ARKIV FOR MATEMATIK Band 5 nr 21

ARKIV FOR MATEMATIK Band 5 nr 21

1.63091 Communicated 22 MayA963 b y AKE PLEIJEL a n d LARS G~-RDING

A g e n e r a l i z a t i o n o f a T a u b e r i a n t h e o r e m b y Pleijel By SVEN SPANNE

1. Introduction

I n [4], [5] Pleijel proved the following T a u b e r i a n theorem. (The factor 121 -a in [4] is here included in the measure da.)

I / 8 < 1 , a ( 2 ) E T s, a ( - 2 ) E T s and

f +:r (2 A- t) -~ d(r(2) = t-l p(t -1) -4- o(tS-1), ⁽¹⁾

- - o 0

when t--> ~ along all non-real hal/rays/rom the origin, then a(2) E I s and a( - 2) E I s.

I / s is an integer, I s ( a ( 2 ) - ( - 1 ) s a ( - 2 ) ) E ~ o ~

I n (1) p is a real polynomial, ar~l the relation ~ E ~ s means t h a t ~ ( 2 ) = c o n s t a n t + o ( 2 s) when 2 - ~ + c~. I k is defiffed b y dlkcp=2kdq9. I f Ik~0Eeo k+s for one value of k, t h e n the same relation is valid for all k * - s a n d we write

~0EI s. W h e n s:~O, I S = t o s, b u t w ~ is a - p r o p e r subset of I ~ The integral in (1) is convergent if a n d only if 1-10*E(D0. Finally, c f E T + ( T _ ) ~ s means t h a t there is a constant C such t h a t dq + C2 s ld2 is >~ 0 ( ~< 0) for sufficiently large positive 2, a n d T s = T~ U Ts-.

I n the proof in [4] of the T a u b e r i a n theorem, the a s y m p t o t i c relation (1) is used only along the imaginary halfrays except in two exceptional cases, which occur when s is an integer. I n these cases ( 1 ) i s used for the four ha[frays arg t = 41~r + n . 89 ~, n = 0, 1, 2, 3. Thus the conditions in the theorem are unne- cessarily strong.

A closer s t u d y of the question has shown t h a t it is sufficient to h a v e the a s y m p t o t i c relation along an a r b i t r a r y pair of non-real halfrays which are sepa- r a t e d b y the imaginary axis. This includes the case when one or b o t h ha[frays are purely imaginary. Moreover, the use of a larger Tauberian class t h a n T s led to a more n a t u r a l proof of the Tauberian theorem.

We can suppose t h a t the halfrays are the i m a g i n a r y ha[frays. F o r if (1) is valid along a n o t h e r pair of rays, the relation holds also along the complex con- jugate rays as the polynomial p is real. The integral a n d the polynomial in (1) increase so slowly when t--> c~ t h a t the Phragmdn-Lindel6f principle can be used in the angles formed in the u p p e r a n d in the lower ha[fplane b y the four halfrays. These angles contain t h e imaginary ha[frays.

S. SPANNE, A generalization o f a Tauberian theorem by Pleijel 2. Tauberian classes

W e use T a u b e r i a n classes which are closely connected with t h e slowly de- creasing functions.

Definition. A f u n c t i o n q) belongs to the class US+ i f lim lim in/ x-~(qD(y)-q~(x))>~O.

/~-->1+0 x---)r162 x<~y~t~x

W e p u t U ~ = - U ~ +, U s = U+ U ~ U ~_ and U ~ = U + N ~ US_.

T h e class U ~ consists of t h e slowly decreasing functions a n d t h e class U0 ~ of t h e slowly oscillating functions (on t h e i n t e r v a l (0, ~ ) ) .

T h e classes US+ a n d Us are c o n v e x cones, U~) is linear a n d if ~1 a n d ~2 belong t o U s, t h e n either ~1 + ~ 2 or ~ 1 - ~ % belongs to U ~. I f ~ E U ~ a n d yJ E U~), t h e n ~ + y~ belongs t o U ~.

T h e classes T~ (T~_) are included in t h e classes U+ (U~), for take, for instance, E TS+. T h e n dq~ + C;t ~ 1 d;t >~ 0 for ;t >~ 40, a n d we m a y e v i d e n t l y a s s u m e t h a t C > 0. T h e n

x - ~ ( ~ ( ; t ) - ~ ( x ) ) > ~ - C x - ~ ; ; t ~ - l d 2 > ~ - C ; u ~ - ~ d u , w h e n ;to ~< x ~< y ~</~x. T h e last integral t e n d s to 0 w h e n #-->1 + 0.

W e shall need t h e following p r o p e r t i e s of t h e classes U.

Proposition.

(a) I f qgEUq+, then IPq)eU~+ +n /or p>~O.

(b) I f cs E Uq+, then cf(;t ~) E Uq+ ~ /or ~ > O.

(e) u'+ = u~+ i/ p < q.

(d) I q = U~.

(a), (b) and (c) are valid also /or the classes U _ . Proof.

(a) Suppose ~ E Uq_. A simple calculation shows t h a t

(IPcf(y) - I % f ( x ) ) = x q (q~(y) - of(x)) + x -p q f / ( o f ( y ) - cf(;t))

x - p ^{~ q} d(;tv).

T h u s if

X-q(q)(y)--~(X))~--C

;to~X~y~x,

w h e n ;t o ~< x ~< Y ~< r This p r o v e s (a).

312

ARKIV FOR MATEMATIKo B d 5 mr 21

(b) S u p p o s e ~c E U~. E v i d e n t l y

x :'q (qo(y ~') - qo(x~')) = X - q ( q : ( Y ) - ~f(X)) >~ - c,

w h e n ;t o ~< X ~< Y ~< # X , w h e r e X = x% Y = y% t h a t is, w h e n ;tl/~ ~< x ~ y ~< #11~ x, a n d (b) follows because /, -+ 1 + 0 w h e n /,1/~_+ 1 + O.

(e) Obvious.

(d) I f q~=O, t h e n I q=~o q, a n d if ~oE1% t h e n

Ix -q ( ~ ( y ) - of(x)) 1 = l ( y / x ) q q ~ ( y ) / y ~ -- c f ( x ) / x q l <. ( ( y / x ) q § 1) o(1) --> 0

w h e n x --> § ~ a n d x 4 y ~</~x. Suppose secondly t h a t ~ E/0. T h e n x - 1 So ;t d~(;t) = o(1) w h e n x--> + ~ . E v i d e n t l y

x

~(x) = x -1 f ~ o ( ; t ) d2 + o(1), w h e n x - > + ~ . a n d t h u s

; fo

W e get q~(y)-~v(x)=o(1) + y -1 q)(;t)d,~-x 1 ~o(;t)d2

= o('l) + y-1 fff ~(;t)d;t § (x/y -- l)X -1 f~ ~9(;t)d~

= o(1) + y - ~ f ; ~ ( ; t ) d;t + ( x / y - 1) ~(x)

= o(1) + y-~ ( ~ (~(;t) - ~(x)) d;t, g x

w h e n x--> + c~, x ~< y ~</~x. This gives

inf ( q o ( y ) - - c f ( x ) ) > ~ o ( 1 ) + ( 1 - # -1) inf ( ~ o ( 2 ) - ~ o ( x ) ) .

x<~y<<ux x<~<~y<~,ux

T h u s inf (q(y) - q~(x)) ~>/~o(1).

T h e s a m e reasoning gives a n u p p e r bound. T h e result is lim sup '] q(y) - q(x) l = 0,

X--.>+~ x ~ y ~ t l X

which m e a n s t h a t ~c is slowly oscillating.

S. SPANNE, ~4 generalization of a Tauberian theorem by Pleijel

3. T h e u n i l a t e r a l T a u b e r i a n t h e o r e m

W e shall use t h e following unilateral T a u b e r i a n t h e o r e m . T h e o r e m . I[ 8 <~ 1 and ~rE U s, then

j

0

implies that (rEI s. I1 s is an integer, the result can be improved to I - ' G E ~ ~ F o r s = 0 a n d for 0 < s < 1, t h e t h e o r e m is a n n o u n c e d b y H a r d y a n d Little- w o o d [1] a n d b y K a r a m a t a [2], b u t no explicit proof is given. W e s k e t c h t h e proof. L e t 0 < s < 1. A p a r t i a l i n t e g r a t i o n (justified b y t h e f a c t t h a t I-laE~o ~ gives

io o

t -1 (2/t) s (1 + 2/t) -2 (~(2)/2 s) d2 = o(1) + K, (2) where t h e c o n s t a n t K = 0 if s > 0.

T h e kernel g ( x ) = x S ( l + x ) -2 is a W i e n e r kernel since

f /

z~(s +

^ui)

I f s = 0, t h e t h e o r e m follows f r o m a t h e o r e m b y K a r a m a t a [3]. I f s > 0 , a E U~

gives t h a t a(2)/2 s is b o u n d e d below ( K a r a m a t a [2] p. 36). F r o m this c o m b i n e d with (2) we get t h a t Stoa(2)d2=O(t s+l) as t--->+ oo. H e n c e

(it -- 1). ta(t) = ((#t - t ) a ( t ) - f ~ t ( r ( 2 ) d 2 ) + O ( t s~) f f

(a(2) - a(t)) d2 + O(t s+l) ~ C. tSd2 + O(t s+l) = O(t s+l) as a E U~. W e conclude t h a t a ( t ) = O(tS). I f a E U s a n d a ( 2 ) / Y is b o u n d e d , it is easy t o see t h a t a(2)/2SE U ~ T h e t h e o r e m b y K a r a m a t a t h e n gives t h e w a n t e d result.

T h e case s < 0 is r e d u c e d to t h e case 0 ~ s < 1 in t h e s a m e w a y as in Pleijel [4]

p. 565. This r e d u c t i o n uses t h e properties (a) a n d (c) of t h e T a u b e r i a n classes.

4 . T h e b i l a t e r a l T a u b e r i a n t h e o r e m

Suppose t h a t (1) holds along t h e i m a g i n a r y halfrays. I n t h e same w a y as in [4] it follows t h a t

~ : ( 2 + t) idA(V2) = t-lpl

(t

ARKIV FOR MATEMATIKo B d 5 nr 21

: ( ) . 4- t) -1 ]/~ dS(V~) = t - l p 2 (t -1) +

O(t(s+l)12-1),

when t-> oo along the positive real axis. In these formulas Pl and p , are real polynomials and

S ( ~ ) = a ( ~ ) + a ( - ~ ) , (5)

A(~)=a(~)-~(-~). (6)

We replace the Tauberian condition in the theorem in section 1 b y the con- dition a(~) E U s, a( - ~) E U 8. F r o m this follows t h a t either S(~) or A(~) E U s.

Suppose t h a t S, say, belongs to U s. Then IS(k)E U s+l b y the proposition (a), and IS(V~)E U 89 b y (b). The unilateral theorem now can be applied to (4).

This gives I S ( ~ ) E I 89 which is the same as S(~t)EI', and b y (d) it follows t h a t S(~t) E U~.

From (5) and (6) follows t h a t A(2) = 2a(2)-- S(2), where 2a(2) E U s and - S(2) E U ~.

This implies t h a t A(2)E U s and A(I/]}EU s/2. F r o m (3) we obtain, using the uni- lateral theorem, t h a t A ( 2 ) E I s, and when s is an even integer, I - S A E~o ~

The case in which A(2)E U s follows from the original Tauberian conditions is treated in a completely analogous way, starting from (2) instead of from (3).

We have the following bilateral Tauberian and Abelian theorem.

Theorem.: Let s< 1 and a(~), a ( - ~ ) E U s. Then the [ollowing conditions are equivalent

1. The relation (1) is valid along one pair o[ non-real hal]rays /tom the origin separated by the imaginary axis.

2. The relation (1) is valid along all non-real hal/rays /rom the origin.

3. a(2) and a ( - 2 ) E P , and I s ( a ( 2 ) - ( - 1 ) s a ( 2 ) ) E e o ~ i[ s is an integer.

We have proved the Tauberian part, 2. ~ 1. ~ 3. The Abelian part, 3. ~ 2 . , is the Abelian theorem in Pleijel [4] p. 568.

R E F E R E N C E S

1. HARDY, G. H., and LITTLEWOOD, J. E., Notes on t h e t h e o r y of series (XI). On Tauberian theorems. Proceedings of t h e L o n d o n Mathematical Society, Series 2, vol. 30, p a r t 1, 23-37 (1930).

2. KARAMATA, J . , Neuer Beweis u n d Verallgemeinerung der Tauberschen S~tze, welche die Laplacesche u n d Stieltjessche Transformation betreffen. J o u r n a l ffir die reine u n d ange- w a n d t e Mathematik, B a n d 164, H e f t 1, 27-39 (1931).

3. KARAI~IATA, J., Bemerkungen zur Note: ~ b e r einige Inversionss~tze der Limitierungsver- fahren. Publications Math~matiques de l'Universit@ de Belgrade, Tome IV, 181-184 (!935).

4. PLEIJEL, A., A bilateral Tauberian theorem. Arkiv fSr Matematik, 4, 561-571 (1962).

5. PLEIJEL, A., R e m a r k t o m y previous p a p e r on a bilateral Tauberian theorem. Arkiv f6r Matematik, 4, 573-574 (1962).

Tryckt den 9 juni 1964

Uppsala 1964. Almqvist & Wiksells Boktryckeri AB