A convergence rate analysis for markov finite approximations to a class of Frobenius-Perron operators

Nonlinear Analysis, Theory, Methods&Applications, Vol. 31, No. 5/6, pp. 765-776, 1998 © 1998ElsevierScienceLtd Printed in Great Britain. All rightsreserved 0362-546X/98 $19.00* 0.00

Pergamon Plh SO362-546X(97)O0438-O

A CONVERGENCE RATE ANALYSIS FOR MARKOV FINITE APPROXIMATIONS TO A CLASS OF FROBENIUS-PERRON OPERATORS JIU DINGt and TIEN YIEN LI:I: t D e p a r t m e n t of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, U.S.A.; and ~:Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.

(Received 30 July 1996; received for publication 18 February 1997) Key words and phrases: Frobenius-Perron operators, invariant measures.

1. I N T R O D U C T I O N

In various fields of physical science, such as neural networks, condensed matter physics, and reaction-diffusion systems, one has to explore the statistical property of chaotic discrete dynamical systems. This problem, in many cases, is often reduced to the fixed density problem of the Frobenius-Perron operator associated with the deterministic system [1]. Originated with Ulam's pioneering work [2] on the numerical analysis of this operator, the computational fixed density problem of the invariant measure has attracted tremendous attention recently (see [3-6] and the references therein). The purpose of this paper is to give a rigorous proof of a theorem on the convergence rate of Markov finite approximations of Frobenius-Perron operators for a class of piecewise stretching mappings of [0, 1]. Let S: [0, 1] -, [0, 1] be a nonsingular transformation in the sense that S is Lebesgue measurable and m(S-~(A)) = 0 whenever m(A) = 0, where m is the usual Lebesgue measure. Let L~(0, 1) be the space of complex integrable functions f with Ilfll = Jo~ Ifl dm and denote by Llr(0, 1) the corresponding real space. The Frobenius-Perron operator P: L~(0, 1) --' L~(0, 1) associated with S is defined by

i'APf dm = I's_ .

.

.

.

,

I(A)

f dm

(1)

for all Lebesgue measurable subsets A of [0, 1]. It is clear that Pf~L~r(O, 1) for f ~ Lit(0, l). The motivation of introducing Definition (1) is that Pf* = f * for a nonnegative f * ~ L~,(0, 1) of unit norm if and only if the absolutely continuous probability measure p, whose Radon-Nikodym derivative with respect to m is f * , is S-invariant, that is, p(S-I(A)) = p(A) for all A. This f * is called a fixed density of P. The following properties of P can be proved easily and their proof is referred to [1]. PROPOSITION 1.1. (i) J~Pfdm = j~fdm for a l l f e Lt(0, 1). (ii) P is a Markov operator, that is, P f > 0 and IlPfll = Ilfll for all functions f_> 0. Thus IIPII = 1. (iii) If Pf = f for f ~ Ltr(0, 1), then Pf* = f+ and P f - = f - , where f ÷ = maxlf, 0l and f - = m a x l - f , 01. 765

766

J I U D I N G a n d T I E N Y I E N I.I

Remark 1.1. The Frobenius-Perron operator P is usually only defined for real functions.

That we extend its definition to complex functions is for the purpose of using its spectral information for the convergence rate analysis. See [7] for more details. In 1976, for a class of piecewise C 2 and stretching mappings of [0, 1], Li [3] proved a conjecturer by Ulam [2] on the convergence of a piecewise constant approximations scheme for numerically solving the fixed density problem P f = f . Because of the slow convergence of Ulam's original method which was obtained from a probabilistic motivation, a class of higher order Markov finite approximations methods were proposed in [5] and the convergence for the same class of mappings was proved. In [8] and [9] some efforts for the error estimates of Markov finite approximations were given. The Dunford integral technique was used in [8], while the spectral approximation idea as employed in [9]. It turns out that the main result of [8] cannot be proved under the condition mentioned in the paper, and the assumption that 1 is an isolated eigenvalue of P in [9] is in general not satisfied, since P is not quasi-compact on L~(0, 1) although it is true on the space of functions of bounded variation for the class of piecewise stretching mappings. Thus, a rigorous convergence rate analysis of Markov finite approximations methods is in need. This paper aims to achieve this purpose. Our approach is still based on the idea of Cauchy integrals as in [8], but a difference between our paper and [8] is that all discretized linear operators are not restricted to the corresponding finite dimensional spaces, but considered as operators on the whole space L~(0, 1). This seems more natural from the view point of spectral theory in functional analysis. In Section 2, we shall prove our main result that expresses an upper bound of the L ~ error of approximate fixed densities in terms of the variation norm. Some consequences will be presented in Section 3, and we conclude in Section 4. 2. E R R O R

ESTIMATES

We assume that S: [0, II ~ 10, 1] is piecewise C 2 and 2 = infls'l > 2 as in [10] or [31. From the Lasota-Yorke theorem (see [1]; Theorem 6.4.1), there are two constants a = 2/2 < 1 and b > 0 such that for all functions f of bounded variation, 1

1

V P f < a V f + bflfll. 0

(2)

0

Let

with the norm I

IlfllBv = Ilfl[ + V f ,

v f ~ BV(O, 1).

0

Then BV(0, 1) is a Banach space, and any closed bounded subset of BV(0, 1) is strongly compact in Ll(0, 1), due to the famous Helly theorem. Based on this compactness argument, it was proved that for the class of mappings considered in this paper, there is a fixed density to the corresponding Frobenius-Perron operator and all fixed densities are uniformly bounded in variation [10], and the Ulam method and more general Markov finite approximation method are convergent [3, 5].

Convergence rate analysis

767

For the sake o f convergence and the convergence rate analysis of numerical methods, we further assume that the fixed density f * of the Frobenius-Perron operator P is unique. Thus, by (iii) of Proposition 1.1, the null space N ( P - 1) is one dimensional, where I is the identity operator. In the following we write 2 - P to denote 21 - P. Now we briefly describe the class o f Markov finite approximations methods. Divide the interval [0, 1] into n equal subintervals I~ = [x,_,, xi] with the length h = l/n. Denote by A °, A~, A~ respectively the corresponding subspaces of (discontinuous) piecewise constant, continuous piecewise linear, and continuous piecewise quadratic functions of L'(0, 1). A basis for A° is

e°~(x) = X~,(x),

i = 1. . . . . n,

where XA is the characteristic function of A, a basis for A~, consists of i = 0, 1, ..., n, where

w(x)

=

(I

Ixl)xt_~,,l(x),

-

--0o
oo,

and a basis for A2. can be chosen as

e2i(x) = u ( ~ - - ~ J )

,

i = O, 1, ..., n,

and

eZi_,(x) = v ( ~ - - ~ )

,

i = 1,2 . . . . , n ,

where

u(x) = (1 - IXl)2Xt_~.II(X),

--oo
oo,

V(X) =

--QO (

o0.

2X(I

- - X)X[O.I](X),

X (

Let fi = Jl, f d m / h be the average value o f f over li. Now define QO: Ll(O, 1) ~ A ° by QOf__ ~ L e o,

(3)

i=|

Q~: L'(0. I) ~ At by n-I

O ~ f = A e ~ + E fi + fi+te] + f.e~, i=t 2

(4)

and Q~.: L'(0, I) --* A.2 by

~nnf = fle~O + "~ I f i + fi+' e;i-, + ~. fief,-, + f,,e~,,.

i=l

2

i=,

(5)

768

JIU DINGand

TIENYIEN

LI

Remark 2.1 QO is related to Ulam's original piecewise constant approximation scheme [2], while Q~ and Q2 were constructed in [5] initially for computing invariant measures of piecewise C 2 and stretching mappings. It was shown in [3] and [5] that Q~£ are Markov operators of finite rank, limn~Q~f=f strongly in LI(0, 1), and V~ Q~f<- V ~ f for each j = 0, 1,2. It was further shown that P.J = Q~P has a fixed densityf~ • A~, and for the class of piecewise C 2 mappings with inf]S'l > 2 such that P has a unique fixed density f * , lim f~ = f *

n~oo

for allj. Now we begin to investigate the speed of the above convergence. In the following we let [Q.] be one of the three sequences of Markov finite approximations defined above corresponding to the sequence of finite dimensional finite element spaces [A,,]. Then P~ = Q~P: Ll(O, 1) --* L1(0, 1) is a Markov operator with the range R(P.) = A.. It is well-known that for the class of Lasota-Yorke mappings considered in this paper, both P and Pn map BV(O, l) into itself. Moreover, since V~ Q . f < - V~f, by (2), we have 1

I

V P,f < a Vf + 0

bllfll.

(6)

0

LEMMA2.1. Let a complex number 2 ~e 1. If f • N(P - 2), then j ~ f d m = 0. Similarly, if P,f~ = 2f,,, then j ~ f ~ d m = O.

Proof[8l. Integrating P f = 2 f over [0, I1 yields 2

f dm=

P f dm =

.JO

0

f dm. , 0

Thus j ~ f d m = 0. The proof for Po is the same.

•

The following lemma is basically due to [8]. LEMMA 2.2. For n large enough, the eigenspace V,, of P,, corresponding to the eigenvalue 3. = 1 is one dimensional.

Proof. V~ always contains a density function f,, for each n. If the lemma is not true, then there is a subsequence [f~,] C {f,,] such that for each f ~ , there is a fixed point g~, of P~, with unit norm that is linearly independent off~i. Let c~, = j~g~i dm and h~, = Cn,fn, - gni II%f~, - g J "

IIh ,ll =

Then j~ h,,, dm = O, P,,,h,,~ = h~, and I

1

1. By (6),

1

1

V h . , = V P . h . , ~ a V h . , + b]]h.,[[ = a V h . , + b, 0

0

0

0

from which we have 1

b

Vh,,, < o

-l-a"

Convergence rate analysis

769

By Helly's T h e o r e m , the sequence h,~ has a convergent subsequence, still denoted as h,~ for the simplicity o f notations, which converges to h ~ Ll(0, 1). Now Ilhll = lim IIhJI = 1. M o r e o v e r , h is a fixed point o f P since i~

IIh - Phil <- IIh - hJI + IIh~,- P~,h~,ll + IIP.,h~, - P.,hll + IIP~,h - Phil - ' O. Since N ( P - 1) is one dimensional by assumption, f r o m (iii) o f Proposition 1.1, h = f * or h = - f * . H e n c e l~h dm = 1 or - 1, which contradicts the fact that ~ h~, dm = 0 imply ~ h d m = O. • LEMMA. 2.3. N ( P , - 1) = N((P~ - 1) k) for all positive integers k and n. Therefore, for n large enough, the generalized eigenspace o f P~ is one dimensional. Proof. Given n, since liml~®l[P~[I/I <_ l i m l ~ 1/I = 0, and since Pn is c o m p a c t , the l e m m a follows f r o m T h e o r e m VII.4.5 and L e m m a VIII.8.1 o f [11]. • N o w define F = 12 # l : P n g = Ag, g # 0 1 and y=

12 # l : P g =

2g, g E B V ( O , I ) , g ; " O ] .

LEMMA 2.4. 1 * 1= LI :p, where ,4 is the closure o f A. Proof. Suppose 1 ~ ITM. Then there is a sequence 12~,1 C F of nonzero numbers such that ,a.,, --+ 1. Let gn, be a corresponding eigenvector with unit n o r m . Then g~, e A~, since ;%, # 0. By L e m m a 2.1,

!

lg,,dm

= O.

0

N o w f r o m (6),

,

:jl

P~, g,, -< ~

V g,, = o o~

l(ao

g"i + b

)

.

Since for i large enough, I;%1 > a, so 1

V g"i <- - 0

b

12n, I - a"

Therefore, we can extract a subsequence of Ini], which is still denoted by Inil, such that gn~ converges to some g e Ll(0, 1). Since IIPg - gll -< IIg - gn, II + I1 - ,L,I IIg~,ll + 112~,g~,-

Pn, g~,ll

+ IIP,,g,, - Pn, gll + IIPn,g - Pgll - ' O, g is a fixed point, and g is either f * or - f * since Ilgll = I. This leads to a contradiction that l = II~ f * dml = Ilimi~.~ ~ g,, dml = O.

770

JIU DING and TIEN YIEN LI

N o w suppose 1 • p. Then

Pgn = ~'ngn for

a sequence 2 n • y such that

gn • BV(O,

1),

IIgA = l, and limn~oo2n = 1. The same contradiction will appear f r o m L e m m a 2.1 and (2). LEMMA 2.5. Suppose ,;t ¢ F and 2 ;~ 0. Then ). does not belong to the spectrum o f Pn.

Proof. We show that P, - 2: L1(0, 1) --* LI(0, 1) is one-to-one and onto. Given any g • L~(0, 1). Since An is finite dimensional, LI(0, 1) = A n O A,~ for some closed subspace A" o f U(0, 1). Let g = g~ + g2 with g~ • An and g2 • A,~. Then, since Pn - 2: An ~ An is one-to-one and onto, we see that f = f~ + f2 where fl = [(pn_ 2)[a,]_l(g , + ~Png2 1 ) ,

f2 = - ] g1z ,

is the unique solution o f the equationn (P, - 2 ) f = g.

•

N o w let d = ~ m i n l d i s t ( l , F U y), l - al. Then d > 0. Let a region o f the complex plane be defined as

n= By L e m m a 2.5, for all n,

l

z:~_
ll_<

"1 .

(Pn - Z) - I : LI( 0, 1) --+ Ll(0, 1) are well-defined and b o u n d e d for all z • f~. LEMMA 2.6. There is a constant M such that for any sequence fn • BV(0, 1),

II(Pn - z)-lfnll ~ MIIfnlIBv,

Proof.

v z • ta.

If the lemma is not true, then there is a subsequence Inil o f positive integers

such that lim Ilgn, ll = oo, where

= (Phi- Zn,)- I fn, g"' Ilfn, llBv and z,i • ~ . Let gn i

~n,--IIg,,ll

and

Lt ~

~fnl

IIg,,ll IIf~,ll~

Then fn, = (Pn, - Zn,)gn, and

IIf~,ll~ i-~oo lim

= lim

1

=0.

Convergence rate analysis

771

N o w since [z,~] C f2 is precompact, without loss o f generality, z,, --' z e f). Then f r o m (6) and the definition o f d.

V~.,=

-

0

0

P.,~.,

i

a

1 -

i

2d

that

P.,~.,+

<-- r~

Zn i

- -

--<

- f.,)

we assume

b

V gni o

+

- 1 -

2d

+

1

for i large enough. Hence i Vg.,

o

<

b+l-2d ,

-l-2d-a

¥ i >- N o .

It follows that there is a subsequence o f ~,,, which is again denoted by ~ , , such that lim high, - gll -- 0

i~oo

for some g ~ BV(O, 1) and Ilgll -- l i m i . ® l l ~ , , l l Ilzg -

Pgll

<- Iz -

= 1. N o w

from

z.,lllgll + Iz.,llLg - ~.,11 + Iltz.,- P.,)~.,II

+ IIP~,(~., - g)ll + IItP., - P)gll -~ o. we see that z e y, a contradiction to the fact that ~ A y = Q .

•

Remark 2.2. In fact, using the same idea, we can prove that if the sequence

Illf~ll/V~f,l

is u n i f o r m l y b o u n d e d , then 1

II(P, - z)-~f~ll -< M Y f , ,

¥z ~ ~2.

0

Remark 2.3. Since in general the sequence Ill(P, V~f, -~--i/oll 1 is not k n o w n to be b o u n d e d , we c a n n o t prove that

II(P. - z)-'/.ll -< MIILII,

z e f~.

N o w we are ready to apply the C a u c h y integral o f linear operators to the convergence rate analysis o f M a r k o v finite approximations. Let C be a circle centered at z = 1 such that C is contained in the region ft. For each n the resolvent o f P~ is defined as R ( z , P . ) = (z - p ~ ) - i

if the inverse exists and is b o u n d e d on L1(0, 1). Then R(z, P,) is well-defined for z e ~ and is analytic in f2 with the L a u r a n t expansion at z = 1 co

R(z,P,) =

~, AgtP.)(Z - 1)*, k=

--a~

772

JIU DING and TIEN YIEN LI

where the bounded linear operator Ak(P,): integral

l

LI(0,

1) ---' LI(0, 1) is given by the Cauchy

,i'c (ZR(z,P.) - 1)n+ldz"

Ak(Pn) = ~ni ,

By Lemma 2.3, for n large enough, z = 1 is a simple eigenvalue of Pn, thus Ak(Pn) = 0 for k _< - 2 . The next lemma is standard in functional analysis and will play a key role in the following. LEMMA 2.7. A_l(Pn) = (2hi) -j JcR(z, Pn) dZ is a projection operator from Ll(0, 1) to m(Pn - I). LEMMA 2.8. For n large enough, iffn e An is the fixed density of Pn, then for all densities f e LI(0, l), A-I(Pn)f

= fn"

Proof. Since N(Pn - 1) is one dimensional, A _ l ( P n ) f = c . f n , where Cn =

I A _ l ( e n ) f dm ,,0

since j~ f d m =

I. Now A - I ( P n ) f dm =

~i

,,0

R(Z, Pn)f dz

drn

0

= -2hi ~c

o

R(z, Pn)f dm

dz.

Let g. = R(z, Pn)f, then (z - Pn)gn = f , so

t"* 1 fdm •

0

=

Z

i~| gn dm - t"| P. g. dm

,,0

= z

i

'1

,,0

i' I

g. dm -

g. dm = (z - 1)

,,0

,,0

g. dm, ~0

from which we have c. =

('

i' 'l g. dm )

A _ ~ ( P . ) f dm = ~

,0

- 2hi

,

c~-

ldZ

t 0

fdm

=

1

0

=~

~-

1 z = 1.

That is, A_l(Pn) f = fn. COROLLARY 2.1. Under the same condition, if f * is the unique fixed density of P, then A_ i(Pn)Qnf* = fn.

Convergence rate analysis

773

THEOREM 2.1. Let f . e A . be a sequence of fixed densities o f P. that a p p r o x i m a t e the unique fixed density f * o f P. Then for n large enough,

IIf* -f~ll

Q.f*IIBv.

~ 3M[[f* -

Proof. By C o r o l l a r y 2.1, it is enough to estimate f* =

~n/,c~dZ

IIf* -

f* = 2hi,

A-I(P.)Q.f*II. since ~dz'

we have

f* - A_l(P.)Qnf* =

1

2hi 1

2hi 1

2hi

c(Z@lf*-R(z,

R(z, P.)

[(z - Q . P ) f * - (z - I ) O . f * l dz

- cZ-l

z c ~-1

Pn)Q.f*)dz

R(Z, Pn)(f* - Q . f * ) dz.

Since f * - Q . f * ~ BV(O, 1), L e m m a 2.6 implies that for all z e C, IIRtz, Pn)(f* - Q.f*)!l <- M l l f * - Q.f*llev. Hence, noting that d _> Iz - I I -> d/2, 1 l+d Ilf* - A_,(P.)Q.f*II <- . . . .

2n

d/2

2rtd. M . Ilf* - Q,,f*llev

= 2(1 + d)Ml[f* - Q,f*llBv <- 3mll f * - Q,,f*]]nv. This proves the theorem.

•

Remark 2.4. T h e o r e m 2. l indicates that the " g l o b a l " error IIf * - fnll o f the a p p r o x i m a t e solution f . depends on the local truncated error Ilf* - Q~f*lIBv o f the discretization o f the exact solution f * , which is consistent with the general principle in numerical analysis. COROLLARY 2.2. If 1If* - Q.f*llnv = O ( n - k ) , then Ilf* - f . l l

= O(n-k).

Remark 2.5. In general, the a p p r o x i m a t i o n order o f Ilf* - Qnf*llnv is 1 less than that o f Ilf*-Q.f*ll, hence the a p p r o x i m a t i o n order of Ilf*-f, ll is 1 less than that of Ilf* - Q~f*ll, a usual fact in the error analysis for numerically solving differential equations. Remark 2.6. Using R e m a r k 2.2, we have the following estimate I

[if* - fnll --< 3 M V ( f * - Q . f * ) . 0

N o w we further show that [If*

-fn[IBV

=

O(llf*

- Onf*llBv).

774

JIU DING and TIEN YIEN LI

THEOREM 2.2. Under the same condition of Theorem 2.1, (3M + l)b + 3M IIf*

IIf* - f~llBv

-

~.,,,~J*llnv.

I -a

P r o o f . Since f * + f,, = f * - P n f * + P , , f * - Pnf,, = P n ( f * - f , , )

+ f* - Q,,f*,

by (6), we have 1

1

I

V ( f * - f . ) <- V P n ( f * - fn) + V ( f * - Q n f * ) 0

0

0

I

I

_< a V ( f * - f . ) + bllf*

- f~ll

+ V (f* - Qnf*)-

0

0

Hence V (f*

0

I l l * - f.ll +

- f ~ ) -< - -

l -a

0

(f* - Q ~ f * )

,

and together with Theorem 2. l, we have 1

IIf* - f,,llBv = IIf* - f,,ll + V (f*

-

f~)

0

-<

- -

l-a

IIf*

-

LII

+

b+l -< - -

1 -a

o

(f*

-

Onf*) +

b IIf* - f,,ll +

1 -a

IIf* -

[If*

-

LII

f,,llBv

_< (3M + l)b + 3 M l l f . _ Q , , f * l t B v . l-a

•

3. A P P L I C A T I O N S

The previous section shows that both Ilf* - f . l l and Ilf* - f.llsv are upper bounded by a constant multiple of the quantity Ill* - Q , f * [ [ a v which depends on the smoothness of the fixed density f * and the specific Markov finite approximations scheme. Using standard results in approximation theory, we can deduce error estimates for the piecewise linear and piecewise quadratic Markov finite approximations. Let Lnf(x)

= ~ f(xi)e~(x) i=0

be the piecewise linear Lagrange interpolation function o f f .

Convergence rate analysis

775

LEMMA 3.1. Let f e C2[0, 1]. T h e n I

V (f-

L n f ) <- h m a x I f " ( x ) [ = O(h).

0

x e [0,11

Proof. D e n o t e B = m a x x e i 0 . 1 1 l f " ( x ) l . C o n s i d e r g(x) + f ( x ) - L,(x) o n [xi_l,xi]. Since g(x~_~) = g(xi) = O, t h e r e is z ~ [xi_ t, x~l such t h a t g'(z) = O. F r o m g"(x) = f " ( x ) , we h a v e

Ig'tx)l

=

<_ IIxg" dm IIX, ,.Z

I dm

<_ Bh.

,~Z

Thus,

V (f - L.f)

=

0

V

Ig'l

g =

i = 1 Xl_ I

<

B h d m = Bh

ldm =Bh.

i= 1 ~xi_ l

LEMMA 3.2. L e t f e

dm

i= 1 ,,xi_ I

0

C2[0, 1]. T h e n

I V (Q~ - L n ) f = O(h). 0

qi = Proof. Since Q ~ f =Y.i=oqiei ~ 1 a n d L , f = ~7=of(xi)e~, w h e r e q o = f l , (fi + fi+l)/2, i = l . . . . . n - 1, a n d q~ = f ~ , f r o m the fact t h a t V~I_, eJ = 1, we h a v e

1

~

V (Qlf_

xi

,V [qi -f(xi)le}

L n f ) <-

0

i= 1 xi_ I n

xt

E

=

I q i - ftx,)l

V

i=l

e~

=

xi-I

Iq,-

ftx,)l.

i=l

N o w the T a y l o r e x p a n s i o n gives

f ( x ) = f ( x i ) + f ' ( x i ) ( x - xi) + O(h2), so we see t h a t Iqo - f ( O ) l

=

O(h),

Iqi - f ( x 3 l

Iq~ - f ( 1 ) l

= O(h2),

=

O(h) a n d i = 1. . . . . n -

1.

Hence, I

V (Q~f-

n-I

L ~ f ) = O(h) + ~

0

O ( h 2) -~- O ( h )

i= I

PROPOSITION 3.1. Let f e C2[0, 1]. T h e n Ilf-

Q~fllBv = o(h).

-.~

Oth).

776

J I U D I N G a n d T I E N Y I E N LI

Proof. Since [ I f - Q~f[[ follows from

O(h2) by

=

1

V (f-

1

1

Qlnf) < V ( f -

0

by Lemmas 3.1 and 3.2.

direct integration (see also [81), the proposition

L . f ) + V ( Q ~ f - Lnf) = O(h)

0

0

•

Similarly we can prove the following.

C2[0, 11,

PROPOSITION 3.2. For a n y f e

[ I f - Q~fllsv = O(h). The next theorem follows from the above propositions and Theorem 2.1. THEOREM 3. I. Supposef, e A, be a sequence of fixed densities of P, from either the piecewise linear or the piecewise quadratic Markov finite approximations method and f * e C2[0, 1]. Then

Remark 3.1. As a by-product, we have also proved the convergence of the Markov methods in the BV-norm, an improvement of the result in [5].

Remark 3.2. Since [If - Q2,fl[Bv has the same order as [If - Q~fI[Bv, we do not expect a higher order in the convergence rate for the piecewise quadratic Markov approximation method. This was also reflected from the numerical results in [5]. The main reason of this drawback of Q~, is that Q]~f is only continuous, but not continuously differentiable.

Remark 3.3. Under a milder assumption that f * ~ w2'l(0, 1), the conclusion of the theorem is still true. This can be done using the Fourier analysis technique and the continuous imbedding of/.2(0, 1) in L1(0, 1). To end this section, it should be pointed out that the result in this paper cannot be used for the convergence rate analysis of Ulam's piecewise constant approximation method, which is a consequence of the following result. PROPOSITION 3.3. For a n y f ~ BV(O, I), I

I

V ( f - O_°f) >-- V f. 0

0

Proof. Since QOf = F.7=, fix,,, V(f0

Q~f) =

(f-fi) i=1

0

+ ~ Ifi+, - f i ] -> i=1

Vf= i=I

Ii

f. 0

Convergence rate analysis

777

4. CONCLUSIONS Based on the idea o f the p r e v i o u s w o r k [8] on e r r o r e s t i m a t e s o f the piecewise linear M a r k o v finite a p p r o x i m a t i o n s m e t h o d for c o m p u t i n g the fixed p o i n t o f the class o f F r o b e n i u s - P e r r o n o p e r a t o r s t h a t satisfy the v a r i a t i o n i n e q u a l i t y (2) for piecewise stretching m a p p i n g s o f [0, 1], in this p a p e r we presented a r i g o r o u s p r o o f o f the c o n v e r g e n c e rate result. T h e result is c o m p a t i b l e with the general c o n v e r g e n c e t h e o r y o f n u m e r i c a l d i f f e r e n t i a l e q u a t i o n s . Based on the e r r o r estimate, we e x p l a i n e d w h y the piecewise q u a d r a t i c m e t h o d c o n s t r u c t e d in [5] is not a big i m p r o v e m e n t to the piecewise linear m e t h o d , thus a d i r e c t i o n o f c o n s t r u c t i n g new higher o r d e r m e t h o d s is via s m o o t h e r piecewise p o l y n o m i a l f u n c t i o n s spaces. Lastly, the idea o f the p a p e r can be directly e x t e n d e d to the piecewise linear M a r k o v a p p r o x i m a t i o n s for m u l t i - d i m e n s i o n a l piecewise C 2 a n d e x p a n d i n g m a p p i n g s [6]. REFERENCES 1. Lasota, A. and Mackey, M., Chaos, Fractals, and Noise. Springer-Verlag, New York, 1994. 2. Ulam, S. M., A Collection o f Mathematical Problems. Interscience, New York, 1960. 3. Li, T. Y., Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture. J. Approx. Theory, 1996, 17, 177-186. 4. Ding, J., Du, Q. and Li, T. Y., High order approximation of the Frobenius-Perron operator. Applied Math. Compt., 1993, 53, 151-17 I. 5. Ding, J. and Li, T. Y., Markov finite approximations of Frobenius-Perron operator equations. Nonlinear Analysis, 1991, 17(8), 759-772. 6. Ding, J. and Zhou, A., Piecewise linear Markov approximations of Frobenius-Perron operators associated with multi-dimensional transformations. Nonlinear Analysis, 1995, 25(5), 399-408. 7. Ding, J., Du, Q. and Li, T. Y., The spectral analysis of Frobenius-Perron operators. J. Math. Anal. Appl., 1994, 184(2), 285-301. 8. Chiu, C., Du, Q. and Li, T. Y., Error estimates of the Markov finite approximation of the Frobenius-Perron operator. Nonlinear Analysis, 1992, 19(5), 291-308. 9. Hunt, F. and Miller, W., On the approximation of invariant measures. J. Stat. Phys., 1992, 66, 535-548. 10. Lasota, A. and Yorke, J. A., On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc., 1973, 183, 481-488. I 1. Dunford, N. and Schwartz, J., Linear Operators, Part I, General Theory. Interscience, New York, 1958.