Aequationes Mathematicae 41 (1991) 111 - I t8 University of Waterloo

0001-9054/91{0t0111-08 $1.50 + 0.20}0 © 1991 Birkhfiuser Verlag, Basel

Some regularity properties of algorithms and additive functions with respect to them

Z. DAROCZY, GY. MAKSA and T . SZAB6

Dedicated to the memory o f Alexander M. Ostrowski on the occasion of the tOOth anniL,ersctry of his birth

Summao'. The paper deals with the behaviour of the so-called "'algorithms with respect to interval filling sequences" A connection is established between the uniquely representable points and the continuity points of the algorithms; also strong continuity properties on monotonic algorithms are proved. Finally lhe results are applied to additive functions. The theorems extend some former results by the authors, by I. Kfitai and by A. Jfirai.

I. Introduction

Let A be the set o f the strictly decreasing sequences 2 = (2 . ) of posit ive real

numbers for which L(~.):=~.~'= i 2 . < + o o . A sequence (,~) ~ A is called interval /filing if. for any x ~ [0. L(2)]. the re exists a sequence (6,,) such tha t 6. ~ {0, 1} for

all n ~ N and x = ~ = ~ 6.2.. Thi s concept was in t roduced and discussed in [1]. It

is known from [ 1] that 2 = (2,,) ~ A is in terval filling if a n d only i f 2n ~< L,, + ~(2) for

all n ~ N where Lr.()~) = Y, TL-.~ 2~, m ~ N. Th e set o f the interval filling sequences will be denoted by IF.

An algorithm (with respect to 2 = ( 2 ~ ) e l F ) is defined as a sequence of functions a,,: [0, L(2)] ---r {0. 1 } (n ~ N) for which

x = ~ ~,,(X)L,

AMS (1980) subject classification: Primary 39B20. Secondary 65D15, 65Q05, 68Q25.

~Icmuscript rec'eived June 27. 1990, and in .final form, November 15, 1990.

I l l

112 z. DAROCZY et at. AEQ. MATH.

for all x ~[0, L(~)]. We denote the set of the algorithms (with respect to ). = (2.) ~ IF) by ~'(2). Obviously, d ( ) 0 ¢ ~ for all 2 ~ IF. Namely, it was proved in [1] and [2] that, if 2 = (2.) e IF and

I i n--I if x < ~ e,,(x),~i+2.

~.(x) = ~= n-- i if x >>- ~ ei (x)).~ + 2.. i=l n E N, x ~[0, L(2)]

or

f l rt--I if x ~< ~ ~* (x )L + ,~.

e*(x) = i=~ n--| if x > ~ e * ( x ) 2 , + 2 . , i=1 n ~ N , x ~ [ 0 , L(2)],

then e = ( e . ) , e * = ( ~ * ) ~ d ( 2 ) . The ~ and ~* are called regular and quasi- regular algorithms, respectively. It can be easily seen that e. is continuous from the right and ~* is continuous from the left at each point of x ~ ]0, L(2)[ for all n e N .

I f (X, II II)is a (real or complex) Banach space, (a~) is a sequence in X such that Ila° II < + 0o, --- ~ IF, do c d ( 2 ) , d 0 4= ~;0, F: [0, L(2)I -~X and

F(x) = ~ ~.(x)a. x ~ [0, L(2)] n=l

for all (c~.) ~ d o then F will be called an ~¢o-additive function (with respect to 2). It is known that the real-valued d(2)-addi t ive functions are linear [3] and a complex-valued {e }-additive function is continuous if and only if it is {e, e*}-addi- rive [2].

In this paper we give a description for algorithms, establish their continuity points, introduce the concept o f monotonic algorithms and discuss their regularity properties. Finally we present some results for Jo -add i t ive functions as conse- quences.

2. Continuity points of algorithms

First we give a characterization for algorithms.

Vol. 41, 1 9 9 1 Regularity properties of algorithms and additive functions 113

T~EOREM 1. L e t 2 = (~,,) ~ IF. A sequence o f f u n c t i o n s ~. : [0, L(2)] ~ {0, 1 } (n ~N) is in ~¢(2) if" cmd only i f there ex i s t s a sequence o f f u n c t i o n s

~,, : [0. L00] ~ {0. 1 } (n ~ N) such that

0 i f x < y~ a,(x))o, + ;.,, t = t

n - - I n I

a . ( x ) = 0.(x) i f ~ e , ( x l 2 ; + 2 . < ~ x < ~ ~ e , ( x l 2 t + L . + ~ ( 2 ) (1) 1=1 i=1

1 i f x > ~ 0~,-(x)2~+L.+I(2 ) l = l

f o r all x ~ [0, L(~)] a n d n ~ N .

P r o o f Let

n - I

s . ( x ) = ~ ~,(x)2, ( s l ( x ) = 0 ) (2) i = 1

if x ~ [0, L(,~)], n ~ N. First we prove tha t (a,,), as defined in (1). is an a lgor i thm. This will fol low f rom

the inequalities

0 <~ x - s . ( x ) ~ L~(2) x E [0. L(2)], n ~ N (3)

Let A = {n ~ N: 0 <<. x - s . ( x ) <~ L.(2) for all x ~ [0, L(,~)]}. Obvious ly 1 6 A. Sup- pose that n E A.

I f x < s . ( x ) + 2. then, by (1), a . (x ) = 0 therefore x - s . + t ( x ) = x - s . ( x ) >1 0

since n ~ A and x - - s . + l ( X ) = X - - S . ( X ) < 2 . ~ < L n + l ( 2 ) since . ~ E I F . Thus n + 1 ~ A in this case.

I f s . (x) + 2 . <~x <~s . (x ) + L~+ ~(2) then, by (1),

x - s . + ~(x) = x - s . ( x ) - e . ( x ) 2 . 1> .~. - ~.(x); t . i>0

and

x - s n + 1 (x ) = x -- s . ( x ) -- ~o.(x).~. ~< L,, + 1 ()~) -- e . (x )2 . ~< L . + ~(2).

114 Z I)AROCZY et al, AEQ~ MATH

Thus n + 1 E A in this case. too.

I f x > s.(x) + L .+ I (2 ) then, by (1),

x - s . + ~(x) = x - s~(x ) - 2~ > L . + l(2) - 2. i> 0

since 2 ~ I F and

x - s,,+ ,(x) = x - s,~(x) - ~.. ~< L.(~) - 2~ = L .~ ~(~.)

since n ~ A. Thus n + 1 e A in all possible cases which proves (3).

Conversely, if (%) ~ M(2) let Q~(x) = % ( x ) , n E N and x e [0, L(2)]. If x < s . ( x ) + 2 , for some n o n then ~ ; L l ~ , ( x ) 2 , < s . ( x ) + ' ~ , therefore

~ = . ~ ( x ) 2 ~ < ~ . . , thus % ( x ) = 0 . I f x > s . ( x ) + L . + j ( 2 ) for some n C N then

~F= l ~(x)2~ > s, ,(x) + L~+ ~(2), therefore ~ = . ~,(x)2i > L . + i (,~.), thus ~ ( x ) = 1, consequently (1) holds for (~.). D

REMARKS 1. I f J = (3.~) e IF, Q.(x) = 1, x ~ [0, L(2)], n u N in (1) then we have

the regular algori thm. 2, Theorem 1 shows that, if (%) ~ d ( 2 ) , then the function ~. is continuous at

0 and L ( 2 ) for all n e N.

In what follows, i f2 = (2.) ~ IF, a ~ [0, L(A)] and ~.(a) = ~.(a) for all (%) ~ d ( 2 ) and n e N (where (e.) is the regular algorithm), we say that the number a is uniquely

represen tab le . The set of uniquely representable numbers will be denoted by U(2). Obviously 0, L ( 2 ) ~ U(;~). The set U(2) will play a role in describing of the set of

continuity points of algori thms as the following theorem shows.

THEOREM 2. L e t 2 = ()..) ~ I F a n d a ~ [0, L(2)]. T h e n % is c o n t i n u o u s a t a f o r all

( % ) ~ d ( ~ ) a n d n ~ N i f a n d on l y ~f a ~ U(2).

P r o o f . Suppose that a ~ U(2), (%) 6 ~¢(2) and define s. as in (2). Theorem 1 implies that

a - s. (a) ¢ D-., L . +, (2)l (4)

for all n e N. Since s l (a ) = 0 it follows from (1) that ~1 is continuous at a. Suppose

that n > 1 and the functions ~ , . , . , ~ . z are continuous at a. Thus the map t --, t - s . ( t ) (t ~ [0, L(,:.)]) is cont inuous at a. Therefore, by (4), t - s~(t) < 2. (or

t - s . ( t ) > L . + 1(2)) in a ne ighborhood o f a in [0, L(2)], that is, % ( 0 = %(a) in this

neighborhood. Thus, by induction, we obtain that ~. is continuous at a for all

n ~ N .

Vol, 41, 1 9 9 1 Regularity properties of algorithms and additive functions 115

Conversely, suppose that a ¢ U0.). Then 0 < a < L(~.) and there is a sequence (6.), ~ ~ {0,1} for all n ~ N such that a = y ' ~ = ~ 6 . 2 , and (~.(a)) 4: (6.). Let k =inf{n ~N: ~3. ~ e~(a)} and

= ~e. (x) if x # a a,(x) (6 , i f x = a , x ~ [ 0 , L(2)], n ~ N .

Obviously (c~,)~ ~¢03. Suppose that ~k is continuous at a. Then there exists a positive number 6 such that ak(t) = ~k(a) for all t ~ ]a - 6, a + 6[ ~ [0, L(2)]. This implies that ak(a) = ~k(t) if a < t < a + 6 therefore 6k = ~k(a) = ak(a + ) = ek(a) since ek is continuous on the right at a. This contradiction shows that ~k is not continuous at a. D

3. Monotonic algorithms

If )~ = 0 - ~ ) e l F then ( ~ , ) ~ 1 0 . ) is said to be monotonic if, for any 0 <<. x < y <~ L()~), there exists k e N such that ~,(x) = ~e(y) if i ~ N and i < k and aAx) = 0, ax(y) = 1. A family of monotonic algorithms can be constructed in the following way.

EXAMPLE. Let )~ = (2,,) ~ IF, (#,~) and (t,) be real sequences such that

)~<g, ,~<L~÷,(L) and t~E{O, 1}

for all n ~ N. Define ~.: [0. L(2)] ~ {0, 1} (n e N) by

I r/--I 0 if x < ~ 0q(x)).;+/2, i=1 n--1

i~l 1 if x > ~ a~(x)2~+~. i~l

Theorem i implies that (~,) ~ ~¢(~). We show that (~,) is monotonic. Suppose that 0 ~< x < y ~</-(2) and for the smallest k E N for which ak(x) 4: c~k(y), c~k(x) = 1 and Z~(y) = 0 hold. Define s~ as in (2). Then ak(x) = 1 implies that x >1 s , (x ) + #k and zk(y) = 0 implies that s~(y) + ~k <~ Y. Since sk(x) = sk(y) and x < y, this is a contra- diction. Thus (an) is monotonic. Consequently, 8 and e* are monotonic algorithms.

116 z DA.ROCzY et al. AEQ. MA'IH

In the following theorem we collect some regularity properties of monotonic algorithms.

THEOREM 3. Let 2 = (2.) ~ IF and (a,,) ~ .~¢(2) be monotonic. Then:

(a) a. is continuous at 0 and at L(A.)for alt n ~ N; (b) the one-sided limits a . ( x + ) and a . ( x - ) exist at each x ~]0, L(2)[ for a[t

n ~ N ;

(~) V

a(,,~)(x ) = ~'a,,(x + ) if 0 ~< x < L(2) [a,,(L(2)) if x = L(2)

and

I'a.(O) if x = 0 a(/)(x)

( a n ( X - ) i f O < x ~ < L ( 2 )

(d)

for all n ~ N then (~t},~)), (a~t)) ~ ~¢(2) are monotonic, a(2 ~ is continuous on the

right on [0, L(2)[ and a~ ~ is continuous on the left on ]0, L(2)] f o r all n ~ N; for any n ~ N there exists a finite subset D, o f [0, L(2)] such that ~ is

continuous at the points o f [0, L(2)]\D,,.

Proof (a) Since 0, L(2) e U(2), (a) follows from Theorem 2. (b) We prove only the existence of an (x+) , the existence of a , ( x - ) can be

proved in a similar way. This will be done by induction on n. Let ),j = inf{al(t): t e Ix, L(2)[}. I fy l = 1 then al (t) = 1 for all t e Ix, L()J[. I f yl = 0 then there exists t~ ~ ]x, L(2)[ such that a! (t~) = 0 therefore the monotonicity implies that a~ (t) = 0 if t e Ix, t~ [. Thus ~ (x + ) exists and a! (x + ) = y~. Suppose that n > 1 and a ~ ( x + ) . . . . . a ,_ ~ (x+) exist. Then there is a number tn_ ~e]x, L(2)[ such that a , ( x + ) = a , ( t ) i f l ~ < i ~ < n - I and t ~ ] x , t n _ l [ . L e t y ~ = i n f { c ~ ( t ) : t e ] x , t , i [} , I f ~ , ,=1 then a , , ( t ) = l for all t ~ ] x , t , ~[. I f ~ n = 0 then a , ( t n ) = 0 for some t,, E ]x. t,, ~ ~[ and the monotonicity implies that an(t) = 0 if t E ]x, t,[. Thus a , ( x + ) exists and a,,(x + ) = ~,,.

(c) Notice that the series ~ff=~ c%2, is uniformly convergent on [0, L(2)]. Therefore (a}7~), (~[)) ~ .~(.~) follows from the equality

x = ~ e,,(x)2,, (x e ]0, L()0D , n ' = I

by taking the one-sided limits on both sides. The one-sided continuity of (a~ ~)) and _(/}~ (~,, , is obvious. The monotonicity of (say) (~,~)) can be proved in the following way:

Vol. 41, 1991 Regularity properties of algorithms and additive functions 117

Suppose that 0 < x < y < L(2), k ~ N and o~r)(x) = o~!r)(y) if i < k, i E N. Then there exist f i ~ ] x , y [ and t2~]y,L(2)[ such that c tAx+)=c~, ( t ) if t ~ ] x , fi[ and ~ , (y+)=c t , ( s ) if s ~ ] y , t2[ for all i < k , i ~ N . Thus ~;(t)=ct,(s) if x < t < t ~ < y < s < t 2 and i < k , i ~ N . Because of the monotonicity o f (ct.) ~k(t) ~< c~k(s) if t ~ ]x, fi[ and s ~]y, t2[. Therefore ctk(x + ) ~< ct~(y + ) which proves that (~(.~)) is monotonic.

(d) Let n s N be fixed. Then (b) implies that for any x s [0, L(2)] there is an open interval I(x) = ]a(x), b(x)[ containing x, such that ~.(s) = ~t~(t) if

s, t ~ ]a(x), x[n[0 , L(2)] or s, t e ]x, b(x)[c~[0, L(2)].

Obviously [0, L(2)] = U~t0J~(~)l I(x). Since [0, L(2)] is compact there are numbers x~ . . . . . xm in [0, L(2)] such that [0, L(2)] c ~'~11(x~). Let D~ ={x~ . . . . . Xm}. If x e [0, L(2)] \1). then

x ~ ]a(xi), x~[c~[0. L(2)] or x E ]Xi, b(xt)[('~l[O, L(2)]

fbr some 1 ~< i ~< m. Thus ~. is continuous at x. []

REMARK. It can easily be seen that for the regular and quasiregular algorithms we have ~[) = e~ and ~ ) - e . - * for all n e N.

4. Consequences for ~o-additive functions

If (a.) is a sequence in a Banach space such that E~=~ Ila. ]1 < + ~ then the series ~ _ ~ ~.a. is uniformly convergent for any algorithms (~.). Thus the following results are simple consequences o f Theorem 2 and Theorem 3.

THEOREM 4. I f 2 = (2~) e IF, (~.) e ~¢'(2), F is a Banach space-valued {(ct.)}- additive function (with respect to 2) and a ~ U(2) then F is continuous at a.

REMARK. The example 4.2 in [2] shows that this statement is sharp: if2~ = 1/2", n e N and F(x) = ~ = 1 e.(x)/n ~-, x ~ [0, 1] then F is discontinuous at every point which is not uniquely representable.

THEOREM 5, Let 2 = (2n) ~ IF, (~.) ~ ~ ( 2 ) be monotone and F be a Banach space-valued {(a.) }-additive function. Then

(a) F is continuous at 0 and L(2), furthermore F(x +) and F(x - ) exist at each x E ]o, L(;O[.

It8 Z DAROCZY eI al. AEQ MA'I'H

(b) There is a countable set D c [0, L(2)] such that F is continuous at each

x ~ ]0, L(,~)[\D.

(c) F is continuous on [0, L(2)] i f and only i f F is {(~(~), ~,,--oq, }-additire.

REMARKS 1. The set D in (b) can be chosen as the union o f the sets Dn in

T h e o r e m 3(d),

2. The s ta tement (c) can be considered as a general izat ion o f T h e o r e m 7.1 in

[21.

REFERENCES

[ 1] DAROCZY, Z., JARAI, A. and KATAI, I., Intercallfiillende Folgen and volladditire Funktionen. Acta Sci. Math. 52 (1988), 337-350.

[2] DAROCZY, Z. and KA'rAt, 1., Intert, al filling sequences and additive functions. Acta SoL Math. 52 (1988), 337-347.

[3] DAROCZY, Z,, KATA! I. and SZaBO, T., On completely additive functions related to interval filling sequences. Arch. Math. 54(1990), 173 179.

Department of Mathematics, L. Kossuth University, H-4010 Debreeen, Pf. 12, Hungary.