abel transforms of tauberian series

ABEL TRANSFORMS OF TAUBERIAN SERIES

BY RALPH PALMER AGNEW

1. Introduction. Let Uo + ul + be a series of complex terms satisfyingthe Tauberian condition

(1.1) n lun < K (n O, 1, 2, .-.).

Let a(t) denote the Abel transform of un

(1.2) (t) tu (0 < < 1).-0

Let L denote the set of limit points of the sequence So, sl, of partial sumsof 2un. Let LA denote the set of limit points of the Abel transform a(t); z" LAif there is a sequence t, t2, such that 0 < tn < 1, tn --* 1, and a(t.) -- z’ asn --, o. If a(t) -- a as -- 1, then Zun is summable to by Abel’s method A;but it is not assumed that 2un is summable A.Hadwiger [1] proved that each of the following assertions is true when

(1.3) p 1.0160-.-

and false when

(1.4) p < .4858 ....ASSERTION 1.1. If un is a series satisfying the Tauberian condition n un < K,

then to each z L corresponds a z" L such that

(1.5) z’ z"l

_p limsupn u I.

ASSERTION 1.2. IfUn is a series satisfying the Tauberian condition n un < K,then to each z" L corresponds a z’ L such that (1.5) holds.

As Hadwiger pointed out, his result implies that L LA when un is a seriessatisfying the Tauberian condition nun -- 0. A Tauberian theorem of Littlewood[2] states that if 2un satisfies the weaker Tauberian condition (1.1) and 2;u. issummable A to , then 2u. converges to . This implies that if Zu. satisfies(1.1) and LA contains exactly one point, then L L. Hadwiger’s result showsthat (1.1) does not imply universal identity of L and L for otherwise theassertions would be true when p 0.

It is the object of this note to prove the following theorem which gives theleast constant p for which Assertion 1.1 is true.

Received August 4, 1944; presented to the American Mathematical Society August 12, 1944.

27

28 R.P. AGNEW

TttEOREM 1. Assertion 1.1 and Assertion 1.2 are true when p >_ pl and Assertion1.1 is false when < 1, where

(1.6) pl T loglog 2 + 2 f:g e--x dx.

Approximate values are

(1.7) pl .5772157 .3665130 + .7573421 .9680448;. is Euler’s constant, the logarithms have base e, and the integral is evaluatedby use of a table [3] of values of the exponential-integral.A treatment of real series is given in 6. An example very similar to one of

Hadwiger is given in 7.

2. Formulas and notation. We shall use the following formulas and notationinvolving Euler’s constant ,. For each n 1, 2,

1 1+ + dk n .1

where

By use of the geometric interpretation, in terms of areas, usually given to themembers of these equalities, it is easy to see that

(2.3) 0 < t, < 1/2n (n-- 1, 2,...).

Letting ,, "r + ., we have

log n + y.,

where , -- ,,/as n -- o.For ech z > O, let

(2.5) E(x) da;

thus E(x) -Ei(-x) where Ei(x) is the familiar exponential-integral function.We shall use the fact that

(2.6) lim [E(x) + log x -k-’] 0;x-0

ABEL TRANSFORMS OF TAUBERIAN SERIES 29

this follows from the known formula

1 1 da lim(2.7)

for , (see, for example, [4]). Let

(2.8) E(x, q)a Le /’- l-ida (x, q > 0).

The quantity in brackets is positive and less than 1 so

(2.9) E(x, q) < E(x) (x, q > 0).

For each x > 0, the integrand in (2.8) is dominated by the integrable functione-"/a; hence, by the criterion of dominated convergence for taking limits underintegral signs,

(2.10) lim E(x, q) E(x) (x > 0).

The same argument shows that if x(q) is a function of q such that x(q) convergesto a positive limit Xo as q --. , then

(2.11) lim E(x(q), q) E(xo).

The substitution B e-"/ gives, when q > 0 and 0 < < 1, the integral formula

(2.12) 1--- d E(q log t-’, q).

3. Proof that the assertions are true when p pl

satisfying (1.1). Let

(3.1) nu,, x,.

Let Zu. be a series

Then Xn is a bounded sequence for which

(3.2) lim sup Ix. lim sup n u. <For each n 1, 2, 3,

(3.3)

where

..and0<t< 1,. ,(t) u _, t%k-.1

1-

(3.4)a(n, t) (1

-t/k

3O R. P. AGNEW

Let

(3.5)

Then

and hence

F(n, t) _, a,(n, t)].

F(n, t) 1-trk,-I k

a’-’da+k-I k,,,,

=.+logn-- 2 lOgl_ 1-Bd/ +lOgl_n=3,.+log[n(1- t)] +2 1-#dt

so that, by (2.12),

(3.8) F(n, t) n + log [n(1 t)] + 2E(n log -t, n)..

For each n 1, 2, 3, the derivative of F(n, t), computed from (3.6), isfound to be negative when 0 < < Tn 0 when Tn, and positive when> T where T (1/2)1/. Hence, for each n 1, 2, 3, the’function

F(n, t) assumes its minimum value when T.. This suggests that, for eachin the interval 1/2 < < 1, F(n, t) should be near its minimum when n is the

greatest integer

_log 2/log -1. For each in the interval 1/2 < < 1, let n(t)

denote this greatest integer. As increases continuously from 1/2 to 1, n(t) as-sumes all positive integer values in increasing order. Since

(3.9) lim n(t)(1 t) log 2

(3.10) lim n(t) log -t log 2,t-l

it follows from (3.8) that

(3.11) lim F(n(t), t) ,t--,1

where p is the constant in (1.6).Setting b(t) a(n(t), t), we put (3.3) in the form

(3.12)k-1

Since

(3.13) lira b,(t) 0 (k 1, 2, ...)

ABEL TRANSFORMS OF T&UBERIAN SERIES 31

it is well known and easily proved that

(3.15) lim sup s.,,) #(t) -< P, lim sup x. I.

To show that Assertions 1.1 and 1.2 are true when p pl, suppose that z isa limit point of s. [or of #(t)]. Then there is a sequence tl t. such thatt -. 1 and s.(,) -. z [or a(t) -- z] as --. 1 over the sequence. This and (3.15)imply that (t) [or s.(t)] remains bounded as -- 1 over the sequence. Hencethe sequence has a subsequence T1, T2, such that a(t) [or s()] has a limit,say w, as --. 1 over the subsequence. Letting --. 1 over the subsequence, wefind by use of (3.15) that

(3.16) )z w < pl lim sup Ix. pl lim sup n u. I.Thus the two assertions are true when p p and hence also when p >_ p.

4. The Abel transform of a special series. As a prelude to the proof thatAssertion 1.1 is false when p p, we obtain properties of the Abel transformof the series (4.6) which depends upon two integers ), and p which we now define.

Let e 0 and let e/8. When k is a large positive integer, the functiony(O) defined by

(4.1) y(O) 1 20 + OTM (0 _< O _< 1)

has a derivative which is negative, 0, or positive according as 0 is less than, equalto, or greater than [2/(X + 1)]/x. Since also y(0) 1, y(1/2) (1/2)x+l > 0, andy(1) 0, it follows that y(0) has exactly one zero, 0o, in the interval 0 < 0 < 1and that 0o > 1/2. If ), > 3, then y’(0) < -1 when 0 < 0 < ]; this and the factthat y(1/2) ()x+ imply that 0o -* 1/2 as X -- o. Hence we can choose an integer), so great that

(4.2)

(4.3)

(4.4)

(4.5)

Let p be an integer so great that p- < ,.1

0+0+ +O+,L,-4p+l

(4.6) 1 1Ap+ 1

E((log 01)/(, 1)) < -log((log 01)/(h 1)) x + et

loglog 0 < loglog 2 +E((X log 0;’)/(), 1), X) > E(log 2)

E((X log O")/(X- 1)) < el.

Let Zu.(p) be the series

1 1p+ 2 + +x-

1xp+2 xx-- +++’’’

32 R.P. AGNEW

uk(p) 0 when < p and when k > p; u(p) - when ;u(p) -- when p p; and (p) G where G s the constantwhich maes 0. Thu

and hence, by (2.3), p G < 1 and G < The terms of the series u(p)satisfy the Taubean condition

(4.8) n u,(p) 1 (n O, 1, 2, ...).

The sequence s(p) of partial sums of Zu.(p) is such that

(4.9) s.(p) < log X + p- (n 0, 1, 2,...)

(4.10) s(p) > log p-.The Abel transform a(t, p) of Zu.(p) is

(4.11) a(t, p) Gt + F(t),

where

F,(0=k-+ k-hp+

Derentiation gives

(.13) F(t) (t 2t + ’)/(1 t).

Defining 0 in terms of p and by the formula

(4.14)

we obtain

(4.15) F(t)

The first factor in brackets is positive over 0 ( ( 1; the second is the functiony(0) in (4.1). It follows that F(t) assumes its maximum value when has thevalue t, determined by

(4.16) 0o t-), p log t; (log 0)/(X 1).

Integrating (4.13) gives

(4.17) F,(t)1 d,

and use of (2.12) gives

(4.18) F,(t) E(p log -1, p) 2E(hp log -1, hp) + E(X2p log -, h2p).


Using in turn the fact that F(t)

_F(t) and the inequality (2.9), we obtain

(4.19) F,(O <_ E(p log ;’) 2E(,p log t’’, )p) -I- E(,p log

Use of (4.16), (4.2), and (4.3) gives

E(p log t;’) E((log ’)/(, 1))

_< , log [(log 0’)/(), 1)1(4.20) , loglog - -b log (h 1) ,_

2 loglog 2 + log ,),.

Use of (4.16) and (4.4) gives

E(Xp log t; x, ),p) E((), log 0’)/(), 1),(4.21)

> E((X log 0)/(), 1), X) > E(log 2)

Use of (4.16) and (4.5) gives

(4.22) E(hZp log t;) E((X log 0)/(h 1)) < .These estimates and (4.19) give

(4.23) E(t) < 5ea A- log ), -loglog 2 , 2E(log 2).

This and the fact that < e7 enable us to conclude from (4.12) that

(4.24) (t, p) _< 6 -{- log h p, (0 < < 1),

where p is the constant in (1.6).

5. Proof that the Assertion 1.1 is false when o < o Let , x and befixed as in the preceding section. Let 2:u be the series obtained by the termwiseaddition of the series Y,u,(p), Y,u,,(p.), where px p is an increasingsequence of integers satisfying conditions given below.

Let to 0. Let p be an integer such that pX < ex, let Y,u.(p) be the corre-sponding series defined in the preceding section, and let (t, p) be the Abeltransform of the series. Since Y,u,(px) converges to 0 and Abel’s method A ofsummability is regular, (t, p) --, 0 as --, 1. Hence we can choose t such thatto < t < land

(5.1) I(t, P,)I< e, (t < < 1).Choose an integer p. such that p > p and

< (o < <_

Since (t, px) --, 0 and a(t, p.) --, 0 as --, 1, we can choose t. such that the in-equalities

(5.3) t_a < t, < 1

34 . P. AGNEW

(5.4) ’ a(t, P)I < , (t, _< < 1)

hold when r 2. Then choose pa such that the inequalities

(5.5) p+l >

(5.6) 2 / < , (0 <

hold when r 2. Then choose, in order, ta p t p such that (5.3),(5.4), (5.5), and (5.6) hold for each r 1, 2, and t 1 as r . Theinequalities (5.5) imply that all nonzero terms of Zu,(p) have subscripts lessthan the smallest subscript of a nonzero term of Zu(p+). This and (4.8)imply that n ]u 1. Hence, when r is a positive integer and t_ t,,the Abel transform a(t) of u= is such that

k-O --1 k’pr

(5.7) (t, ) + (t, ) + t%

2, + (t, p,) 8 + log X

Therefore

(5.s) (t) + og x- p, (o < < z).

The sequence s of prti] sums of Zu, is such that, when

s s(p,).

It follows from (4.9) nd (4.10) that Jim sup s log X. Thus the point zodefined by Zo log k is a limit point of the sequence x,, that is, Zo is point ofL. It follows from (5.8) that if z" is limit point of e(0, that is, a point of L,,then

(5.9) z" + Zo p,

nd, since lim sup n u 1,

(5.10) Zo z" (p, ) lira sup n u [.

This shows that Assertion 1.1 fils when p p 2e. This completes the proofof Theorem 1.

6. leal Series. Let Zu be a series of real terms satisfying the Tauberiancondition n u < K. The sequence s of partial sums of 2u is such that


Let, where a and b may be finite or infinite,

(6.2) a lim inf s., b lim sup s..

Then, as is well known, the Abel transform a(t) of 2;u. is such that

(6.3) a < a lim inf r(t) < lim sup r(t) f < b,

where and/ are defined by the equalities.If a and b are not both or both , then 1.1) implies that the set L of

limit points of x. is the entire interval from a to b, closed at such of its ends asare finite. Since (t) is continuous over 0 < < 1, the set La of limit points issimilarly the entire interval from to/. If a b o, then /and both L and La are empty. If a b , then a =/ o and again L andL are empty. Hence, since a _< a _</ <:/, the set L is in every case a subsetof L. This means that to each z" L corresponds a point z’ L, namely thepoint z" itself, such that z’ z" O. Hence, for real series, the conclusionof Assertion 1.2 holds for each p >_ 0.

It is not necessarily true that L is a subset of L, even when 2;u. is real. Thefollowing assertion differs from Assertion 1.1 in that the hypothesis involvesonly real series.

ASSERTION 6.1. If ZU. is a series of real terms satisfying the Tauberian conditionn u. < K, then to each z’ ,. L corresponds a z" ,. L such that

(6.4) z’ z"l _< p limsupnlu, I.

That this assertion is true when p >_ pl the constant in (1.6), follows fromTheorem 1. That the assertion is false when p < pl follows from the fact thatall of the series constructed in 4 and 5 are real.

7. The series with partial sums s. n’. Hadwiger [1] proved that Assertions1.1 and 1.2 are false when p < .4858 by use of the series 2;u* for which theterms, partial sums, and Abel transform are

(7.1)

(7.2)

and

(7.3)

(-O)u.* -ioio (n + io)t

n -k- iO n!(iO)!

iO n,o(1 + o.) iO e,O...(1 + o.),n+O n

s.* (1 + i0)(1 + i0/2)... (1 + loin)

(n+iO)t 1 + o.n,_ 1 + o.n!(iO)! (iO)! (iO)!

e

,*(t) ( t)-" e’*’’t’/(’-’),

36 R.P. AGNEW

being a real parameter. Here and hereafter, o denotes a function of n andwhich converges to 0 as n --, .The same result is obtained by use of the series Zu. for which So 0 and

(7.4) s,, n’ e,Oo (n 1, 2, ...),

8 being a real positive parameter. The set L of limit points of s is the unit circlez 1. For eachn> 1,

(7.5) nu. n(s. s._,)

where h log n log (n 1). The quantities in braces approach 1 as n --*hence

(7.6) nun

and

(7.7) limn]un 0.

Since sn (1 -t- On)(iO)!S* and Abel’s transformation is linear and regular,

a(t) =or- (iO) !a*(t)(7.s)

ot

The set La of limit points of a(t) is the circle ]z] (iO)![. From

(7.9) (iO)!l (iO)!(-iO)! 0/sinh 0 < 1,

obtained with the aid of the formul z!(-z)! rz/(sin z), we see that Assertion1.2 holds for the series with partial sums (7.4) if and only if

(7.10) 0-’[1 (v0/sinh 0)

The maximum value of the left side of (7.10), attained when 0 1.1909 is.4858 .... This gives Hadwiger’s result that Assertion 1.2 is false whenp < ,4858 ....

REFERENCES

1. H. HADWIGER, ber ein Dis$anz-theorem bei der A-Limitierung, Commentarii MathematiciHelvetici, vol. 16(1944), pp. 209-214.

2. J. E. LITTLEWOOD, The converse of Abel’s theorem on power series, Proceedings of the LondonMathematical Society, Series 2, vol. 9(1911), pp. 434-448.

3. NATIONAL BUREAU OF STANDARDS, Tables of sine, cosine, and exponential integrals, NewYork, 1940.

4. E. T. WHITTAKER AND G. N. WATSON, Modern Analysis, Cambridge, 1920.

CORNELL UNIVERSITY.

abel transforms of tauberian series

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