etween the analy tic functions with t he method of Euler-lCn opp. In the last section the in verse of the L-integral '" u transform is constructed and Llsed to obtain the f (z) = ~ ,; (5) inverse m atrix of the Lototsky series transformation. v= 1 Z and 1 'r his wo rk was carried ou t in part under a Nation al B ureau of Standards 00 contract with T he America n University. n 2 P resent address: U n ivers it. y of Ztir"ic h, Zurich, Swi t zerland. ¢ ( a)= ~ U na . 3 Figures in brackets ind icate the literature references a t the end of this paper. n= l 1 To obtain the kernel of this transformation we From (7 ) we obtain between the coefficients U v and define the functions P ,, (z) by Un the rela tions co 1 1 p ,Jz) = L::; an V+ I Z ' (n = l , 2, . .. ). rf: t/>(a) ( )2rf:rf: K(z,a) n ~ O U n= 27r'i J an+1 da= 27ri J'j an+ 1Z' j(z)dzda Because of (1 ) these series are convergent for every = ( _1 "K(z ,a) v d Z· . )2pp L..; n+ l U• zea. (8) z and thus P ,, (z) are entire functions of z. 27r' ~ v a Z It follows from (3) that the functionj(z) is regular outside of a certain circle, Izl= p. Now we integrate Hence the integral transformation (7) defines a series the product, P n(z)j(z) along a circle Izl=R (R > p), transformation whose matrix is given by and obtain 1 rhrh K(z,a) l d t/>P,,(z)J (z) dz=rh L::; a "V+ lu vz '~d z nv ( )2 J V,jl a = 27ri 'j 'j an+ 1z' e Z a. = 27r'i ~ a n ~11~ = 27r'iUn ' (6) I' ~ l It can be shown that under certain hypotheses about ' the kernel the series transformation (8) is regular in UndeI' the h)'pothcsis that the function the following sense: If the series :L:uv is convergent, v the series L::; Un is Abelian summable (A-summable). n K(z, a)= ~ P n(z)an n ~ l This means that the limit is regular III th e domain Izl< oo, la l< l , we derive (a real) from (6) exists and the relation Hence, ddilling the function t/>(a) for la l<l by CO ) co lim ( L::; U n an = L::; 11 . a - H - O n ~ J v ~ l t/> (a) = ~ unan, n ~ J holds. The hypotheses about the kernel are: (i ) K(z,a) is a regular function of z and a in the we obtain thc integral transformation 4 domain Izl < 00, Ia l< 1 and therefore can be expanded in the form '" K(z, a)J(z)dz, (7) K (z ,a) = :L: an+ l(a)Zn. (9) </> ( a)= 2~i ~ n~O which maps every function j(z) which is regular (ii) For every fixed n we have outside of a certain circle into a function </>( a) which is regular within the unit circle. lim an (a)=l (a real) . Conversely, let (7) be an integral transformation <>-71- 0 whose kernel K(z, a) is regular in the domain Izl< 00 , !a l<l and fulfills the conditionS K(z,O) = O. (iii) There exists a cons tan t M not depending on ex Substituting forj(z) a function6 regular for Izl>l, such that co L::; !an+1 (a) - an(a ) I ~M. '" u n~ l J(z)= L::; ~ , v ~ l Z , To prove the above statement we consider the trans- i we obtain for t/>(a ) a function which can be expanded formation into a power scries within the unit circle, </>(a )= 2~i PK (z ,a)j(z )d(z) </> ( a) = ~ U nan• n~l and substitute for K(z,a) and f(z) the expansions (9) and (5), respectively. So we obtain the series­ to-function transformation • The path of integration in the z-plane will always bc a circle Izl =R (R > I) . • This is not an essential restriction because it can be achieved by replacing K (z, a ) by K (z ,a)-K (z,O) . Then the function .p(a ) is only changed by a con­ stant. '" • It is assumed that there is no constant term in the expansion ; in other words, t/> (a) = L::; u vav(a). t hatf( co) ~ o . v~ l 2 U sin g a wcll -known I'cgul al'i Lv Lh eor em we concl udc scn es ~ U n is convcrgcnL rcspectivcl.\T A-summable from conditions (ii) and (iii) Lh at this transformation n is regul ar ; th is m cans, th c limit lim 4>(0' ) exists pro- the origin al series ~, ?j" will bc call cd L- ummable a - H-O v idecl Lhat th c series ~u , is conver gent and the r cspcctively AL-summable, and wc wri tc r elalion 00 lim 4> (0') = ~ u,. (11 ) a - H - O v=l and h olds. lim (~ U nan) = AL (~ u ,, ). Thc statement which has just becn provcd can bc a ~ 1-0 n v formulated as the following: TlmoR EM 1. L et K (z,a) be a function with the 00 For the integral kernel, K (z,a)= L. P ,,(z) a", of the properties (i), (ii), and (iii), and let the matrix anv n=l be defined by L-transformation, the functions 1-',,(z) JIaVC thc rol­ lowing form: 1 )2rh rh K (z ,a) a",= ( 27ri J J an+ 1zv dzela. P 1(z ) =Z Then the series transformation induced by this matrix 1 (_ 1) 11 (l -Z ) carries every convergent sel'ies ~ 1L , into a11 A -summable P,,(z)=rJ P n- I (z)= z - 1 n (n= 2, :) , ... ). v series ~ Un with the same sum. (15) n It is worth notin g lha t thc sc ri cs ~ Un n ee d not This lcads to 11 be convergent itself. However, if the mat ri x an, fulfills additional ill equalities of the form K (z, a) = _ 1_ ± (_ I )" (l -Z ) a ,,=~- a )I -Z - I . z-1 n= 1 n z- l "';;-"L..J 1a nv I=<M- (11 = 1, 2, ... ) (12 ) v=l n Hence , defining the a nalytiC' (unction cf> ( a ) (10' 1< 1) b.v wJlcr c J.1I1 is a fi xcdnumber , it foll ows th al ] p(l _ a)I-Z_1 cf> (a)= -2 .