Asymptotics IVb

MATHEMATICS

ASYMPTOTICS

IVB

THE MULTIPLICATION-INTERPOLATION METHOD BY

J_ G. VANDER CORPUT (Communicated at the meeting of December 17, 1955)

Section 5 *). Invariant properties. In this section we consider couples A= (.x, 2 a,.), where the first element is a real or complex number and the second element is a formal series 2 an, formed by real or complex numbers a,. (n > 0). If A= (.x, 2 a,.) and B = ((3, 2 bn) denote such couples, then the couple (.x(3, 2 c,), where

is called the product AB of the two couples; furthermore

where 0.::;;0.::;; I, is called a couple interpolated between A and B. We call a property P of such couples invariant with respect to multiplication and interpolation if it satisfies the following condition: if two couples A and B possess the property P, then also their product AB and also each couple interpolated between A and B possesses the property P. *) Under the titles: Asymptotic Expansions I, II, III and IV the author has published four self-contained technical reports. These reports, the first three of which were facilitated by the support of the United States Air Force and the fourth facilitated by a grant of the National Science Foundation, have the following subtitles: I. Fundamental theorems of Asymptotics, June 1954, IV + 66 pages; II. Elementary Methods, November 1954, III + 54 pages. III. The Asymptotic behaviour of the real solutions of certain second order differential equations, June 1955, III + 171 pages. IV. The Multiplication-Interpolation Method, November 1955. Summaries of the reports I, II, III have been published under the titles: Asymptotics I, These Proceedings, Series A 57, 206-217 (1954) and Indagationes Mathematicae 16, 206-217 (1954). Asymptotics II, These Proceedings, Series A 58, 139-150 (1955) and Indagationes Mathematicae 17, 139-150 (1955). Asymptotics IliA, These Proceedings, Series A 59, 1-10 (1956) and Indagationes Mathematicae 18, 1-10 (1956). Asymptotics IIIB, These Proceedings, Series A 59, 11-14 (1956) and Indagationes Mathematicae 18, ll-14 (1956). The papers Asymptotics IVA and IVB give a summary of Report IV.

137 The significance of this notion for the problems under consideration is clear. In fact, the couples A:(k) = (
:L ahn(k))

(h = 0, I, ... , r)

by means of multiplication and interpolation. Therefore, to prove that a couple A~ (k) possesses a certain invariant property, it is sufficient to show that each of the r+ 1 couples Ah(k) possesses that property. For instance, the condition lxl 0 we have

lx: (k) I ~ I. This gives us for this problem immediately a convenient upper bound for Ix;(k)l. The condition ( -I)n an 3 0 (n = 0, I, ... ) is also a property which is invariant with respect to multiplication and interpolation. In the Szekeres problem the couples Ah(k) (h=O, 1, ... ) possess that property, so that for each integer k, for each integer n;>O and for each system e= (e1 , e2 , •.• ) formed by a finite number of integers ;:;;,0 we have The property that (n = 0, I, ... )

(-)nan 3 0

and that moreover iX =

a 0 +a1 +

... +an_ 1 +An an

(n

= 0, I, ... )

where 0 0 and for each system e= (ev e2 , ••• ) formed by a finite number of integers ;:;;,0 we have (- )n (k) 3 0 and x: (k) = a:0 (k) + ... + a:.n- 1 (k) +Ana;. (k) ,

a:,.

where 0
n-1

!

m-0

a:,. (k).

3) See for instance G. POLYA, Remarks on computing the probability integral in one and two dimensions, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, presented 1945, printed 1949, 63-78, see 76-77.

138 In this case we even find an upper bound for the absolute value of the remainder term, namely Ja;,(k)l, but this does not help us now, as at the moment we do not have at our disposal a suitable upper bound for ja;,(k)l. How it is possible to find such an upper bound will be explained in the next section.

Section 6. Majorants. In chapter II, section (4), we use a general class of majorants, but for the sake of simplicity we restrict ourselves in this summary to a particular type, namely to majorants of the form

2,, n=O n. 00

(6.1)

ei

= _2

....

where j ~ 0.

We say that a couple A= (ex, .La.. ) is majorized in the weak sense by

i"' Ia,. I,;;:; n!

(6.2)

If A is majorized by (6.3)

ei

lex-

(n

ei

if

= 0, 1, ... ) .

in the weak sense and if moreover n-1

jn

.2 amJ ,;;:; 1n. m=O

(n = 0, 1, ... ),

then we say that A is majorized by ei in the strong sense. We shall prove in Chapter II, sections (4) and (6): if A is majorized in the weak or strong sense by ei and if B is majorized in the same sense by ei', then AB is majorized in the same sense by ei+i' and each couple which is interpolated between A and B is majorized in the same sense by emax(i.i'>. Now it is clear how to find majorants for the couples A;(k) defined in the preceding section. We start from the given couples Ah (k) (h = 0, 1, ... , r) and we determine a number jh;>O in such a way that the couples Ah(k) (h = 0, 1, ... ) are majorized by eih, either all in the weak or all in the strong sense. Each of the couples A;(k) is generated by the couples Ah(k) (h=O, 1, ... , r) by means of multiplication and interpolation. Repeatedly applying the preceding theorem we find a number j > 0, depending on e and k, such that A;(k) is majorized by ei in the weak or in the strong sense. Consequently we find certainly (6.4)

la;,(k)j,;;:;:!

(n=0,1, ... );

if A;(k) Is majorized in the strong sense, we have moreover (6.5) In this way we find the required upper bounds for the left hand side of (6.4) and (6.5). In particular, choosing n=O in (6.5), we obtain again

lex; (k) I,;;:; l. The question arises how to determine in reality this number j. It would

139 be much too encumbrous to apply all the multiplications and interpolations which are necessary for the construction of the couple A;(k) itself, but happily that is not necessary. In order to determine this number j we introduce in Chapter III, section (3), the notion of the degree of a formal power series, which enables us to deduce the number j almost immediately from the given recurrence relation. For instance, in the problem of Szekeres we obtain in Chapter III section 8 (6.7)

As soon as j is known, the formulas (6.4) and (6.5) yield the required upper bounds for j£X;(k)j and for the absolute value of the remainder term £X:(k)-

ft-1

L: a;,.(k).

•-o

Section 7. Evaluation of the coefficients occurring in the asymptotic expansion of F(k) and y(k). In section (4) we have formed for the problem 00

of Szekeres an asymptotic expansion ,2' a,.(k) for F(k). The terms a,.(k) n=O

occurring in this asymptotic expansion are constructed by means of multiplication and interpolation, but this construction is so complicated, in particular for large w and k, that it does not enable us to evaluate the coefficients a,.(k) in this way. We must make an exception however for a 0 (k) which is rather simple, namely a0 (k)='f} (~),where 'f}='f}(X) is a solution of the equation (7.1)

00

'f)= x+

,2 ch(t'f})h.

h=l

If x is given, then this equation possesses one and only one solution, depending on x and t, which is an analytic function of t at t = 0. By this condition the function 'f)( X) is uniquely defined, provided that Itl is sufficiently small. The calculation of the coefficients a,.(k) for n = 1, 2, 3, ... is not so simple. We shall see in Chapter III, section (11) that a,.(k) can be written as w_,. times a function of~' in symbols: Q) a,.(k) = w-"'f},.

(!)·

Then

F (k) "' ..~ow-" 'f} 11

(!),

where 'fJo(x} denotes the solution 17(x) treated above of the equation (7.1). The problem is now to determine the functions 'f),.(x) (n= 1, 2, ... ). To that end we shall express these functions by means of the functions 17(x) and

140

For instance, if we put 1;=1-tlJI(t'f}), then 'f/1 = -t?;-2t~ ljl' (t'f}) ;

(7.2)

~ 'f/2 =

~ +1

i

c-' to 'f/2 ljl' lJI" P'+ 1 c-s t4 'f/2 P"' + 1 c-s ts 'fJ P";

c-6 t6 'f/2 lJI' ljl' ljl'+ i

c-' t4 'fJ P'

the argument in each of the functions lJI', lJI" and lJI"' is t'f}. In general, we find for each positive integer n that 'fJ,.(x) can be written as a linear combination, with constant rational coefficients, of products of the form (7.3)

here X is an abbreviation for a product of the form

where lXv tX 2, ••• , !XLI denote positive integers. Here Ll (X) and .Q(X) denote positive integers depending on X, namely Ll (X)= Ll and .Q(X) = 1X1 +1X2 + ... +!XLI· We call LI(X) the degree of X since it denotes the degree of X with respect to the derivatives of the function lJI. We call .Q(X) the order of X, since .Q(X) denotes the number of differentiations necessary for the calculation of X, provided that we start from the given function lJI(w). In Chapter III, section (12), we shall prove more, namely, that, for each positive integer n the function 'fJn can be written as a linear combination, with constant rational coefficients, of products of the form (7.3), where n,;;;;.Q(X),;;;; 2n-l.

For instance, in the terms occurring on the right hand side of (7.2) the orders are respectively equal to 3, 3, 2, 3 and 2. In this way we have found an asymptotic expansion according to ascending powers of w-1 for the numbers (7.4)

F (k)

= y(k-1)

y(k)

'

under the assumption that k is at most of the same order of magnitude as w. Now of course the question arises whether this result enables us to find an asymptotic expansion for log y(k), when y(k) denotes the coefficient occurring in (1.1 ). It follows from (7 .4) and y(O) = 1, that y(k) = {F(k)F(k-1) ... F(1)}-I,

hence (7.5)

-logy(k)=log F(1)+log F(2)+ ... +log F(k).

Under general conditions this formula gives us the asymptotic behaviour of log y(k), if we know the asymptotic behaviour of F(k). But Szekeres has remarked that in his problem there exists a relation between log y(k)

141

and F(k) which is simpler than (7.5). His result is only a special case of the following general statement: Let P(w) be a polynomial in w of degree m. We can therefore write P'(w)

=

P1+PsW+ ··· +pmwm-l.

Let g(w) denote a function which is analytic at the origin with g(O) * 0 and put eP
00

=

I

k=O

uJc,

y (k, t)

so that y(k, t) is a polynomial intofdegree <:,k. Assumey(k,O) *0. Put F(k t) '

=

y(k-l,t) y(k,t)

Then we have for sufficiently small y~ij

-log y(k,O) (7.6)

=

(k

~

1)

.

iti

t

p1 ~ (F(k,u)-F(k, 0))

~

u

du +p2 f' (F (k, u) F (k-1, u)- F (k, 0) F (k-1, 0))0 u t

.

•

+ ... +Pmf0 (F (k,u) ... F (k+ 1-m,u)-F(k, O) ... F (k+ 1-m, 0)) -u ; here F(k, u)=O for k<:,O. If the asymptotic behaviour of F(k, u) is given, the right hand side gives immediately the required asymptotic behaviour of log;:::~). We have said that (7.6) holds for sufficiently smalljtj. It is sufficient to choose ItJ so small that y(k, u) is 0 at each point u lying on the line segment (0, t). In the problem of Szekeres we have P( w) = ww, hence m = 1 and p 1 = w, so that

*

y(k,t)

(7.7)

-log y(k, O)

1

=

w { (F (k, u)- F (k, 0))

du u-·

The proof of (7.6) is very simple, namely: g (tw) = e-P

Then

I

00

k=O

y (k, t) wk.

t llg(tw) _ llg(tw) llt -Wilw"

This gives, with the dot indicating differentiation with respect to t, te-P

I

00

k=O

y (k, t)

uJc =

- e-P
wP' (w)

I

00

k=O

y (k, t) wk+ e-P

I

00

k=O

y (k, t) kufc.

Multiplying through by eP and equating on both sides the coefficients of uJc, we find ty(k, t)= -ply(k-1, t) -p2y(k-2, t) - ... -pmy(k-m, t)+ky(k, t);

142

here y(q, t)=O for q
If t ~ 0, the left hand side tends to zero, so that 0=k-p1 F(k, 0)- •.. -pmF(k, O) ... F(k+l-m, 0),

hence

_ y(k,t) _ F(k,t)-F(k,O) + F(k,t) F(k-l,t)-F(k,O)F(k-1,0) y(k,t) - Pl t P2 t ~k,t) ... F(k+I-m,t)-F(k,O) ... F(k+I-m,O) + +

•••

Pm

t

Integration gives the required identity (7.6).

·