Analysis Mathematica, 12 (1986), 237--249

Continuous additive functions and difference equations of infinite order

Z. DAROCZY and I. KATAI

(1.1)

1. Let l < q < 2 and L:= = = q - 1

Then ([1], [2], [3])

(1.2)

For xE[0, L] let by induction on n

' if i~ [ - - - ~ - + ~ = x

5.(x) := .=1 i(x) 1 if ,=~ - - - ~ - + 7 > x.

..(x) X=~__ 1 q"

and we call the representation (1.2) the regular expansion of x.

Let a, EC, and ~ la,[<oo. Then we call the function

(1.3) F(x):= ~e.(x)a. (xE[0, L]) ? / = 1

additive, e, (x) denoting the digits 0, 1 from the expansion (1.2). In view of 1 < L the expansion

(1.4) t -- ~ ' ~" (1)

exists. If ~,(1)= 1 for infinitely many values of n then, let

(1.5) 1,:= e,(1) (nEN).

If e,(1)= 1 occurs only finitely many times, the let s > l be the largest index for which e~(1)=l. Then e,(1)=O for n>s . Letin this case be

f~,(1) if n = k s + i (1 < - i < = s - 1 , k=O, 1,...); (1.6) In :=t 0 if n = k s (k = l, 2, ...).

Received June 22, 1985

238 z. Dar6czy and I. K~ttai

In both eases it follows from (1.4) that

(1.7) 1 = ~ ' n'-z- 1

where 1~=1 for infinitely many values of n. We call (1.7) the infinite expansion of the number 1.

In the paper [3] we have proved the following fundamental result:

T h e o r e m 1.1. Let l < q < 2 . The additive function F: [0, L]--*C is continuous in [0, L] i f and only i f

(1.8) a. = ~ . lla~,+~ t = 1

for any nEN, where l s denotes the digits 0, 1 in the infinite expansion of the number o o

1, and F is defined by the sequencea, EC ( Z la.[<~o). In this paper we investigate the following problems:

(A) Let F: [0, L ] ~ C be an additive and continuous function for which [F(x)l<=Cx ~ (0<A<=I) holds, where C > 0 is some constant. Which are the func- tions satisfying the condition just formulated?

(B) Let F: [0, L]--*RcC be an additive and continuous function for which F(x) >=0. Let us determine these functions.

On the basis of Theorem 1.1 these problems can be reformulated as follows:

Let a, EC (.--~x la"l<~~ be a sequence which satisfies the difference equation

(1.8) of infinite order.

(A) Determine those solutions for which la.I -<- C (nEN; 0 < A ~ 1).

(B) Determine those solutions for which a, ~ 0.

We can give the following answers to the problems raised. Problem (A) is, in general, capable of an answer, but further investigations are needed, depending on what we can say about the roots of the power series

G ( z ) : = l - ~ l ~ z a ( Iz l<l) . i = 1

Continuous additive functions and difference equations 239

We obtain as an important result that if

la, I < Cn (naN), - - q .

then the additive and continuous function defined by the sequence an~C is linear (Theorem 3.3). With the help of this we prove the following fundamental result (Theorem 4.6):

I f the additive and continuous function F is differentiable at a point, then F is linear.

This implies that any nonlinear additive and continuous function is nowhere differentiable. We have already constructed such an example in [3], but in this general form the theorem is surprisingly new. The solution of problem (B) is already in a certain sense a consequence, since we show that a nonnegative, additive and con- tinuous function is monotone increasing, so by the theorem of Lebesgue it is dif- ferentiable almost everywhere, and consequently it must be linear. This is a certain generalization of the resuk in [4], where a similar statement is made for completely additive functions.

2. On difference equations of infinite order

For the sequence {x,} (n~N, x,~C) we define the translation operator Exn:=X,+ 1 (nEN). Let E~ 1 be the identical mapping (E~ and let us denote by En:=E(E "-~) (nCN) the powers of the translation operator E. Let

(2.1) a(z) := ~ g~z k k = 0

be a power series convergent on the disc Izl<l, such that gk~0 for infinitely many values of k. Let moreover 0 < a < l .

D e f i n i t i o n 2.1. a,~D(G, a) if

(2.2) G(e)a,, = 0 (nEN) and (2.3) [a,I <= Ca" (n~N)

for some constant value C>0. By the definition, (2.2) states the validity of the difference equation

(2.4) 0 = G(E)an = ( ~ gkEk)a, = gkaa+k k,O= k=O

240 Z. Dar6czy and I. K~tai

(linear, homogeneous and with constant coefficients), which we investigate under the side condition (2.3).

R emark . If [~l<l and G(~)=0 then a,:=~" clearly satisfies (2.2) and in ease I~[ ~ r < 1 also (2.3). The linear combinations of these rOOtS are also solutions.

T h e o r e m 2.2. I f {a,}6D(G, o'), then a, can be written in the form

(2.5) a, = p l ( n ) ~ + . . . +ph(n)~g,

where 41, ~2 . . . . , r are all the different roots o f the function G(z) on the closed disc [z]-<_a. The functions Pl, P2 . . . . , Ph are polynomials with complex coefficients, and their degrees satisfy . . . .

deg (p~) ~ m j - 1 ( j = 1, 2, ..., h),

where m a denotes the multiplicity o f the root ~j.

Proof . Since 0 -<a< l , there exists o ' < r < l , such that G(z) has no root in the ring a<lz]~_r. On the other hand, G(z) can have only finitely many roots

let these be Ca . . . . . ~h, with the respective multiplicities on the dosed disc ]zips; m 1 , . . . , m h .

Let

(2.6)

and

h

A (z) := f f ( z - ~j)'J j = l

(2.7) H(z) := A (z--S"

Then I I (z) is regular on the disc Iz[<l, and H ( z ) # O for ]z[<=r. This yields

tt(z) = ~ h~z ~ k = 0

(2.8)

for [z[<l, and o o

1 2 tl zt (2.9) H(----~ = 1=0

for ]zi <=r. The functions (2.8) and (2.9) are regular in their domain of convergence. By multiplying the two power series (2.8) and (2.9) we immediately get

(2.10)

If {a,}ED(G,a), then let (2.11)

1 for m = O

k~--m for m > 0 . l+ tthk = 0

b , := A ( E ) a , (nEN),

Continuous additive functions and difference equations 241

Since A(z) is a polynomial, from (2.3) we get

(2.12) Ib.I <= Ca" (nEN)

for some constant C>0 . At the same time

(2.13) 0 = G(E)a, = H(E)A(E)a, = H(e)b. (nCN).

Seeing that for arbitrary NEN we have by (2.12)

the series

Z [tthkbn+,+kl <= Can ( ~ Jhl at)( ~ Ihk] a k) <o~ l = O k = O 1=0 k~O

Z ~ hhkbn+t+k l=O k = 0

can be arbitrarily rearranged. From this we get by (2.13) and (2.10)

0 = t=0~h{H(E)bn+z}= ~oh{k~ohkb~+,+k} =

l~=Ok~otlhkbN+l+k ~ 30bN+m{l+k~=mtlhk} = bN

i.e. bu=0 for any NCN. Thus from (2.11)

(2.14) A(E)a, = 0 (n~N)

and this implies that {a,} satisfies a difference equation (linear, homogeneous and with constant coefficients), having A(z) as its characteristic polynomial, Therefore (2.5) holds and the properties enunciated in the theorem are valid. []

3. Continuous additive functions

Let l < q < 2 and let F: [0, L]-~C be an additive continuous function. Then by

Theorem 1.1 the difference equation of infinite order

(3.1) an = ~ lia,+i (nCN) i = 1

holds, where l~ denotes the digits 0, 1 in the infinite representation of 1. Clearly /i = 1 and l, = 1 for infinitely many Values of n.

Now (3,1) says that the power series

(3.2) G(z) := 1-11z'12z~-... = 1- ~ l~z t i=l

242 Z. Dax6czy and I. Khtai

(which is dearly convergent on the disc [z[ < 1) satisfies, as generating function, the difference equation of infinite order

(3.3) G(E)a, = 0 (nEN).

If for the continuous additive function F the inequality IF(x)l <=cx ~ (0<A <- 1) is also satisfied, then

= la.I <-c -~- (nEN)

1 i.e. (2.3) holds with tr.-'---qa (0< t r< l ) . Thus a, ED(G, a), whence by Theorem 2.2

(3.4) a, = p l (n )~+. . . +ph(n)~

where ~1 . . . . . ~h denote all the different roots of the generating function G on 1

the closed disc Izl<=~=-b-z, and the degree of the polynomials pl is not greater . L

than mi -1 , where m~ is the multiplicity of the root ~i (/=1, 2, ..., h), From this we get

o o

(3.5) F ( x ) = Z s , ( x ) a , / I = 1

where a, is of the form 0.4). With this we have proved the following result.

T h e o r e m 3.1. Let l < q < 2 and F: [0, L ] ~ C an additive and continuous

function defined by the sequence anEC ( Z lanl<=). I f there exists a constant O<A <= 1 such that ,=1

(3.6) IF(x)[ <= Cx ~ (xE[0, L])

where C > 0 is some constant, then F has the form (3.5), where a, is given by the for- mula (3.4). In (3.4) {1, {= . . . . . {h denote all the different roots o f the generatingfunc-

1 tion (3.2) on the closed disc [z[-<_--27, moreover deg(pi)=<m~-I where m, is the

q multiplicity o f the root {i ( i= 1, 2 . . . . , h).

As a consequence we immediately get the

T h e o r e m 3.2. Let l < q < 2 and F: [0, L ] ~ C an additiveandcontinuousfunc- tion. I f

(3.7)

for some constant

(3.8) for any xE [0, L].

C > 0 ,

IF(x)l-< Cx (xC[0, L])

then there exists a complex number aE C such that

F(x) =

Continuous additive functions and difference equations 243

Proof . On the closed disc Izl~_l/q the generating function G has only one simple root, namely ~1:= 1/q. Taking into account (3.4) we get from this that a,=pl(n)~=e/q n, whence (3.8) follows. []

The following result will play a fundamental role later on.

T h e o r e m 3.3. Let l < q < 2 and F: [0, L ] ~ C an additive and continuous

function defined by the sequence a,E C ( ~ la.I < ~o). I f n = l

Cn (3.9) la.I <-- (C > O, nEN) q"

then there exists c~EC such that F(x)=~x (xE[0, L]).

Proof . For 0 < e < 1-1 /q arbitrary there exists a constant K(e)>0 such that

n - - < K(8 +e (nEN). qn

Since the generating function G has on the disc Izl <= 1/q a single simple root ~! = l/q, there exists 0 < 8 < l - 1 / q such that on the disc Izl<=l/q+~=:a<l the function G has no other root than I1= 1/q. Therefore by

la.l<= q. <CK(e +~ =C'a"

we have a, ED(G, a), and in view of Theorem 2.2 this implies a,=pl(n)~7=~/q". From this F(x)=~x (xE[0, L]) immediately follows. E]

4. Additive functions differentiable at a point

Let l < q < 2 and F: [0, L ] ~ C an additive function defined by the sequence

a.EC ( 2 la.l<~176 �9 In what follows we suppose that there exists an xE[0, L] at ?1=1

which F is differentiable, i.e. F'(x) exists. From the paper [3] we know the following.

Def in i t ion 4.1. We call the number xE[0, L] finite if there exists NEN such that in the representation (1.2) e ,(x)=0 for n>N. If x is finite and ~m(x)=l, moreover ~, (x) = 0 for n >m, then we say that x has length m, and we write h (x) = m. Also, h(0)=0.

Let NEN fixed and

(4.1) VN := {xlx~[0, L], h(x) ~_ N}.

Clearly, Vs is a finite set.

244 Z. Dar6czy and I. K/ttai

D e f i n i t i 0 n 4.2. Let x E V N and x>0. We caU the number

x_ (N) := max {tlt~VN, t < x}

the left hand neighbour of x in VN. Clearly x_ (N)CVN and x_ (N)< x.

In [3] we proved the following result.

Lemma 4.2. Let xC[0, L] be finite and m=h(x)>=l (i.e. x>O). Let N>=m. Then xEVN and

m.=3 en(x ) 0 It 12 + IN-m (4.2) x_ (N) = ,2=~ - 7 ~ + ~ + ~ + ~ - - 4 g "" "~ qN

is the regular expansion of the lefthandneighbour of x in VN, where li (i= 1, 2 .. . . . N - m ) denotes the digits 0, 1 in the infinite expansion of the number 1.

Lemma 4.3. Let F: [0, L]-~C be an additive function, differentiable at the finite point xC[0, L]. Then (4.3) lira an q" = F'(x).

Proof . Let h(x)=m and

~. (x) x U = 1).

Then there exists no>m . such that for N>=no we=have

1 ~ . ( x ) + 1

1 7 where on the right hand side there stands the regular expansion of x+ 1/q N. From this

F'(x) = lim F ( x + ~ ) - F ( x ) = l ima N qU

qN

follows, i.e. (4.3) holds. []

'Lemma 4.4. Let F: [0, L ] ~ C be an additive function, differentiable at the non-finitepoint xE[0, L]. Let moreover N1 := {nlnCN, ~,(x)= 1}. Then

(4.4) lim anq" = F'(x). n E N x n ~ o o

Proof . Here N1 is an infinite set and 0 x=L. By the differentiability of the function F at the point x there exists a function

Ex:[O,L] ~ C such that limEx(y) = 0 y--~:r

Continuous additive functions and difference equations 245

and (4.5) F(x) - F(y ) = F ' (x) (x - y) + Ex (y) (x - y)

for any y~[0, L]. Let

x = .~1 T and SN (X) = .=1 qn

Then the regular expansion of SN (x) is itself. Let now be NEN1 arbitrary, i.e. eN(x)=l. Then from (4,5)

F(x) - F ( S N (x)) = F ' (x) (x " Szr (x)):~- ex (SN (x)) (x -- S N (x)) and

r ( x ) - F(Su_I (x)) = r ' (x) (x - SN - i (x)) + E~, (SN-1 (x)) (x -- SN-1 (x))

whence by NEN1 (,4.6) au = F(SN(x ) ) -F (SN_ i ( x ) ) =

= F" (x)---~+ E x ( S N - a ( x ) ) ( x - S N - I ( x ) ) ' E x ( S N ( x ) ) ( x , S z ~ ( x ) )

follows. In view of

x - s~ (x) = ~ e~(x) 1 ~- B~(x) 1 ,=u+l q~ =T,=N~++lqTZ-Y=~ "~/~+l(x)

we get from (4.6)

(4.7) aNq u = F'(x)+qE~(SN_I(x))qu(x)--Ex(SN(x))~I~+I(x) .

Since (4.7) is satisfied for any NEN 1 and E~(Su(x))~0 fo r N-+~ (NCN1), we have auqU-~F'(x) in case NCN~ and N~oo. []

L e m m a 4.5. Let F: [0, L]~C. be an additive and continuous function, dif- ferentiable at the non-finite point xC[0, L]. Let moreover N0:={n[nEN , ~,(x)--=0}. Then

Cn (4.8) Ia,l <-- - ~ (nENo)

with a constant C>0.

Proof . If xE(0, L), then N o is an infinite set. Let

= ~ ~.(X ) g ~.(x) + =~+l~.(x) x ,~=lq" '--':,~=1--~- 2 q,, =SN(x)+~IN+I(X).

If SN(x)>0 and NEN0, then eN(x)=0 and there exists l~=/c<N such that

u-k-1 e ix) 1 (4.9) y!0) := SN(x) = ~" ,t , . S ,-@1 q, . + ~ = .-k(x).

246 Z. Dar&zy and I. K~ttai

Then L

(4.10) 0 < x - y (~ = r/N+x(X ) < ~ - .

Let y(~):=y~)(N), i.e. the left hand neighbour o f y (~ in V N. Then by Lemma 4.2

N--k--1 61 (X) 11 lk" (4.11) y t l )= t=l ~ q'i -~ oN-k+1 I-. . .+ qN ,

where la, 1~ . . . . . l k denote the first k digits in the infinite expansion of 1. Then

1 11 lk lk+l Ik+~ L 0 -<- y(O)_y(t)= ... = _ . ~ + ~ + , , . < qN-k qN--k+l q. . , - q. . , -

By Lemma 4.2 yO) has the regular representation (4.11), i.e. eN(y(1))=lk. I f 8n(y(1))= 1, then the process has ended; otherwise let y(~):=y~ i.e. the left hand neighbour of yO) in Vn. In the same manner as previously

L 0 ~_ y (1 )_ y(~) < q--

and if ~N(y(~))=l then the process has ended. After finitely many steps (e.g. in the r-th step, where r<-k~N) , we get es(y(~ with

(4.13)

Then

L 0 <= y ( O _ y f f + l ) < ~ " (i = O, 1 . . . . , r - 1 ) .

N e, (y( , ) ) y(O : Z

n~l qn

is a regular representation and SN(y(0)= 1. Since

N--~ ~. (y( , ) ) z~O := y(O___~ = n=l "~ q"

is also a regular representation, differentiability at the point x implies

F(x) - F (y (0) = F ' (x) (x -y( ' )) + Ex (y(O) (x _y(O) and

f ( x ) - F(z (')) = F" (x) (X - z (O) + Ex (z tO) (x - zr From tiffs

14) a N - ~ F( y tO ) - F(z cO) = F" (x) ~ + Ex (z tO) (x - z t')) - E~ (ytO) (x - ytO). (4. q-.

Continuous additive functions and difference equations 247

Continuity of F implies the boundedness of Ex: [0, L]-+C (i.e. IE,(y)I<K for yE[0, L]), and so, taking into account (4.13), we infer from (4.14) that

laNI ~-IF'(x)l +K (N )L ~- K(N+I)LqN <-- qN

where C > 0 and NENo. []

T h e o r e m 4.6. Let F: [0, L ]~C be an additive and continuous function, dif- ferentiable at some point xE[0, L]. Then there exists aEC such that F(x)=ax (xE[0, L]).

P roof . Lemmas 4.3, 4.4 and 4.5 together imply that

Cn [ I < a. = qn"

for any n. An application of Theorem 3.3 now immediately yields our statement. []

Theorem 4.6 shows that if an additive and continuous function is nonlinear, then it is nowhere differentiable. We have already constructed such a concrete example in our paper [3].

5. Nonnegative continuous and additive functions

L e m m a 5.1. Let F: [0, L ] ~ R c C be a continuous additive function, such that F(x)>0 for x>0. Then F is monotone increasing in [0, L].

P roof . Let O<=a<fl<=L be arbitrary. Then, by the continuity of F

min F(t) = F(~)

~ t h ~ f l . We now show that a=~, i.e. F is monotone increasing. Suppose the contrary: then 0<~a<~<_-fl. We distinguish two cases. (i) Let

be a finite number. Then, by N=h(~)=>l we have

N -1 e . (~ ) . 1 r = Z

n = l

Now there exists kEN, such that

�9 ~ x , . ( ~ ) + 11 + (5.1) c~ < Ck .--- ~--1 - - ~ ~ "'" +qN+k

since ~k-*~ (k~ oo), where li denotes the digits of the infinite expansion of 1. By

6 Analysis Mathemafiea

248 Z~ Dar6czy and I. Khtai

Lemma 4.2, (5.l) is a regular expansion of ~k, hencebyF[~)=a,>O and by (1,8)

N ' I N - 1 o o

F(~k) = 2 en(~)a~+lla-N+l+...+lka-N+k< ~e~(~)a~+~l~a-N+~= n = l n = l i = 1

N - - 1

= ~ e. (~) a . + an = F(r / ' /=1

which is clearly a contradiction. (ii) If r is a non-finite number, then let

where 8,(~)= 1 for infinitely many values o f n. Then for the sequence

one has ~-N~ (N~oo), i.e. there exists No such that c~<~n0<~. On the other hand, the regular representation of ~-N0 is known, so by a ,>0 we get

-N O oo

F(~No) = Z e,(~)a, < Z e , ( ~ ) a , = F(~), n ~ l n = l

a contradiction. []

T h e o r e m 5.2. Let F: [0. L] -~RcC be a continuous additive function, satis- fying F(x)~-O for any xC[0, L]. Then there exists a constant aCR (a>=O) such that F(x)=ax (x~[0, L]).

Proof . Let Fl(X):=x+F(x) (xC[0, L]). Clearly, F1 is an additive and con- tinuous function, moreover Fl(X)>0 for x>0. Thus by Lemma 5.1 F1 is monotone increasing, and consequently, by the theorem of Lebesgue, differentiable almost everywhere. By Theorem 4.6 this implies that Fl(x)=~x, i.e. F(x)=ax, where a_-->O. []

The theorem just proved generalizes in a certain sense the result of the paper [4], since it is known from [3] that there exists an additive and continuous function which is not completely additive.

Continuous additive functions and difference equations 249

References

[1] Z. DAROCZY, A. J/;RAI and I. K.~TAI, IntervaUfiillende Folgen und volladditive Funktionen, Acta ScL Math. (Szeged) (to apear).

[2] Z. DAR6CZY, A. J~RAI and I. KA, TAI, Intervalfilling sequences, Ann. Univ. ScL Budapest. Sect. Comput. (to appear),

[3] Z. DAR6CZY and I. KA, TAI, Additive functions, Anal. Math. 12 (1980, 85--96. [4] GY. MAKSA, On completely additive functions, Acta Math. Hungar., 48 (1986).

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DEPARTMENT OF MATHEMATICS L- KOSSUTH UNIVERSITY DEBRECEN 4010, H U N G A R Y

DEPARTMENT OF MATHEMATICS L. EOTVOS UNIVERSITY BUDAPEST, MI2ZEUM KOR13T 6--8 1088, H U N G A R Y

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