RHOADES, B. E. Math. Zeitschr. 81, 62--75 (1963)

A method

of Hausdorff

summability*

By

B. E. RHOADES

Let # = {/~} denote a sequence. Define a linear difference operator /J, operating on/~, by A / , k = / ~ - - / ~ + z, /l~#~=A(~-l/zk); k, n=O, t, 2, .... A Hausdorff matrix H=(h~k ) is a matrix defined by h~k=(~)A~-~#k for \tv!

k=n. Hence # is called the generating sequence for /~/. The sequence/~ can also be obtained as a solution to the classical moment problem; i.e., 1

a.=/tnd~(t),

~--o,t,2 .....

9(t)~.BVEo, II,

0

where 9 (t) has been normalized so that 9 (t) = E9 (t + 0) + 9 (t-- 0).] ~ for 0
denoted by (A), consists of the set of sequences x such that A,, (x) is r Two matrices are said to be equivalent if they have the same convergence domain. Let A and B be two regular matrices. We say that A is stronger than B, or A includes B if (A)~ (B), If A and B are equivalent this will be denoted by (A) -----(B). In speaking of a Hausdorff matrix H generated by a sequence/,, I shall denote the matrix by Ha, and the convergence domain by (H,/~)~ except that I shall denote the convergence domain of the Ces~ro matrix of order t by (C, t). For other basic properties of a Hausdorff matrix see ~2, -Chapter XI~. Several authors have considered transformations which carry moment sequences into moment sequences. SCOTT and WALL in ~10~ consider transformations of the form 1

g(x) =

f

dg(t)

1 + xt '

9(t) monotone,

9(t) -- ~(0) = t .

0

Let /-/g denote the Hausdorff matrix generated b y {g(n)}. They prove the following. A portion of this work was performed under the auspices of the U. S. Atomic Energy" Commission.

A method of Hausdorff summability

63

Theorem A. Let (C, O) denote ordinary convergence. Then (C, O)(=(H, g (n) ) (= (C, t); (H,g(n))=(C, 0) i/ and only i I 9(t) is discontinuous at t = 0 ; and 1

(H, g(n)) = (C, t)) # and only q f dv)(t)/t is linite. 0

In [11 GREEXBERG and WALL show that, if 9) (t) is any function of bounded variation on the interval 0 < t < oo and 9 (oo) -- 9 (0) = 1, then the function (30

ot(z) = f d ~ (t)/(t + zt) is a regular m o m e n t function. If 9 (t) is further restricted 0

to be monotone, then (C,O)<((H,o~(n))<=(C, t). The conditions for equivalence are analogous to those stated in Theorem A. Also in [1], the transformation ~ l ( z ) = (t--c~(z))/qz is defined, where q is a normalizing factor such t h a t ~1 (0) = 1. Using a (z) = (t + ~)-~, k = t, 2, 3, -.., GI~rENBeRG and WALL show t h a t (H, c q ( n ) ) = (C, 1). JAKIMOVSKI [31 establishes a sufficient condition on a function ~(z) in order t h a t H~ be a regular Hausdorff matrix. Specifically, let/* be a regular m o m e n t sequence; i.e., 1

/*~=ft~dq~(t),

n---=0, t, 2 . . . . .

0

where 9 ( t ) E B V [ 0 , 1 l, 9 ( 0 + ) = 9 ( 0 ) and 9 ( t ) = t . of all regular Hausdorff transformations for which (i)

~ (t)/t is Lebesgue-integrable on [0, t ], and

(ii)

C=C(c?(t))= f

Let H denote the set

1

*P(t~)

0

T h e o r e m J 1. Let {t,} be the trans/orm o/ a sequence {s,} obtained by the regular Hausdor// trans/ormation T, generated by {/*~} (and q~(t)). I] T E n then the trans/orm {%} o/{s~}, where

vo---so;

sk -- tk

v , = s o+

C-k '

n>O

k=l

is obtained by a regular Hausdor]] trans/ormation generated by {/,~}, where ,

,

/'~

/*"--

I --#n C n

n~O, '

and I

I

/*'o=f t" *(" 0

,

t

- ;todla; a 0

too

.

du} -

This paper deals with transformations of the form ~o= 1, ~ = (t--#k)/ck, k > 0, where /* is a regular m o m e n t sequence and e is a non-zero constant. The approach here is to deal with the generating sequence directly, r a t h e r t h a n to use the integral description as in [1] or [3] or to deal with the n-th linear transform as is done in [3].

64

B.E. RHOADES:

The terminology here adheres closely to that used in [8] and [6]. A sequence/, is said to be totally monotone if all the successive forward differences are non-negative; i.e., A~k~0 for k, n = 0 , 1, 2 . . . . . If, in addition, the sequence # generates a regular Hausdorff matrix, /, is called a totally regular sequence. T h e o r e m 1. Let # be a totally monotone sequence with i,o<=t. Then the sequence t defined by l o = 1 , lk=(t--#k)/(ck), c any positive constant, has the property z]"tk__ 0, n-----0, t , 2 . . . . ; k = t , 2 , 3 . . . . ,

and A"to is monotone decreasing in n. Before proving the theorem we shall note the truth of the following lemmas. L e m m a 1. Let ~ and fl be any two sequences. Then zl" (~k ilk) =

~=o(n)(A'-J~,+i) (Aifl~), j

L e m m a 2. Let y k = l/k, k = 1, 2, 3 . . . . .

Anyk=n!Hyk+i,

n , k = 0 , 1 2, . . . . .

Then

n=0, t,2 .... ; k=1,2,3 .....

i=0

Both lemmas are easily proved b y mathematical induction. Both lemmas are well-known results. For L e m m a t, see [6, p. 2641, and for L e m m a 2 see [6, p. 2461. To prove the theorem, let Yk= l/k, 8k= t --/~k, k>= t. Then, from Lemmas t and 2 n

i=0 n

= n ! H y k + , " (t -/~k) ' =

"lr~0

=

*

~"

Let

2) Then

ek = l -- /~k -- k (k + / -- ~) AQ~k. j=l i

A method

of Hausdorff

summability

65

But

Therefore -

+

j

Using the identity

in the first summation, and replacing J-- 1 b y ] in the second leads to

Combining the first and third summations, and replacing j - - t the second summation, we have Aek=--

.

+

n

A"+i/~k--A#~+:+A~k--

Hence Ae~<=O; i.e., ek~ek+ x for k = t , 2, 3 . . . . . monotone a n d / * o ~ t, (3)

by 1' in

But, since /~ is totally

el ----t --/~1 -- Y, A i ~ = l -- ~ Ai#~ = i + A"+l~o -- ~o >=A"+~/~o >~O. i=x i=0

Thus A~ak-->0 for k = t , 2, 3, ...; n-----0, t, 2 . . . . . That A%~0~ in n follows from [6, Lemma 2]. For future reference we note that, i f / % = t,

(4)

,=o

= As l%

A.,,=I

s=l

65

Before continuing with the next theorem we define some additional terminology. A matrix A is said to be conservative if it is convergence-preserving; i.e., x convergent implies {A~(x)} convergent, but not necessarily to the same limit. Necessary and sufficient conditions for a matrix A = (a,~) to be conservative are (i) [JAIl= sup ~, ]a,a[ is finite, (ii) t = lim ~. a,~ exists, and n

k

n

(iii) a k exists for each k, where ak= lira a~k. Associated with each conservative Mathemati.~he Zeitsehrift. Bd. 8~

5

66

B.E.

RHOADES:

matrix A is a number z(A)----t--~, a k. If z(A)4=O, A is called coregular; k

if z ( A ) = 0 , A is called conull. A conservative matrix A is called multiplicalive if a~----0 for each k. We now quote a result from [6] which will be utilized in the theorem to follow. L e m m a 3. Let H~ be a coregular HausdorJ] matrix. Then H~ is ~otally coregular i / a n d only i / h , k > O /or k = t , 2, 3 . . . . ; n----0, t, 2 . . . . . T h e o r e m 2. Let ~, 4, c be as in Theorem t, with # a totally regular sequence.

I/, in addition, 0 < / 6 < 1, and ~ (As#o)/s converges, then H a is a totally coregular Hausdor/] matrix. ~=1 The hypothesis of the theorem and (4) guarantee that A~4o is bounded. Using the result of Theorem t,

[IHa[l=sup Y, Ih.~[ =sup It~"401 + s~ .n

k

n

k =1

1 J

= sup []A" 4ol + 1 -- A ~40] < 0r n

Any Hausdorff matrix with finite norm is automatically conservative. Since A ' 4 o is monotone decreasing, and bounded, it possesses a limit. Call it h o. Since A , ~ 0 = # x < t , h 0 < t . Thus )~(A)=t--ho>O, and H a is coregular, hence totally coregular b y Lemma 3The restriction/~,< 1 cannot be removed from the hypothesis of Theorem 2, since the sequence {1} is totally regular and, for this sequence, with c----t, 4-----{t, 0 . . . . }, which is totally monotone, but H A is conull. Similarly, #-----0, c = 1, leads to the eounterexample 40= 1, 4k-=--t/k, which is not conservative. One cannot replace "totally regular" b y "totally monotone with ~0----1, / h < l " since, for the sequence #----{t, }, ~ . . . . }, c = t , 40=1, 4k=(2k) -~, and H~ is not even conservative. It does not follow that e v e r y / z satisfying Theorem 2 leads to a totally multiplicative Hausdorff matrix, unless an additional restriction is placed on c. For, consider the sequence # ~ = 2 / ( k + 2 ) and use c = t . Then ~ o = t , 4,---- (k + 2)-1 and H a is totally coregular but not multiplicative by Theorem t 0

of [6]. In order to guarantee in advance that the value of c will give rise to a multiplicative Hausdorff method for every sequence ~ satisfying Theorem 2, it is obvious that we choose c so that A*40-+0; i.e., choose

(.)

c = 2 A~_~o. 9 s=l

S

T h e o r e m 3. Let H 1 denote the set o] all regular sequences such that (.) exists and is not zero. Then the sets H~ and H are equivalent in the sense that every i~ in H1 gives rise to a trans/ormation T ~n H, and every trans[ormation T in H is generated by a sequence l~ in H 1.

A method of Hausdorff summability

67

To establish the equivalence, let # be a regular sequence with associated mass function 9 (t). Then 1

s=l

S

_

s=l

s=z

's f 0

X

8

s=l~"

l

= f [1- (1 --t)"j 9(t)t dr. 0

Hence (,) exists and is non-zero if and only if ~0 satisfies (i) and (ii). Also it is clear that the c of (,) is equal to the C of (ii). Throughout the rest of this paper, c will always denote the value of (,) or (ii). It is possible to provide another proof that the Hv, of Theorem J l is regular, which we now do. Case I. c > 0 . Since/~ is regular, ~ = ~ - - f l , where ~ and /5 are totally monotone and e0--/~o=t. Making this substitution in (t) we have

i=o

/

/

Using (2) with #k replaced b y ~k we arrive at the same result for e~; i.e., ek i s non-decreasing in k. Using this fact and (3) leads to

Since/~ is totally monotone and ~0--/~0=1, A'~>--0 for n, k = 1 , 2 . . . . . In particular, A~k>__0. Therefore ~k~in k. But ~k-+0. Hence each ~k>=0. Also, A ~ o ~ i n k and zl~;tk-+0. Therefore H a is totally regular. Since c is a constant for each/~, and so chosen that A~'2o-+O, if c < 0 we need show only that H a has finite norm. From the preceding discussion it is clear that /-/ca is totally coregular, and INall finite implies tl/~A finite. In addition to having proved Theorem J t, we have shown that every T E H with c > 0 leads to a totally regular method H a. In particular, every totally regular transformation T C H will have a positive value of c, and hence H a will be totally regular. However, it is possible to obtain a totally regular method H a from a regular sequence # with c negative. For example, consider the sequence tz~=2(k-l-l)[(k+2), which is clearly regular, but not. totally regular. A straightforward computation shows that c = -- {, and that ~ = 2 / ( k + 2), which is totally regular. 5*

68

B.E. RHOADES:

It is of interest to note another derivation of the mass function corresponding to the 2~ of Theorem J 1. Let 2 , = (l--t,~)/cn. Then 2,=fl,_1, where

fl = c(n+O '-m+~

- - T t ( n +i ~ - - # . + i ' ~ .

I

)

Thus fl has a corresponding mass function ~9(u)=[~91(u)--~$(u)~/c, where ~ l ( u ) = u , and ~95(u) is the mass function corresponding to the product /4~+1/(n + t). Let 9 (u) denote the mass function of/~. Using the composition formula (see, e.g., [1, p. 7811) l

o,(u) = o~ (u) + f ol(u/v) dO3 (v), u u

where v%(u)=ft rig(t), we have 0 u

1

vaz(u) = f t dcp (t) + f (u/v) v d9 (v). 0

u u

Integrating' the first integral b y parts we obtain va~(u) -= u -- f 9 (t) dt. Then 0

O(u) = and hence

:[(/ u-

u-

~(t) dt 0

)1 =

X

f

f

0

1

0

0

1

0

__ f u.

~(t) dt,

u

tx .,r ""', e , } au . 0

1

Therefore (H, 2) is regular if and only if f ~ (t)/t dt exists and is non-zero. 0

From the form of the constant c in the transformation 2.=(t--~.)/cn, it is clear that if we define a function # (t) such that # ( n ) = # . , then (5)

c = lim

~--~0+

i t

Theorem 4. Let # E 111. Then # ' ( 0 + ) exists i/and only i[ ~m+ 9 (u) log u = 0, where 9 (u) is the mass [unction/or iz. Proo[. Since/zC.Hx, c exists and is not zero. Using (5), if/z'(O+) exists, c= --#'(0+).

But l

# (t) = f u' d 9 (u), 0 1

/z'(t) = f u' log u dq~ (u), 0

A method of Hausdorff summability

69

which gives us 1

1

# ' ( 0 + ) = f log ~ d g ( u ) ~ --~-~o+lim9(**)log u - - f 0

Since c = - - # ' ( 0 + ) ,

9(u)u du.

0

it follows that

(6)

lira 9 (u) log u = 0.

u->0+

Conversely, if (6) is satisfied and # E H 1, c exists and is non-zero. Hence c =

-

~'(o+).

The following extends Theorem J 1. T h e o r e m 5. Let T E H . Then (H, 2) )=(C, 1). Proo/. Let vk= (k + 1)~k, where ~k is as defined in Theorem t.

vk --

c

+ ~k,

Then

k : 0, t, 2 .... ,

and therefore

A'~ vk : A" 2~ -- ! An #k,

n, k : O, t , 2 . . . . .

c

Since A and # are both regular, A'*v~-+O for each k, H v has finite norm, and % = t . Hence Hv is regular. Q.E.D. The converse of Theorem 5 is false; that is, T E H does not imply (C, t) (H, A). A counterexample is provided b y [1, pp. 780--781], using the EulerKnopp method, i.e., # k : r k, 0 < r < l . For two regular matrices A and B, B is said to be totally stronger than A (written B t.s. A) if, for each sequence x for which A,,(x)-+l, then B,~(x)--> l ( ] / [ ~ o o ) . If A and B are regular matrices for which A,~(x)-+l implies B,,(x)--~l, l finite, but there exists a sequence x such that A,~(x)-->+ o~ but B,,(x)-[-~+oo, then we say that B is not totally stronger than A (written B n.t.s. A). The definition of not totally stronger is also meant to include the case where A~(x)-+--c~ and B~(x)-[-~--o~. In all cases where one is determining the total relative strength of two matrices A and B, one must first have the condition that B is stronger than A. If H a and H A denote the Hausdorff matrices generated b y the sequences # and A, and if A~:t: 0 for any n, then the statement Ha t.s. HA reduces to showing that the sequence/~]A, is totally monotone and t h a t / * o : A0. If H a and H A are equivalent, then to show that Han. t. s. H Ait is sufficient to show that #~/A~ is not totally monotone. One might conjecture that a totally regular T in H would give rise to a A such that H At.s. (H, t). To see that such is not the case, let # ~ : 2 / ( k + 2), c = 8 9 Then H a is totally regular, but ~-----2/(k+2) and H a n . t . s . H 1. The sequence defined b y (7)

# o - ~ t , /,~-- [/(/~)_i(~)]

k

~

k

/~

'

70

B.E. RHOAD~S:

where

and

-- I < ~ < f l

1

0

generates a Hausdorff summability method referred to in [8] as the method. The sequence defined by (8)

/~;=t

'

#;--

(k+~)-~--(k+~)-r (r - ~ ) k

I(e, ~)

k>o, o~
'

is referred to in [31 as the f(~, fl) method. Also, the CesAro and H61der matrices of order ~ are denoted, respectively, by (C, ~) and (H, ~). It should be noted that the sequences # for which (1--l%)/ck=#k are those of the f o r m / , k = ( c k + l ) - l , for some c=t=0. This fact provides another simple proof that I(~, ~ + 1)= (C, ~ + t) and J(~, m + t ) = ( H , ~ + t ) . (See [8, pp. 352, 380].) For, using (7) with f l = ~ + t we may w r i t e / ~ in the form t

where ~ is as in Theorem t, using / ~ = ( e - + - l ) / ( k + e + t ) .

Since (t--/~k)/

(ck) =~k,

Similarly, using (8) with/3 = ~ + 1, we can write/4~in the form/~ = (k + 1) -~-i. A regular transformation T fails to be an element of H if either c = 0 or the integral of (i) does not exist. For the sequence {t}, c = 0 . For the sequence (9)

#,=ft"dg(t),

= /0, where

~(t)

0

(t=o)

[_log 2 log (#:2) '

(0
q0 is a positive non-decreasing continuous function of bounded variation on [0, 1], but 1

f q~(t) dt -t fl

fails to exist. Thus H is a proper subset of the set of regular Hausdorff transformations and does not include all of the totally regular Hausdorff transformations. I am indebted to JOHN ROmNSON, Lafayette College, for the second example above.

A method of Hausdorff summability

71

T h e o r e m J2. I8, p. 357]. The I(o'~,fl)-trans/ormations are regular Hausdor/] translormations whenever -- 1 < ~ < / 5 . T h e o r e m J 3. [8, p. 363]. For any three real numbers c~, fl, y with -- t < x < f l , the I(o~, fl) and I(a, 7) methods o] summability are: (t ~ equivalent to each other, (2 ~ to the (C, o~+ t) method; while (3 ~ the method I(o:, fl) includes I(o~', ~) i / a n d only i / = > c d > -- t. T h e o r e m J 4. [3, p. 373]. Let {tn} and {t*} be two Hausdor]] transiormations o/ a sequence {s,,} generated by {#,} and {#*}, respectively, with # o = # ~ . For any constant D #= 0 the sequence {%} defined by v o = s o;

v~=s o+2

G--t*

D

n;>O,

'

is a Hausdorli transiorm o/{s.} generated by {2,~} where 2o = 1 ;

~ =- I*~ - l~* D n

n > O.

'

l] in addition there exist two ]unctions go(t) and go*(t) o] bounded variation in [0, 1] and such that 1

1

#.=ft"dgo(t);

#* =ft"dgo*(t),

0

n>=O,

0

and that 1

f q, (t) -7 9*(t) d t 4= 0 0

exists as an L-integral and we choose 1

= f

dt

0

then I

I

I

DI o

and {v.} is a regular SSausdor/l trans/orm o] {s.}. T h e o r e m J 5. [8, p. 379]. For any pair o~, fl o/ real numbers, with c~
H a includes H+,. (H, t).

Dk

'

72

B . E . RHOADES:

Proo/. For k>O, since #k#:0 for each k, (k + t) 'lk//~k= El -- ~/~k)] (k + I) Dh As # * I # E H i, Hal ~ includes (H, 1) from Theorem 5. Hence, H A includes H , . ( H , t). Clearly Theorem J 2 and the last part of Theorem J 5 are special cases of Theorem 6. If we now consider Theorem J3, Theorem 6 provides a proof that/(m, fl) includes ( C , a + l ) . However, to show the converse, one is forced to use Lemmas 5.t to 5.3 of [3], or some equivalent technique. Having established part (2~ of Theorem J3, part (1 ~ follows immediately since both the Imethods are equivalent to (C, 0c+t). Part (3 ~ of Theorem J3 and the first part of Theorem J 5 are proved similarly. Theorem 6 also aids in proving analogous results for the methods defined in

The ~ " method (see [7]) has a moment generating sequence t*

i

lz,, = ,,]

tn a~ ta-1 log F(~)

dt ,

a > O.

0

From the form of the mass function for the Gamma method, it is clear that the method is totally regular for all positive ~. A straightforward computation shows that (i) is satisfied and that c =~]a. It is possible to show that all of the results proved in [3] and [4] which involve the H61der method hold also when the Gamma method is used instead. Let/~ and #* be two totally regular sequences in H 1 and let ~ = (t --#k)/ck, ~=(t--#*)/c*k. Then, by Theorem 2, H a and Ha. are totally regular. A natural question then arises. If H a and Ha. are equivalent, is H a t.s. H~. or H an. t. s. Ha. ? We show by means of two examples that no general statement is possibl m Let/~k= (k + 1) -I,,+x),/~ = (k + t ) - 5 m a positive integer. In [1] it is shown that H a and Ha. are each equivalent to (H, t). However, it is shown in I8, Theorem 4.9, p. 407] that H a t. s. Ha. for r n ~ 4 and in [9, Theorem 3] that H a n.t.s. Ha. for m > 4 . I conjecture that the converse of Theorem 4 holds for every sequence /, in H 1 for which H~ is equivalent to (H, ~) for some a > 0 . Consider the set of transformations T~ defined by T0(/,k)=/z k, Tx(/zk)----(1 --[~k)/cak, and, in general, T~(/~)---- [ t - - T,~_x(tzk)]/c,,k. Let H 1 denote the domain of T1, H, the range of T1. Use H, for the domain of T~., and in general use H , for the domain of T,. These transformations then generate oo

a set of sequence spaces of the form Hi >H~ > . . . . Let H* = N H , . H* is not empty since it contains at least all transformations generated by secluences of the form {(cn+ t)-1), c>O, which are invariant under the T,.

A method of Hausdorff summability

73

I f / z ~ = (k + t)-% ~ a positive integer, then

T,~(#k) -~

o~+ n - t

~=o ( ~ ) ( k + ' l

' ~

which can be established b y induction. Clearly T~ (#k) is regular and is stronger than (H, 1) for n = t, 2 . . . . . Since lim/~k-----0 for k -~ 1, 2 . . . . . lira T. (/~k)= 0 for k ----t, 2, 3 . . . . ; n ----1, 2 . . . . . But the sequence {t, 0, 0 . . . . } is not in any of the H . or H*. Thus none of the H . , H* is sequentially closed. Let S denote the set of regular sequences # such that #~-->0. Then, clearly, H~ ( S. However, H~ & S. For, let lu be defined b y (9). Then/~ C S, but/z ([ H 2 . We also note that the inclusion H . ) H . + 1 is proper for each n. For, let #(~+1) be a sequence and 9.+1 (t) its corresponding mass function. We wish to find a sequence #(*1 in H~, with corresponding mass function 9 . (t) such that t

I fy.(u)

du.

0

Then, for O < t g t , 9'.+l(t)=9.(t)/tc~. If we then use the 9(t) of (9) for 9.(t), then #(.+1)(~H.+I ' b u t / f l ' ) C H . . With a little diligence one can derive the mass function 91 (t), corresponding to a sequence #(1) and show that #(1)C//1An interesting unsolved problem is to find necessary and sufficient conditions on 9(t) for the corresponding sequence to be an element of H* or any of the H A. A set of sufficient conditions is the following. Let 9'(t) exist everywhere in some e-neighborhood of t = 0 . Then in the interval [0, e~ we may write 9 ( t ) = 9 ( O ) + t g ' ( ~ g t ) for some v~ satisfying 0 ~ v ~ g l ; and, since 9 ( 0 ) = 0 , (i) is automatically satisfied. Additional restrictions can then be placed on 9 (t) to ensure that (ii) is satisfied. However, the above set of conditions are not necessary, as can be illustrated with the following example. Let 9(U)----u% 0 < c r Then 9'(u) does not x

exist for u = 0 , but f g ( u ) / u d u exists and has the value I/:r 0

Let T. (/z) denote the sequence {T. (/%)}~--0- The following theorems provide information on the Hausdorff transformations generated b y T.(#). In each instance it is assumed t h a t / , is a sequence such that T. (/~)CH.+ 1. T h e o r e m 7. Let P denote the set o] sequences # such that 9 ' ( 0 + ) exists. Then T,, (#)C P / o r every n. Proo/. By induction. The mass function for T1 (#) is 91 (u) = ~

j ~ ( Ot 0

~ (u)

-

v(u) c~u

dr.

74

B.E. RHOADES:

Hence q~(0-F) = lim ~(u) _ ~--->0+

~ lim ,T'(u)-- ~'(o+)

Cl'b~

C1 U - + 0 +

C1

Assuming the induction hypothesis we have

dt.

q%+l(u) = - - ~ f -r cn+ 1 J t 0

Then %~+t(O+) = lira

9 , , ( u ) = ~p,do+)

u.-r

cn+ 1 12

c~+ 1

Theorem 7 says, in effect, that if # is a sequence in P, then c~ exists for every n. I f , in addition, c,=k0, then T,(/,) E/-/,+I. T h e o r e m 8. (H, S,(#))_)_(C, 1) /or each, n. Pro@ The case n = l is Theorem 5. For n > I, T~(/~)

--

~-

T~-~(t,)~ cn h

Let vo= t, V~ ~- (k + 1 ) T n (/~#) - - ~ - Tn-1 (/'*k) + T,~ (F#),

k ~- O.

Cn.

Then H~ is regular, since it is an appropriate linear combination of regular methods. T h e o r e m 9. For all n ~ t , (H, T.+I(/~))Jp(H,~ ) /or any :~>1~ Proof. A necessary condition for (H, T.+I(f)))(H,~) is that the corresponding moment generating sequence be convergent. Let,5. = lira T. (/~k). Then~-Vor each n > 1, 5 . = 0 . (k + If'T,,+1 (,uk) -- t - Tn(,uk) (k + t) a. cn+1 k

Hence, for all k sufficiently large, (k + t) T,+ 1(#~) ---(k + t) ~-l, which is unbounded for 0~> t. T h e o r e m 10. For all # E H~ with fi =#1, wherefi = 1Lm,u~, (H, T~Cu)) ~ (H, ~)

/or any o~> 1. The proof is the same as that of Theorem 9, since ( k + 1)TI(/~)---('|--/7).

(k+t)~-X[cl, and 1 - - / 7 ~ 0 . For /7 = t no general statement can be made. Consider the following two examples. Let/4~= (k + 2)/2 (k + t). Then/7 = t, k+2 2(k + ~) l ;t,~ =

k/2

=

h+ t

Hence (1t, 2)= (H, 1). Now l e t / ~ ~ [(k + 2)/2 (k + t)] v, y :~ 1. Then ('k+2 /v - ~2(~-;;j cz k

(k + t%

A method of Hausdorff summability

75

and, since ( k + l)/k-+l as k-+oo, we are concerned with evaluating i lim k~oo

fk+2/' - \ ~ 1 (h + :):-~

(

-I

/ 2(i+-t) 2 ) = lim - ~ t2~G-~-] 4-.00 (1 - ~) (k + 1)-~

if t < ~ < 2, (k + t ) ~ 0 , whereas (k + 1)~2~ is unbounded in k for x > 2 . T h e o r e m 11. I[ /z is a sequence such that (H, T ~ ( # ) ) = ( H , 1) for some m>=t, then ( H , T . ( # ) ) = ( H . 1) /or all n ~ m , or equivalently, ( H , T , , ( # ) ) = (H, Tp(#)) /or all n, p>=m. Proo]. The equivalence is apparent. Also, it is sufficient to show that (H, T,~+: (#)) = (H, T,~ (#)). Tm+:(lzl~) Tm~u,)

t -- rrn(#~) cm+xk

(k + I) t

I (k + 1) Tm(t*~) "

Hence, Tin+: (l~k)/T,,,(#4) can be expressed as a product of two moment generating sequences, each of which is equivalent to convergence. References [1] GREENBERG, H . J . , and H. S. WALL: Hausdorff means included between (C, 0) and (C, 1). Bull. Amer. Math. Soc. 48, 774--783 (t942). [2] HARDY, G. H.: Divergent Series, 1949. [3~ JAKI~aOVSKI, A.: Some relations between the methods of summability of ABEL, ]3OREL, CESXRO, H6LDER, and HAUSDORFF. J. D'Analyse Math. 3, 346--381 (1953/54). [41 -- Some Tauberian properties of H61der transformations. Proc. Amer. Math. Soe. 7, 354--363 (t956). [8] KNom', K. : Theory and Application of Infinite Series, t928. [63 RHOADES, ]3. E.: Some properties of totally coregular matrices, illinois J. of Math. 4, 518--525 (t960). [71 -- Total comparison among some totally regular Hausdorff methods. Math. Z. 72, 463--466 (1960). [8] -- Hausdorff summability methods. Trans. Amer. Math. Soc. 101, 396--425 (1961). [9] -- Hausdorff summability methods. Addendum. Trans. Amer. Math. Soc. (to appear). [10] SCOTT, W. T., and H. S. WALL: The transformation of series and sequences. Trans. Amer. Math. Soc. 51, 255--279 (1942).

Dept. o[ Math., La[ayette College, Easton, Pennsylvania (U.S.A.) (Received September 17, 1962)