An aymptotic problem on iterated functions

An aymptotic problem on iterated functions

Citation for published version (APA):

Bruijn, de, N. G. (1977). An aymptotic problem on iterated functions. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7706). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1977-06. Issued July 1977.

An asymptotic problem on iterated functions

by

N.G. de Bruijn.

University of Technology Department of Mathematics P.O. Box 513, Eindhoven. The Netherlands.

An asymptotic problem on iterated functions.

by

N.G. de Bruijn.

I.Introduction. Recently-A.Odlyzko ~tudied·the function F defined by the functional equation

•

, 2 3

F(x)

=

x + F(x + x ). (1. 1)

He con.i.ectured that its power series coefficients tn satisfy tn ....

"'an-J q,n v(log n). where a,is a constant, $

-~

(1+15), and v is a positive

pe~iodic

function with period 10g(3-$-I).

A related problem was treated in [l J, viz. the asymptotic behaviour of the power series coefficients of the function

rk -1 H(x)

=

log ~=o (1 - x ) , which satisfies 'r H(x) :;: -log(1-~) + H(x ) (t.2) (1. 3)

~

(r is an integer> 1). This was achieved by studying the asymptotic behaviour of (1.2) when x approaches the singularity at the point I, and deriving the behaviour of the coefficeints from what is essentially Cauchy's coefficient formula. Some years later W.B.Pennington DJ gave a shorter derivation by means of. a Tauberian theorem of Ingham •.

The asymp~otic formula for (I. 2) follows from the following exact formula

( 1 1 -1)2 -1 -1

H(x)

=

og og x _ ~ log log(x ) + W(log log x ) +

2 log r

(Xl -1 n n '

+ 1.: _n= 1 B (log x ) !(n.n!(r -1», _n

where the Bare Bernouilli numbers, and W is periodic with period log r:

n W(y) with 00 =

r

_k- ~ exp(2TIiky/log r), _(Xl k ( 1.4) (1. 5)

(211'ik) (, 21Tik) -1 a_k

=

r

log r S ~ + log r (log r) •

In the present note we study the more general problem of the behaviour of sums of the type

g(x) + g(e(x» + g(e(e(x») + ••• (1. 6)

and this will still contain a periodic function like the above W. Our main result will be (4.4).

If e(x)

=

x2+x3, g(x)

=

x we get the F of (1.1), if e(x)

=

xr, g(x) =

-log(l-x), we get the H of (1.3).(lt is not necessary that r is an integer, and that was not assumed in [I]. Only, i f r is not an integer, the notion • "coeffic ient of the power series" has to be slightly revised).

2. Conditions on

e

and g. Let b be a positive real, and let e(x) be defined for

o ::;

x s; b, with

(i) e is real-valued, continuous and strictly monotonically increasing, (ii) e(O) = 0, (iii) (iv) (v) (vi) e(b)

=

b,

o

< e(x) < x (0 < x < b),

there is a constant c with 0 < c < 1 such that e(x) < cx for 0 < x < Ib, 6 is differentiable at b, with e'(b) > I, and 6(x)-b-(x-b)e'(b)

=

= O(x-b)2) (x < b, x

~

b).

~

On account of (i),(ii),(iii), there is an inverse function and there is a doubly infinite sequence {en} nEE with 6

I-S, Sn+l(x) == e(en(x» for all nEil.

So 6_ 1 is the inverse of e~ 6

0 is the identity, and if n > 0 then en is the n-th iterate of

e.

If 0 ~ x < b, and x is fixed, then

e

(x) decreases exponentially if x is

n

fixed and n ~ 00. Actually we have e (x) == O(cn) (see (v». Similarly, b-e (x)

n . n

decreases exponentially if n ~ -00, since e'(b) > 1. (For a general discussion

on these iteration questions we refer to [2J, ch. 8).

The function g will be assumed to be real-valued and continuous on the

interval 0 ~ x < b, with g(O) = 0, and such that g(x)/x is bounded on 0 < x < ~b.

We shall also use on 0 ~ x < b an auxiliary function Z which has to have the following property : if h is defined by

(2.2)

converges for every x in 0 < x < b, and uniformly in every interval a) < x < b with 0 < at < b (note that it suffices to require uniformity in an interval

e(x_O) ~ x ~ Xo with some Xo E (0, b».

We quote two examples. First, if g(x) ... x for all x, then we can take

Z(x)

=

-b log(b-x)/log(e'(b». (2.3)

It easily follows from (vi) that hex) ... O(x-b), and that guarantees the con-vergence of (2.2).

Secondly, if b=l, g(x) ... -log(t-x) then we can use

2

Z(x)

=

(log(t-x» -

i

log(l-x),

2 log e t (1)

(2.4)

which again leads to hex) ... O(x-b).

In general, the existence of Z (such that (2.1) and (2.2) hold) is no problem ( we can prescribe Z(x) arbitrarily on some interval 8(x

O) < x ~ Xo and continue it such that (2.1) holds with hex)

=

0 for all x ~ xO; cf. the discussion on (3.1) in section 3). But what we want, of course, is a function Z that is easy to handle, at least asymptotically.

3. Two related functional equations. We consider the functional equations

L(e(x» ... L(x)

M(e(x» ...

e'(b)

M(x).

(3.1) (3.2)

It is easy to construct all solutions of (3.1) on 0 < x < b. We take

an arbitrary Xo in that interval and prescribe L(x) arbitrarily for 6(X O)<

< x ~ xo. Since 6

n(xO) -+ 0 ifn .... +oo e_n(xO) -+ b i f n+-oo , this function can be extended to a solution of (3.1) for 0 < x < b : for every XE (O,b) there is a unique n E

'I"

with e (x) E (S(x ) ,x J.

n 0 0

As to (3.2) it suffices to produce a single positive solution on (O,b), since every other solution is the product of that positive solution and a solution of (3.1).

Equation (3.2) 48 directly related to the Schroder equation: if we

-1

define w, ai'_ f by w{x) • M(b-x)~ f(x) = b ,-

a_I

_{(b-x)., a 1 ...}

(e.'

(b» ,

we get the Schroder equation w(f(x» ... atw(x) for which an infinite product solution was described in [~, section 8.3J. In our present notation it amounts to the following. If n is defined by

b - 6_ 1 (x)

fleX)

=

et(b)

b - x

(0 < x < b)

we have n(x) ~ ] + O(b-x) by (vi, a function MO by

section 2). It follows that we can define

(3.3) (note that b - 6 (x) tends exponentially to zero). It is easy to verify that

-n

MO satisfies (3.2).

If L satisfies (3.1) then there obviously exists a periodic function v with period 1 such that

(

, log MO

(X)~

L(x)

=

v

.

log et(b)

(3.4) As MO(x) ~ b-x if x < b, x ~ b, it requires only light smoothness conditions on L in order to get from (3.4) to

(x < b,x .... b). (3.5)

It suffices to assume that L is continuously differentiable on[6(x

O),xO

J.

4. The sum F. Let e and g satisfy the conditions of section 2. We define

g

(O::;;x<b). (4.1)

The series converges rapidly since 6 (x) tends exponentially to zero, and

n

g(x)

=

O(x). Obviously

F (x)

=

g(x) + F (S(x»

g g (O::;;x <b). (4.2)

We want the behaviour of F (x) for x~b. Let us assume we have a

g

function Z as described in section 2, i.e. with uniform convergence of (2.2) for every interval a < x < b (if 0 < a < b). For 0 < x < b, we now define L(x) by

L(x) = lim «~oo g(e. (x») - Z(e (x»).

n .... + GO "K=-n -k -n (4.3)

The existence of the limit follows from the convergence of (2.2), and we can write

00

L(x)

=

Fg(x) - Z(x) + _En=} h(e_n(x». (4.[')

L(x)

=

L(e(x» (O<x<b), i.e. L satisfies (3.1), and has the form (3.4).

Because of the uniform convergence of (Z.2) we have

(x < b,x -+ b) ,

00 00

since Ln=m h(e_n(y» = Ln=t h(e_n(x» if y

=

0m(x), and y€ (a,b) as soon as x € (e(a),b). Thus we have obtained, as our main result,

-m

lim _{x < b,x} (F (x) - Z(x) - L(x» •

o.

(4.5)

-+ b g

Formula (4.4) presents a quite useful representation of L(x). In the special case where g(x)

=

x (0 ~ x ~ b) we can also use the function MO of section 3. We define Z by (2.3), whence

(

MO(x) )

log ,

(0' (b» n

and now (4.3) gives

L(x)

=

F (x) - En (b - e_k(x» +

g k=l

b

log e i (b) log MO (x). (4.6)

Note that the two series, as well as the product expansion of MO(x), are rapidly convergent. This can be used to show the following: if e is

con-tinuously differentiaple for 0

<

x < b, then L is continuously differentiable, and thus we have (3.5).

~

5. Applications. (i) First we take the function F defined by (J.l). According to (4.2) this equals F , where g(x)

=

x, and e(x)

=

xZ+x3, if we take b

=

g

=

~(I

+

1:5),

i.e. the positive root of x = x2 + x3• We have e'(h) = !(7

-is).

By (4.5) we have

b

F(x) = - log et(b) log (b-x) + L(x) + 0(1) if x < b, x -+ b. The term 0(1) can be replaced by

(ii) Next we take a look at H(x) of (1.2). We take b=l, Sex)

=

xr (r

is

a real number > t), g(x)

=

-log(l-x), and Z(x) as in (2.4). By (4.2) we have F

=

H. Now (4.5) gives

g

2

H(x) =

(lo~(]-x»

-

~

10g(l-x) + L(x) + 0(1), 2 log r

and this is in accordance with (1.4). 'Note that i f W(log log x-I) is

~.--"-.. - ..

abbreviated as U(x), then U(x)

=

U(xr) (since W is periodic with period log r).

(iii) Let us take the simpler case where still b=l,a(x)

=

xr, but now g(x)

=

XC with some positive constant c. This means that F (x) _g

=

F(x;c), where F(x;c) is defined by

2

F(x;c)

=

xC + xcr + xcr + •••

We take b=l, and Z as in (2.3) (this works for every g with g(x)

=

1 + + O(x-l», and now (4.5) gives

F(x'c) = - log(l-x) + L(x) + 0(1)

, log r (x -+ I)

with L(x)

=

L(xr). Actually, by the method of [IJ an explicit formula for F(x;c) can be produced; it is just a bit simpler than the one for (1.2).

It is

F(x;c) =

--1

log log x

log r _ y + log log c r +! +

-]

log log x ~ -J n

log r )+ En=1 Sn(log x ) ,

-1 21Tik/log r where uk

=

r(21Tik/log r) (log r) c

and y is Euler's constant. We mention that it is easy to verify, as a check, that F'x;c)

=

XC + F(x;cr).

References.

1. N.G. de Bruijn. On Mahler's partition problem,

Nederl.Akad.Wetensch., Proc. 51, 659-669

=

Indagationes Math. 10, 210-220 (1948).

2. N.G. de Bruijn. '~symptotic methods in analysis:'

North Holland Publishing Co. and Wolters-Noordhoff Publishing, rd

3 ed. 1970.

3. W.B.Pennington. On Mahler's partition problem. Annals of Math. 57, 531-546 (1953).