ARKIV FOR MATEMATIK Band 4 nr 45

ARKIV FOR MATEMATIK Band 4 nr 45

Read 28 F e b r u a r y 1962

A 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

By AKE PLEIJEL

This note contains t h e proof of a bilateral T a u b e r i a n t h e o r e m which in in- complete f o r m was used in m y p a p e r [2].

B y the e l e m e n t a r y proof t h e T a u b e r i a n t h e o r e m is reduced to two well-known t h e o r e m s b y H a r d y a n d Littlewood, see [1].

1. The Hardy and Littlewood theorems

L e t a be a real function of b o u n d e d v a r i a t i o n on e v e r y finite s u b i n t e r v a l of 0~<~t< + ~ .

T h e following Abelian t h e o r e m s are easily proved.

Theorem A {q}. I / 0 < q < 1 and a(,~) = o(~ q) when ,~--> + ~ , then

f / ( 2 + t) -1 da(2) = o(tq-a), t-+ + cr

Theorem A{0}. I / a ( 2 ) = a ( + ~ ) + o ( 1 ) when ~-~ + ~ , then

f/(

2 + t) -1 d a ( ~ ) = (a( + ~ ) - a(0)) t -1 + o(t-1), t - > + ~ .

The conclusions are still valid if t tends to infinity along half r a y s f r o m t h e origin in the c o m p l e x t-plane which are different f r o m t h e real negative axis.

To see this one has only to use the i n e q u a l i t y 12 + t I ~> (~t + It I) sin 89 (~, l arg t I ~< ~ - (~, w h e n the necessary e s t i m a t i o n s are p e r f o r m e d .

To f o r m u l a t e the corresponding T a u b e r i a n t h e o r e m s we need

Definition 1. The /unction ~ belongs to T s i/ there is a real constant C such that d ( ~ ( ~ ) + C ~ - i d 2 is de/inite, >~0 or <~0, /or su//iciently large values o/ ~.

T h e t h e o r e m s are

Theorem T {q}. I / 0 < q < 1, ~ E T q and

AXE PLEIJEL, A bilateral Tauberian theorem

f : ( 4 + t) -1 da(4) = o (tq-1), t---> + 0%

then a(4) = o(4 q) w h e n 4--> + r162 Theorem T {0}. H a E T o and

f : (4 + t) -1 da(4) = H t - 1 + O(t-1), _t--> + cr

then a ( 4 ) = a ( + oo)+o(1) w h e n ,E-->+ oo. T h e constant H is related to a by H =

~( + oo) - ~(0).

2. Certain lemmas

An addition of a constant to the function a of the previous section is evidently irrelevant. The same holds for the functions of this section since we are essen- tially only interested in their differentials. Thus the definition

,kip( ) =

determines Ik• up to an additive constant. I t is always possible to avoid di- vergence difficulties b y taking the lower limit of integration positive. Obviously

IJ(Ikip) = I j + kip .

Definition 2. W e write ip(4)fi to ~ il q ( 4 ) = o ( 4 s) w h e n 4--> + ~ a n d s > 0 or i!

ip()~)=ip(+ c ~ ) + o ( 4 ~) w h e n ),--> + ~ a n d s<~O.

Consider Ik~p when tP fifo'- I t is no restriction to assume ip = 0(4 ~) also when s ~< 0. F r o m the relation

I k f p ( 4 ) - / k i p ( A ) = ~ 4 k+~ tP~A 7) A k + s _ k f a

_{J A ~ -} ip(~) #k+S-id#

₍₁₎

it follows when k + s > 0 t h a t

Ikip(4) = o(4k+'), 4 - ~ + oo.

I f k + s < 0 relation (1) shows the convergence of Iklp(4)when 4-->+ c~. The transition to the limit A - + + cr then leads to a formula from which one con- cludes t h a t

Ikip(4) - F @ ( + o0) = o(4k+'), 4 - + + oo.

Thus we have proved

ARKIV FOR MATEMATIK. Bd 4 nr 45 Lemma 1. I / ~ 6 c o ~ then

Ikq~6~o~+s

provided b 4 - s .

T h a t the lemma is not valid without the condition It=t=- s is seen b y the ex- ample r log log t. This function belongs to r when s 4 0 but I - S v ( t ) = log log ~t is not in r ~

Definition 3. W e write q ) 6 I s i / I k q ) 6 w ~+s /or all k ~ = - s .

Observe t h a t it follows from l e m m a 1 t h a t if Ik~ fi co k+s holds for an arbitrary value of k it holds for all values k * - s. This leads to

Lemma 2. to s = I s i/ s # 0 and to ~ c I ~

The function log log i belongs to I ~ b u t is not in r ~ L e m m a 3. ~ 6 I s i m p l i e s I k q ~ 6 I k+s.

L e m m a 3 is an immediate consequence of the definition of P . The lemma is closely related to l e m m a 1 b u t is free from supplementary condition.

Lemma 4. T h e class I s is linear.

L e m m a 5. I] ~v(~)6I s then q~(~k)6iks.

L e m m a 4 a n d 5 follow from the definition of to s and I s.

L e m m a 6. I/ qp 6 T" then IkqD 6 T

TM.

For if dq~+ C]t'-ld]~ is definite, the same is true for the differential obtained when d q ~ + C 2 " - l d 2 is multiplied b y 2 k.

Lemma 7. I / 9)1, ~f26 T" then either q~l + r 6 T s or (Pl - q~26 T s.

If d~l + C 1 ~tS-ld2 ~> 0, d ~ + C 2 is-1 dt/> 0 it follows b y addition t h a t ~1 + ~ E T ' . I f d~, + C 12s- ld2/> 0, d~2 + C2 t ' - ld2 ~< 0 subtraction yields ~, - ~ 6 T s.

L e m m a 8. I / ~ ( t ) e T" then q~(~k)E T ~`.

L e m m a 9. I / s < r then T" c T r, T s . T r. A l s o oJ" c eor,r 4- w r and I" c I ~, I" ~ I ~.

L e m m a 8 is obtained b y a change of the independent variable and L e m m a 9 is easily deduced from the definitions of T s, m s a n d I s.

3. U n i l a t e r a l t h e o r e m s

L e t a be a function of the t y p e in section 1 and assume t h a t the integral of 2-~da(2) converges absolutely when t a k e n over a right side neighbourhood of the origin. Under this condition consider

.~KE PLEIJEL,

A

bilateral Tauberian theorem

fo ^~-h(~

-]- t) ldo.(,~.). (2)

If this i n t e g r a l converges a t i n f i n i t y for one v a l u e t = to, for i n s t a n c e for t = 0, i t is seen b y i n t r o d u c i n g i n (2)

fl(2~) = f~/LI-h([,~

"~ to)-ldo'(#), a > O,

a n d b y p a r t i a l i n t e g r a t i o n t h a t (2) converges a t i n f i n i t y for e v e r y t. If

f/

~(,~ + 0 -1 d I - h - l a(2)

is i n t e g r a t e d b y p a r t s it follows t h a t (2) is o(1) w h e n t t e n d s to i n f i n i t y along half r a y s different from t h e n e g a t i v e real axis. T h u s we h a v e

Theorem 1. Necessary and su//icient /or the convergence o/ (2) is I - h - l a E o ) ~ The integral is then o{1) when t - ~ c ~ along hall rays /rom the origin which are di]/erent ]rom the negative real axis.

I n t h e A b e l i a n t h e o r e m for (2) which will be considered p r e s e n t l y , it is as- s u m e d t h a t a E I s w h e n s - h 44= i n t e g e r or I sa E eo ~ w h e n s - h = integer. These relations a p p e a r as conclusions i n t h e corresponding T a u b e r i a n theorem. I t fol- lows t h a t it is n a t u r a l t o suppose s < h + 1. If s = h + 1 t h e c o n d i t i o n I - * a E ~o ~ is e q u i v a l e n t to t h e existence of t h e i n t e g r a l (2) a n d also implies t h a t this in- tegral is o(1) w h e n t--> ~ . T h e A b e l i a n a n d T a u b e r i a n t h e o r e m s g i v e n below c a n therefore be e x t e n d e d i n a t r i v i a l w a y to i n c l u d e also t h e case s = h + 1 i n w h i c h t h e T a u b e r i a n t h e o r e m holds w i t h o u t t h e T a u b e r i a n c o n d i t i o n a E T ~. W e re- strict ourselves to t h e n o n - t r i v i a l case s < h + 1.

T h e following A b e l i a n a n d T a u b e r i a n t h e o r e m s are p r o v e d in section 4 a n d 5.

Theorem A ( 0 , ~ ) . I / s < h + l and a E i s or in case s - h is an integer, i/

I - S a E eo ~ then

f/2-h(~+t)-~da(~)-t

~p(t71)+o(t s-h ~), t-> +

Cx3 ,

(3) where p is a polynomial.

Recall t h a t if s 4 0 t h e c o n d i t i o n a E I s coincides with a E co s i.e. a ( ~ ) = o(~ s) w h e n s > 0 a n d a(),) = a( + c~) + o(~ s) w h e n s < 0. T h e c o n d i t i o n I - S a E e) ~ is e q u i v a l e n t to t h e convergence a t i n f i n i t y of

f ~ ,~-S da( ,~ ) .

According to l e m m a 1 a n d d e f i n i t i o n 3 t h e r e l a t i o n I - S a E eo ~ implies t h a t a E I ~ b u t t h e converse is n o t true.

ARKIV F/JR MATEMATIK. Bd 4 mr 45 The conclusion of A(0, ~ ) also h o l d s w h e n t t e n d s t o i n f i n i t y a l o n g a r b i t r a r y half r a y s f r o m t h e origin d i f f e r e n t f r o m t h e n e g a t i v e r e a l axis.

Theorem T ( 0 , ~ ) .

I /

s < h + l ,

i/ n E T ~ and i/, with a polynomial p,

f ~ ~-h(~

⁺ t ) - i d ( ~ ( ~ ) =

t-~p(t-~) §

o(tS-h-1), _t-->

§ ~ ,

(4)

then a E I s. I / s - h is an integer the result can be sharpened to I-~aEeo ~

Remark. T h e a s s u m p t i o n a E T s can be r e p l a c e d b y t h e s l i g h t l y m o r e g e n e r a l c o n d i t i o n t h a t a + ~ E T s for a ~ s a t i s f y i n g ~ E I ~ or, in case s - h = i n t e g e r , I-S~o E co ~

4. P r o o f o f t h e u n i l a t e r a l A b e l i a n t h e o r e m

T h e o r e m A(0, c~) is e a s i l y r e d u c e d t o

A{q}

a n d A { 0 } . I f 0 < s - h < l t h e p o l y n o m i a l

t-lp(t -1)

is i r r e l e v a n t in ( 3 ) a n d c a n b e i n c l u d e d in t h e r e m a i n d e r . W h e n

s - h = O

o n l y t h e first t e r m

alt -1

of

t - l p ( t -1)

is r e l e v a n t . T h e l e f t h a n d side of (3) c a n be w r i t t e n

f o (~ + t)- l dI-ha(2).

A c c o r d i n g t o l e m m a 3 i t follows f r o m t h e c o n d i t i o n a E I s of A ( 0 , ~ ) t h a t

I - ~ a E I s-h

a n d if

s-h=~O

we k n o w f r o m l e m m a 2 t h a t

IS-h=eo s h.

T h u s i - h a E e o s h w h e n

s - h 4 = O

which shows t h a t if 0 < s - h < l t h e o r e m A(0, ~ ) r e d u c e s t o A {q} w i t h

q = s - h

a n d w i t h a r e p l a c e d b y I : h a . W h e n

s - h = 0

i t is r e q u i r e d in A(0, c~) t h a t I - : a E ~ o ~ which is t h e c o n d i t i o n of t h e o r e m A { 0 } if a in t h i s t h e o r e m is r e p l a c e d b y

I - h a = I - S a .

T h u s A(0, c~) is p r o v e d w h e n 0 ~ < s - h < l .

To p r o v e t h e t h e o r e m w h e n s - h < 0 we use t h e i d e n t i t y

k - 1

(~ § -I = ~ (-- ~)it-J-l § (--t)-k ~ ( ~ +t)-I

i=0

w i t h

h - s < ~ k < h - s +

1. T h u s t h e left h a n d side of (3) e q u a l s a p o l y n o m i a l in t -1 w i t h o u t c o n s t a n t t e r m p l u s

( _ t ) - k ~ - h ( ~ + t)-ld(~(~t). (5)

Since a E I s i t follows t h a t I - h +j a E I s - h +j c I s - h + k- 1 = ~os - h + k- 1 c co0 w h i c h shows t h e c o n v e r g e n c e of t h e occurring integrals. I n (5) 0 ~ < s - ( h - k ) < 1 a n d t h e in- t e g r a l can be t r e a t e d in t h e s a m e w a y as t h e left h a n d side of ( 3 ) w h e n 0 ~< s - h < 1. This c o m p l e t e s t h e proof.

AKE PLEIJEL, i bilateral Tauberian theorem

5. Proof of the unilateral Tauberian theorem

W h e n 0 ~< s - h < 1 t h e T a u b e r i a n t h e o r e m

T(O, ~ )

is reduced t o T {q} a n d T ( 0 ) . T h e a s s u m p t i o n

a E T s

gives

l - h a E T €

according to l e m m a 6. Hence if 0 < s - h < l it follows f r o m

T(q}, q = s - h ,

t h a t

I-haEco~-h=I s-a.

L e m m a 3 shows t h a t

a E I s.

^{W h e n}

s - h = O

the theorem T ( 0 ) gives I-h(rEco ~ or

I-~aEco ~

I f

s ~ h < O ,

t h e basic relation (4) implies t h a t

f:(

] t + t ) - l d I - h a = a l t 1-[-O($-1), t-->-~ c~.

Since

I - h a E T s-~

a n d

s - h < O

it follows f r o m l e m m a 9 t h a t

I - h a E T ~

T h u s T{0} shows t h a t

I-~aEco ~

Because of this a n d t h e o r e m 1 formula (4) can be written

f o , ~ - h d a - f o~l-h(~ + t)-lda=al + a2t-l + ... + o(t~-h).

L e t t i n g t t e n d to infinity one finds because of the second p a r t of t h e o r e m 1 t h a t

a n d the formula reduces to

f : ,~-ada = a 1

f : ~.l-h(2 + t)-lda(~.) = _ a 2 t -1 _ ... + O(tS-h).

Thus, provided s - h < 0, formula (4) can be replaced b y a similar one with h - 1 instead of h. I f

s - ( h - 1 ) < 0

t h e procedure is repeated. F i n a l l y a formula is o b t a i n e d in which h is replaced b y h - r with 0 ~< s - ( h - r ) < 1. We are t h e n in t h e case a l r e a d y considered a n d t h e proof i s accomplished.

6. Inhomogeneous theorems

L e t Z = K ~ ~ w h e n s=~0 a n d ~ > ~ a > 0 a n d let

z-=K

log ~t when s = 0 a n d

~ t > / a > 0 . Here K a n d a are constants. F o r 0~<~t<a t h e f u n c t i o n Z is arbi- t r a r i l y defined so as to cause no trouble a b o u t the convergence a t t h e origin of t h e integral

f:~t-h(]t +

t) -1

dZ(]t).

(6)

I t is easy to see t h a t when s < h + 1 the condition I - a - l Z Ew ~ is fulfilled which guarantees t h e convergence at infinity of the integral. Also i~ E T ~.

I f a is replaced b y ~ - Z in A(0, oo), T(0, cr these theorems are transferred into " i n h o m o g e n e o u s " theorems connecting the conditions ~0 - X E I s or I-s(cp _ Z) E co o (when s - h = integer) on one side a n d

ARKIV FOR MATEMATIK. Bd 4 mr 45

f : ~-h( X + t)-ldcp = f : ~_~(~ + t)_ld Z + t-lp(t -1) + o ( [ t [~- h-~)

o n t h e o t h e r . I n t h e i n h o m o g e n e o u s T a u b e r i a n t h e o r e m t h e c o n d i t i o n ~pET s t a k e s t h e p l a c e of a E T ~.

S i m p l e c a l c u l a t i o n s show t h a t in t h e s e i n h o m o g e n e o u s t h e o r e m s t h e i n t e g r a l (6) c a n be r e p l a c e d b y

K 7~8

sin z ( s - h ) - - t s - h - x w h e n s # 0 , s - h # i n t e g e r ,

K s ( - 1)s-at s-h-1

log t w h e n

s#O, s-h=integer,

7~

s m ( - n h ) w h e n s = 0, s - h # integer, K ( - 1 ) - h t - h - 1 log t w h e n s = 0 , s - h = i n t e g e r .

7. A bilateral A b e l i a n t h e o r e m

I n t h e b i l a t e r a l case f u n c t i o n s a(2) a r e c o n s i d e r e d which a r e d e f i n e d on - - ~ o < 2 < + ~ a n d of b o u n d e d v a r i a t i o n o v e r e v e r y f i n i t e s u b i n t e r v a l . I t is a s s u m e d t h a t

fill

c o n v e r g e s o v e r a t w o - s i d e d n e i g h b o u r h o o d of t h e origin.

W e a r e c o n c e r n e d w i t h t h e s t u d y of t h e i n t e g r a l

which is s u p p o s e d to converge. This m e a n s t h a t

I-h-la().)

^{a n d}

I-h-la(-$)

a r e in co ~ w h e n c o n s i d e r e d for p o s i t i v e v a l u e s of ~. A s in t h e u n i l a t e r a l ease i t is n a t u r a l t o a s s u m e s ~<h + 1 a n d t h e case w h e n s = h + 1 is t r i v i a l . W e t h e r e f o r e s u p p o s e s < h + 1.

T h e p a r t of (7) w h i c h comes f r o m - ~ < $ < 0 is t r a n s f o r m e d i n t o a n i n t e g r a l f r o m 0 t o + ~ b y t h e s u b s t i t u t i o n 2 = - / x . A f t e r w a r d s / x is r e p l a c e d b y ~t. W i t h

t h e r e s u l t r e a d s

S(~) = a(~) + a( - ~), A(~t) = a(~) - a( - ~),

f o]tl-h().2 - t2)-l d~q().) - t f : ).-a(~,2 - t2)-i dA(~).

AKE PLEIJEL,

^A bilateral Tauberian t h e o r e m

F r o m this it follows t h a t a relation

f ~ ] ~I-h(~ § t)-ldo-(~)~ t-lp(t-1)+ o ( ] t Is-h-l),

holding w h e n t--> ~ along two opposite n o n real half rays, can be split into f o r m u l a s

f~

^{A h/~ (A} + T ) -~ d A ( V A ) = T - I P I ( T -~) + o ( t T I ~2-ht2-~) (8) f : A n~l (A + = r + o [ / s _ h - 1 1 \

), (9)

valid w h e n T--> ~ along a half r a y different f r o m the n e g a t i v e real axis. H e r o T = - t z, A = ~t 2 a n d t h e polynomials P1, Pz are d e t e r m i n e d b y p ( x ) + p ( - x ) = 2 PI( - x2), p(x) - p( - x) = - 2 x P 2 ( - x 2) when t h e p o l y n o m i a l p is known. Evi- d e n t l y 8 9 1 8 9 2 4 7 8 9 1 8 9 since s < h + l .

T h e bilateral Abelian t h e o r e m is n o w a n i m m e d i a t e consequence of A(0, ~ ) applied to (8), (9).

Theorem A( - ~ , + ~ ) . I / s < h + 1 a n d s - h is not a n integer it ]ollows / r o m a(2) E F , a( - ,~) e I ~ that

f +~12 l-%~ § t)-i d(~(~) = t-l p(t-~) § o( ]tI ~-h-~)

- oo

w h e n t - - ~ along n o n real hal~ rays /rom the origin. Here p is a p o l y n o m i a l . I / s - h is a n integer the same result is valid under the s u p p l e m e n t a r y conditions I - ~ ( ( ~ ( ~ ) § ~ w h e n s - h is odd, I - S ( ~ ( ) . ) - ~ ( - g ) ) e o ~ ~ w h e n s - h is even.

8. The bilateral Tauberian t h e o r e m

Theorem T ( - ~ , § ~ ). L e t s < h § ] a n d a s s u m e that (~(~) E T s, (I( - ~) E T s /or positive values o / L I / with a p o l y n o m i a l p

(10)

as t--->~ along n o n real hall rays / r o m the origin, then a ( ~ ) E I s and o ' ( - - ~ t ) r s /or ~--> § ~ .

Observe t h a t t h e couple A ( - ~ , + ~ ) , T ( - ~ , + ~ ) does n o t show t h e s a m e s y m m e t r i c reciprocity as A(0, c~), T(0, + ) .

I f we w a n t e d t o i~lclude also t h e case when s = h + i we would conclude f r o m t h e m e r e existence of t h e integral in (10) t h a t I "~(~)Ew ~ I - S ( ~ ( - 2 ) E o) ~ which implies a(2) e F , o( - ~) E I s.

ARKIV FOR MATEMATIK. Bd 4 wr 45 B u t for certain exceptional eases occurring only when s - h is an integer the proof of T ( - ~ , + oo) is obtained by replacing t in (10) b y

it,

the new t being real. If the resulting relation is split as in section 7 we obtain (8), (9) with T real positive.

According to l e m m a 7 and 8 either s(V-A) or A(VA) belongs to

T sly.

L e t us first assume S(VA)E T ~/2. Then T(0, o~) can be applied to (9) with the result

S ( ~ A ) E I s/2.

I f s - h is an odd integer T(0, cr gives in addition the convergence of

f ? ,t-~d(a(~) + a( - 2))

which, however, is of no use in the rest of the proof. Provided s - h is not an even integer theorem A(0, oo) yields

j ~ h 8 h

A -~ (A +

T ) - l d S(VA) = T -1P~(T -1) + o(1T [~ --~- 1),

0

(11) where P~ is a new polynomial. Addition and subtraction of (11), (8) lead to separate formulas for a(~A) and a ( - l/A). Application of T(0, ~ ) to these for- mulas finally shows t h a t a(V-A), a ( - ] / A ) E I s/2 and the conclusion of T ( - ~ o ,

+ c~) follows from lemma 5.

I f instead A(VA) E T ~j2 the same method is applied b u t with starting point in (8). The same result is obtained provided s - h is not an odd integer. I f

s - h

is an even integer we find the additional result t h a t

?,F~d(a(,~) - a( - ,~))

exists.

The exceptional cases s - h = even integer, S(~)E T s and s - h = odd integer, A(~) E T s remain to be considered.

9. P r o o f o f the bilateral Tauberian t h e o r e m in the exceptional cases To t a k e care also of these eases we m u s t use (10) not only along one couple of opposite half rays b u t along two. Since it is not more complicated to con- sider n couples we do this.

The point of departure is the partition in partial fractions 2n-1

2nt2n-~-l).~(~.2n +t2n) -1 = -- ~ S~+l'~t~-- ekt) 1,

(12) k=0

where ~ = 0 , 1 , 2 . . . . ( 2 n - l ) and

AKE P L E I J E L , ~4 bilateral T a u b e r i a n t h e o r e m

I n (10) t is now r e p l a c e d b y - r k = 0 , 1 , 2 . . . . ( 2 n - l ) , a f t e r w h i c h t h e re- s u l t i n g f o r m u l a s "are m u l t i p l i e d b y e~ +1 a n d a d d e d . T h e r e s u l t is c o n d e n s e d b y h e l p of (12) a n d o c c u r r i n g i n t e g r a l s f r o m - c r t o 0 a r e t r a n s f o r m e d i n t o inte- grals f r o m 0 t o + ~ . A t l a s t t h e n e w v a r i a b l e s A = ~t ~ , T = t 2~ a r e i n t r o d u c e d .

T h e f o r m u l a s d e d u c e d in t h i s w a y a r e

~ : h - g 1 1 8 h - ~ 1

A 2n ( A + T ) - l d ( a ( A ~ ) _ ( _ I ) ~ a ( _ A ~ ) ) = T - 1 p ~ ( T - 1 ) + o ( I T I ~ - ~ - - ) ,

(13)

~ = 0 , 1 , 2 . . . . ( 2 n - l ) . T h e y a r e c e r t a i n l y v a l i d w h e n T - - > ~ along t h e p o s i t i v e r e a l axis.

A c c o r d i n g t o l e m m a 8 a n d 7 we k n o w t h a t

1 1 8

(A 2 n) - ( - 1) ~ a ( - A ~ ~) E T 2~ (14)

h o l d s e i t h e r for :r o d d or even. A s s u m e first t h a t i t h o l d s for a o d d . T(0, ~ ) can t h e n b e a p p l i e d t o a n y of t h e f o r m u l a s ( 1 3 ) w i t h ~ o d d . r e s u l t is

1 1 s

a (A ~ ) + a ( - A 2~ ) r 12 ~ N e x t chose ~ e v e n a n d 'such t h a t

s h - ~ 2 n 2 n

T h e o r e m T h e (15)

is n o t a n integer. T h i s is p o s s i b l e if o n l y n > 1 i.e. if t h e r e is m o r e t h a n o n e e v e n :r 0~< ~ < 2 n . W i t h t h e c h o s e n e v e n a t h e o r e m A ( 0 , ~ ) shows b e c a u s e of (15) t h a t

f : h-~r 1 1 s h - ~ 1

A ~ ( A + T ) - l d ( a ( A ~ ) + a ( - A ~ ) ) = T - 1 p ' - ( T - 1 ) + o ( [ T [ ~ - 2 ~ - ), w h e r e P~ is a p o l y n o m i a l . This r e l a t i o n a n d f o r m u l a (13) w i t h t h e s a m e e v e n i

l e a d t o r e l a t i o n s for

f : A h-~ 2~ ( A + T ) - l d a ( + _ A 2 n ) 1 f r o m which i t follows b y h e l p of T(0, ~ ) t h a t

1 s

~ ( + A 2 " ) ~ I 2n.

A c c o r d i n g t o l e m m a 5 t h i s is e q u i v a l e n t t o a(~) r 18, a( - 4) ~ F .

T h e case w h e n (14) is v a l i d w i t h ~ e v e n is s i m i l a r l y t r e a t e d w i t h t h e s a m e r e s u l t .

ARKIV FOR MATEMATIK. Bd 4 ar 45 10. I n h o m o g e n e o u s f o r m o f the bilateral Tauberian t h e o r e m

With

when s 4 0 and

K12 s for 0 < a ~ 2

X ( 2 ) = K 2 ( - ~ . ) s for 2~< - a K log 2 for 0 < a ~ < 2 Z(2)= K 2 1 o g ( - 2 ) f o r 2 ~ < - a

when s = 0 the theorem T ( - ~ , + ~ ) takes the inhomogeneous form

Theorem T ( - 0 % + c~). I / s < h + l

and q~(2) E T 8, q~(-2) E T s and i/ with a polynomial p

;: 12]-h(A+t)-ldq~(2)= f;: 1 2 [ - a ( 2 + t ) - l d z ( 2 ) + t - l p ( t - 1 ) + o ([tl s-h-') as t - > ~ along non real hall rays /rom the origin, then

~ ( 2 ) - Z ( 2 ) E I s, ~ ( - ~ ) -

Z( - 2) e I s /or ]~--> + o~.

I n this theorem the integral on the right hand side

f: 2 -h (2 + t)-ldZ(2) + f : 2-h(2

- t ) - l d x ( - ~) (16) can be replaced b y expressions directly obtained from section 6. The expres- sion for the first integral in (16) shall be real when t is real positive (we as- sume K 1 and K 2 real) and the one replacing the second integral shall be real when t is real negative.

R E F E R E N C E S

| . HARDY G. H. and LITTLEWOOD J . E., Notes on the theory o] series (XI): On Tauberian theorems, Proceedings of t h e London Mathematical Society, Series 2, Vol. 30, P a r t 1 (1930), 23-37.

2. PLEIJEL A., Certain inde]inite di/]erential eigenvalue p r o b l e m s - The asymptotic distribution o/their eigen]unctions, Partial Differential Equations and Continuum Mechanics, Madi- son, The University of Wisconsin Press (1961), 19-37.

Tryckt den 23 november 1962 Uppsala 1962. Almqvist & Wiksells Boktryckeri AB