Construction of Continuous Functions with Prescribed ...

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Construction of Continuous Functions with Prescribed Local Regularity ´ VY VE´HEL and Yves MEYER Khalid DAOUDI, Jacques LE

N˚ 2763 December 95

PROGRAMME 5

apport de recherche

ISSN 0249-6399

Unite´ de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Te´le´phone : (33 1) 39 63 55 11 – Te´le´copie : (33 1) 39 63 53 30

Construction of Continuous Functions with Prescribed Local Regularity

Khalid DAOUDI, Jacques LE VY VE HEL and Yves MEYER Programme 5 | Traitement du signal, automatique et productique Projet Fractales Rapport de recherche n2763 | December 95 | 41 pages

Abstract: In this work we investigate both from a theoretical and a practical point of view the following problem: Let s be a function from [0 ; 1] to [0 ; 1]. Under which conditions does there exist a continuous function f from [0 ; 1] to IR such that the regularity of f at x, measured in terms of Holder exponent, is exactly s(x), for all x 2 [0 ; 1]? We obtain a necessary and sucient condition on s and give three constructions of the associated function f . We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally we present a result on the \size" of the set of functions with prescribed local regularity. Key-words: Holder exponent, IFS, Weierstrass function, Schauder basis, multifractals (Resume : tsvp) Khalid Daoudi : e-mail: [email protected] Jacques Levy Vehel : e-mail: [email protected] Yves Meyer : e-mail: [email protected] Unite´ de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Te´le´phone : (33 1) 39 63 55 11 – Te´le´copie : (33 1) 39 63 53 30

Construction de Fonctions Continues  a Regularite Locale Prescrite

Resume : Dans ce papier, nous nous interessons, aussi bien du point de vue theorique que

applique, au probleme suivant : etant donnee une fonction s de [0 ; 1] dans [0 ; 1], sous quelles conditions existe-t-il une fonction continue f de [0 ; 1] dans IR telle que la regularite de f en x, mesur ee en termes d'exposant de Holder, soit exactement s(x), pour tout x 2 [0 ; 1]? Nous obtenons une condition necessaire et susante sur s et nous donnons trois constructions de la fonction associee f . Nous examinons aussi quelques extensions concernant, par exemple, la dimension de bo^te ou de Tricot, ou le spectre mutifractal. Finalement, nous presentons un resultat sur la \taille" de l'ensemble des fonctions ayant une regularite locale prescrite. Mots-cle : exposant de Holder, IFS, fonction de Weierstrass, bases de Schauder, multifractals

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

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1. Introduction Since Riemann [1], a number of authors have been interested in constructing nowhere dierentiable continuous functions. Some use geometrical constructions, of which the best known examples are probably Von Koch's [2], Peano's and Hilbert's [3] curves, while others are based on analytical tools. The very well known example is here the Weierstrass function, which was shown by Weierstrass to be continuous and nowhere dierentiable [4]. This result was later greatly enhanced by Hardy [5] who showed that

f (x) =

1 X

n=0

bncos(an x)

has nowhere a nite derivative, provided that 0 < b < 1; a > 1; ab  1: Hardy also analyzed the Holder conditions satis ed by f (x). If ab > 1, let  < 1 be =b) de ned by  = log(1 log a . Then, for h ! 0,

  jf (x + h) ? f (x)j = O jhj for every x   jf (x + h) ? f (x)j = o jhj for no x:

but

Another example of nowhere dierentiable function, which ts well the main ideas of this paper, is the Takagi function [6] de ned by :

T (x) =  (x)

1 X j =0

2?j  (2j x)

where is the periodic function of period 1 de ned on [0 ; 1] by  (x) = 2x if 0  x  1=2 and (x) = 2 ? 2x if 1=2  x  1. Indeed, we consider in the sequal three dierent constructions of nowhere dierentiable functions : one is based on a generalization of the Weierstrass function, another one on an expansion in the Schauder basis, and the last one on a generalization of IFS theory. The construction of the Takagi function bears some analogy with that of the Weierstrass function. On the other hand, the restriction of T to [0 ; 1] can be written, in its expansion in the Schauder basis, as : X X ?j j T (x) = 2 (2 x ? k) j 0 0k 1; 0 <  < 1, each n is an arbitrary number, and  is a function which has period one. They showed that there exists a constant C > 0 such that, if is large enough, then the Hausdor dimension of the graph of W is bounded from below by 2 ?  ? logC . Several other techniques are now employed for constructing continuous nowhere dierentiable functions. One powerful scheme is to use wavelet decompositions. For instance, Jaard [10] has given a construction of a function with prescribed multifractal spectrum (; f ()). Choosing in such a construction f () such that f ()j ?1 [ 1 = ?1 leads to a nowhere dierentiable continuous function. Another method that has been investigated a lot these past years is based on Iterated Function System (IFS). Although the study of iteration of matrices dates back to Doeblin and Fortet [11] and Dubbins and Freedman [12], it is Hutchinson [13] who really laid the foundations of the IFS theory. Subsequently, several authors have explored this path (see for instance [14, 15, 16, 17] and many others). Barnsley [14] showed that, under some conditions, it is possible to construct an IFS whose attractor is the graph of a continuous nowhere dierentiable function. More precise results are now known, concerning the almost sure Holder exponent of such functions [17] or multifractal spectrum [18, 19]. RRtheir n2763 ]

;0] [1 ;+

[

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 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

We shall hereafter call f the Holder function of f (which associates, to each point x, the Holder exponent of the function f at x). The main objective of the present work is the following: Let s be a function from [0 ; 1] to [0 ; 1]. Under what conditions on s does there exist a continuous function f from [0 ; 1] to IR such that f (x) = s(x) for all x in [0 ; 1].

The motivation for this investigation stems partly from applications in signal processing. Indeed, in some cases, it is desirable to model highly irregular signals while precisely controlling the irregularity at each point. This happens, for instance, when the signi cant information lies in the singularities of the signal more than in its intensity. In such cases, we want to tune the value of f (x) everywhere and not merely almost everywhere. An example in speech modeling is presented in [19, 20]. Our main result is the following:

Theorem

Let s be a function from [0 ; 1] to [0 ; 1]. Then, the following conditions are equivalent : i) s is the Holder function of a continuous function f from [0 ; 1] to IR. ii) There exists a sequence (sn)n1 of continuous functions such that : s(x) = lim inf s (x) ; 8 x 2 [0 ; 1]: n!+1 n

The proof of i) ) ii) is easy and is given in section 3. The proof of ii) ) i) requires more work. For practical purposes, we are here interested in constructive proofs, i.e, we want to derive explicit methods to construct the function f . We present below three such proofs which highlight dierent aspects of the problem. We also investigate related problems, as for instance the evaluation of the local box dimension of f at each point or the computation of the multifractal spectrum of f . Finally, for practical applications, we want to construct functions f with prescribed Holder function that satis es additional constraints, as for instance interpolating a nite number of points (xi; yi ) 2 [0 ; 1]  IR, i = 1; 2; : : : ; N . This naturally leads to a characterization of the set of functions with prescribed Holder function. The remainder of this paper is organized as follows: in section 2, we recallINRIA some

Construction of Continuous Functions with Prescribed Local Regularity

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basic de nitions about the local regularity of functions, the Hausdor, Tricot and box dimension. We also prove a new relation between the local box dimension and the Holder exponent. In section 4 we construct functions with prescribed local regularity s(x) at each point using the Schauder basis. In section 5, we give another solution based on a generalized Weierstrass function. In section 6, we use IFS to give a solution which constructively allows to interpolate a given nite set of equispaced points. In section 7, we propose some desirable extensions that would allow to measure more nely the local structure of graphs of continuous functions. Section 8 shows some implementation results. 2. Recalls and a result relating the local box dimension and the Ho lder exponent

In this section we recall some basic de nitions useful for the sequel. The de nitions are not given in full generality, but only in the form adapted to our problem.

2.1. De nition of the Hausdor dimension. Let E be a non empty set of IR2 .

De ne :

jE j := sup fjx ? yj ; x; y 2 E g x;y

to be theSdiameter of E . If E  Ei with 0 < jEi j   for each i, then fEigi2N is called a (countable) i2N -cover of E . For  > 0 and r  0, de ne +1 X Hr (E ) := inf f jEijr = fEigi2N -cover of E g

i=1

Hr (E ) is a non increasing function of , and we note Hr (E ) := lim Hr (E ) = sup Hr (E ) !0  >0

the Hausdor r-dimensional outer measure of E . The Hausdor dimension of E is the unique value dim H(E ) such that [21] : RR n2763

Hr (E ) =

 +1 if r < dim (E ) H 0

if r > dim H (E )

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

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2.2. De nition of 2the box dimension. For any  > 0, we consider the set of -mesh squares in IR of the 2form [i; (i + 1)]  [j; (j + 1)] with i; j integers. For

any bounded subset F of IR , we denote by N (F ) the number of -mesh squares which intersect F . The box dimension of F is then de ned by [22]:  log N (F )  ; dimB (F ) = lim !0 ? log  whenever this limit exists. When the limit exists, its value is unaected if we change the de nition of N (F ) and take any of the following: (1) the smallest number of squares of size  that cover F ; (2) the smallest number of closed balls of diameter  that cover F ; (3) the smallest number of sets of diameter  that cover F ; (4) the largest number of disjoint balls of diameter  with centers in F . 2.3. De nition of the Tricot (packing) dimension. Let F be a non empty set of IR2, where n  1 is an integer, and :

Pr (F ) = sup

(X

i2IN

jBi jr

)

;

where fBigi2N is a collection of disjoint balls of radii at most  whose centers belong to F: Consider : P0r (F ) = lim P r (F ) !0  this limit exists since Pr (F ) decreases with . De ne now the r-dimensional Tricot measure [23, 22] P r by :

P r (F ) = inf

(X 1 i=1

P0r (Fi) : F 

1 ) [

i=1

Fi ;

then, the Tricot (or packing) dimension dimP is de ned as follows: dimP F = sup fr : P r (F ) = +1g = inf fr : P r (F ) = 0g :

2.4. De nition of the Holder spaces and the Holder exponent. Let I be an interval in IR, f a continuous function from I to IR, and 2 IR+ n IN . De nition 1. f is said to belong to the global Holder space C (I ) i there exists a positive constant c, such that for every x0 2 I , there exists a polynomial Px of degree less than or equal to the integer part of , such that : jf (x) ? Px (x ? x0)j  cjx ? x0j 8x 2 I: 0

0

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De nition 2. Let t0 be in I . Then f is said to belong to the pointwise Holder space

C (t0 ) i there exists a polynomial P of degree less than or equal to the integer part of , and a positive constant c such that, for every t in the neighborhood of t0 , we have :

jf (t) ? P (t ? t0)j  cjt ? t0 j :

Recall that if 2 IN , the space C must be replaced by the Zygmund -class [24].

De nition 3. A function f is said to have Holder exponent at point t0 iff : i) for every real < :

lim jf (t0 + jhh)j ? P (h)j = 0

h!0

ii) if < +1, for every real > :

lim sup jf (t0 + jhh)j ? P (h)j = +1 h!0

where P is a polynomial whose degree is less than or equal to the integer part of . When < +1, this is equivalent to :

f2

\

>0

C ?(t0) but f 2=

[

>0

C +(t0 ):

It is also equivalent to :

= supf > 0 : f 2 C  (t0)g: Notice that f 2 C (I ) does not imply that = inf  (t). As an example, consider t2I f the continuous function f de ned on IR by :

(

1 jtj ) f (t) = jtj sin( 0

if t 2 IR if t = 0

then, f 2 C (IR), but f is C 1 at each point, except at 0 where f (0) = 1. 1 2

RR n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

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2.5. A relation between the local box dimension and the Holder exponent. In this section we propose a new result that links the local box dimension of the graph of a continuous function and the Holder function of this function. Let f be a continuous function from [0 ; 1] to IR. We suppose that s(x) = f (x) 2 [0 ; 1] for all x 2 [0 ; 1]. Let x 2]0 ; 1[,  > 0 such that ]x ?  ; x + [ [0 ; 1] and  2 ]0 ; [. We cover the plane by a -mesh, i.e a grid of squares of the form [i ; (i + 1)] 

[j ; (j + 1)], with i; j integers. Let N be the number of squares that intersect graph f j x? x  . We de ne, respectively, the upper and lower local box dimension [22] of the graph of f at the point x by : log N dimxB graph f = lim lim sup ? !0 log  ]

; + [

!0

and

log N : lim inf ? dimxB graph f = lim !0 !0 log  x When these numbers coincide, we denote by dimB graph f the local box dimension of f at x. For t 2]0 ; 1[ such that ]t ?  ; t + [ [0; 1], de ne :

n

c(t; ) = inf c 2 IR+ : 8u 2 B (t; ) ; jf (t) ? f (u)j  cjt ? ujs(t) and

n

c(t; ) = sup c 2 IR+ : 9u 2 B (t; ) : jf (t) ? f (u)j  cs(t)

o

o

Proposition 1. Let x be a real in ]0 ; 1[. De ne the following conditions : there exists 0 > 0 such that : C (x; ) = sup c(t; ) < +1 for every  < 0 (c1 ) t2B(x;)

and there exists 0 > 0 such that : C (x; ) = t2Binf(x;) c(t; ) 6= 0 for every  < 0 Then, if (c1 ) holds, we have the following inequality :



dimxB graph f  2 ? min lim inf s(t); s(x) t!x

and, if (c2 ) holds, we have :



(c2 )





2 ? max lim sup s(t); s(x)  dimxB graph f t!x

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Proof : Let  be a real such that 0 <  < 0 .We denote : sx = inf fs(t) ; t 2]x ?  ; x + [g: sx = supfs(t) ; t 2]x ?  ; x + [g: Rf [t1 ; t2] = sup jf (u) ? f (v)j: t1 0 = 8x 2 [0 ; 1]; 9a1(x) > 0 = V  a1(x)s (p1) 9s2 > 0 = 8x 2 [0 ; 1]; 9a2(x) > 0 = V  a1(x)s (p2): Condition (p1) implies that : log V (x)  s lim inf 1 !0 RR n2763 log  1

2

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 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

which means that f (x)  s1, for every x 2 [0 ; 1]. In the same way, condition (p2 ) implies that : V (x)  s lim inf loglog 2 !0  which means that f (x)  s2, for every x 2 [0 ; 1]. Then we have the following result, due to Claude Tricot :

Proposition 2. If condition (p1) holds, then : dimP graph f  max(1; 2 ? s1 ):

This result remains true when condition (p1) holds for every x 2 [0 ; 1] except on a set E such that dimP (E ) = 0. If condition (p2) holds, then : dimP graph f  2 ? s2 : This result remains true when condition (p2) holds for every x 2 [0 ; 1] except on a set of Lebesgue measure zero.

3. Characterization of the set of Holder functions of continuous function

Theorem 1. Let f be a nowhere dierentiable continuous function from [0 ; 1] to IR. Then, there exists a sequence fsngn2IN of continuous functions, such that : f (x) = lim inf s (x) 8 x 2 [0 ; 1]: n!1 n

Conversely, let s be a function from [0 ; 1] to [0 ; 1] such that s(x) = lim inf s (x), n!1 n where the sn's are continuous functions. Then, there exists a continuous function f from [0 ; 1] to IR such that : f (x) = s(x):

The rst part of the theorem is easy to prove. Indeed, take ( ?n ) ) log( j f ( x + h ) ? f ( x ) j + 2 : sn(x) = 2?n jinf hj 0, there exists a constant C such that : jc(j; k)j  C ?2?j + jk2?j + j2?j ? x0 j
s: Then : f 2 C s?(x0) 8 > 0: Proof : Let x1 be a real in the neighborhood of x0 , and j0 be the integer such that 2?j  jx1 ? x0 j < 2?(j ?1): We de ne the integer j1 such that w(2?j ) = 2?sj . Then jf (x1 ) ? f (x0 )j  W + X + Y + Z where X X W= jc(j; k) (j;k (x1) ? j;k (x0))j 0

0

0

1

0j j0 0kj0 0k j0. Using proposition 3, we conclude that f (x0 )  s(x0 ). Let us now show that f (x0 )  s(x0 ) ? , for every  > 0. Remark that there exists j such that : s(x0) ?  < Qj (k2?j ) INRIA 0

Construction of Continuous Functions with Prescribed Local Regularity

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for every j  j , and k such that x0 2 [k0 2?j ; (k + 1)2?j [. This implies that : c(j; k)  2?j(s(x )?) : 0

j

Furthermore, since c(j; k)  2? j , it is easy to see that condition (2) holds. Hence, we conclude using proposition 4 that f (x0 )  s(x0 ) ? : log

5. Use of Weierstrass type functions In this section, we show that a simple generalization of the Weierstrass function allows to control the regularity at each point. For a related result, see [24; p:282]. We rst recall some properties of the Weierstrass function, which is de ned by :

W (t) =

+1 X

k=1

?ks sin(k t)

where  > 1 and s 2]0 ; 1[. It is well known [26] that W (t) = s for all t and that dimB graph W = 2 ? s. However the value of dimH graph W is not yet known. Of course : dimH graph W  dimB graph W , and using mass distribution methods depending on estimates for the Lebesgue measure of the set ft = (t; W (t)) 2 Dg where D is a disc, it can be shown [9] that there exists a constant c > 0 such that dimH graph W  s ? c=log. As mentioned in the introduction, several authors have considered generalizations of the Weierstrass function, by replacing the sinus with other type of functions. Here we consider another type of generalization.

Proposition 6. Let s(t) be a function from [0 ; 1] to [a ; b] ]0 ; 1[, which is the lower limit of a sequence of continuous functions. Let a0 and b0 be two reals such that 0 < a0 < a < b < b0 < 1 and consider the sequence IL = (lp)p1 de ned by :

( l =1 1 h 1?a0 i lp+1 =

l +1

1?b0 p

(4)

where [.] denotes the integer part. Then :  there exists a sequence fQngn1 of polynomials such that :

8 s(t) = lim inf Q (t) 8t 2 [0 ; 1] < n! 1 n : jjQ0n jj1 n2ILn 8 n  1 +

where Q0n is the derivative of Qn. RR n2763

(5)

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

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 de ne :

f (t) =

X k2IL

?kQk (t) sin(k t):

Then, provided that  is an even integer large enough, we have : f (t) = s(t); 8 t 2 [0 ; 1]: Proof : The proof of the rst item is similar to that of Lemma 2 ; the only dierence is that we now de ne the sequence qk by : 0 qk  max(Mk ; 11 ?? ab0 qk?1 + 1); for k > 1: Now, we give the proof of the second item. Let t be xed and let  be a positive real such that s(t) +  < b0 and s(t) ?  > a0 . We begin by proving that f 2 C s(t)?(t): There exists an integer k0 such that Qk (t) > s(t) ? , for every k > k0 . Let h be a real such that 0 < jhj < ?k . Then we have : 0

X   ? kQ ( t + h ) k ? kQ ( t ) k jf (t + h) ? f (t)j =  k sin( (t + h)) ?  k sin( t) k2IL  A + A0k + A0 0

where

A=

+1 X (?kQk(t+h) ? ?kQk (t) ) sin(k (t + h)) ;

k=1

A0k = 0

and

A0 =

k X 0

k=1





?kQk(t) sin(k (t + h)) ? sin(k t) ;

+1 X

k=k0 +1





?kQk (t) sin(k (t + h)) ? sin(k t) :

Let us give an upper bound for A. We have :

A but :

+1 X

k=1

j?kQk (t+h) ? ?kQk (t)j

?kQk (t+h) ? ?kQk (t) = ?(log )  [Qk (t + h) ? Qk (t)]  (k?k ) INRIA

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where  2 [min(Qk (t); Qk (t + h)) ; max(Qk (t); Qk (t + h))]. Thus : +1 X A  (log ) k?k jQk (t + h) ? Qk (t)j: Since we have :

k=1

jQk (t + h) ? Qk (t)j  kjhj;

jAj  c1 jhj  c1 jhjs(t)?; P 1 k2 ?ka . with c1 = log  +k=1

Let us now give an upper bound for A0. For this purpose, we consider the integer N such that :

?(N +1)  jhj  ?N :

We have, using the mean value theorem : where and but :

A0  jhjX + 2Y

X= Y=

N X k=1

?k(s(t)??1)

+1 X

k=N +1

?k(s(t)?)

X  1 ? 1s(t)?1 jhjs(t)??1

Y  1 ? 1?s(t) jhjs(t)?: Since s(t) is bounded, there exists a constant c2 > 0 depending only on t and  such

that :

A0  c2 jhjs(t)? :

Finally, it easy to see that there exists a positive constant c3 , which depends only on t and , such that :

jA0k j  c3 jhj  c3 jhjs(t)? : 0

Hence, if c = 3 max(c1 ; c2 ; c3 ), we have : jf (t + h) ? f (t)j  cjhjs(t)? Now we will prove that f (t)  s(t). There exists an in nite set ? = ?(t; )  IL such that s(t) ?  < Qk (t) < s(t) + , for RR n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

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every k 2 ?. Let N be an integer in ? such that N  k0. Let h = c(N ) ?N , where c(N ) is chosen in the set f1; 2g so that : j sin(N t + c(N ) ) ? sin(N t)j > 101 : Hence, if  is an even integer, we have : jf (t + h) ? f (t) ? ?NQN (t) (sin(N (t + h)) ? sin(N t))j  A + A0k + A00 + A000 ; where : X ?kQk (t) A00 =  j sin(k (t + h)) ? sin(k t)j 0

k>k0

k2ILn?

A000 =

X

k k0 , we have : X ?k(s(t)+) A00   j sin(k (t + h)) ? sin(k t)j; k2IN

thus, there exists a positive constant c4 such that :

A00  c4jhjs(t)+ : Let Nl be the highest integer in ? less than N . Then : A000 

Nl X k=0

 jhj 

?k(s(t)?)j sin(k (t + h)) ? sin(k t)j

Nl X

k(1?(s(t)?))

k=0 Nl (1?(s(t)?)) jhj 1?(s(t)?) ? 1 :

Using the fact that 1 ? s(t) +  < 1 ? a0 and 1 ? s(t) ?  > 1 ? b0 , we get : N (1?s(t)?) A000  jhj 1?(s(t)?) ? 1 :

Thus, there exists a positive constant c5 such that :

A000  c5 jhjs(t)+ :

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We can choose  large enough so that the constants c1 ; c3 ; c4 and c5 are less than 801 . Hence we end up with : jf (t + h) ? f (t)j > 201 jhjs(t)+ : In the case where s is a continous function, we have the following result : Proposition 7. Let s be a continuous function from [0 ; 1] to [a ; b] ]0 ; 1[ such that s(x) < s (x) 8x 2 [0 ; 1]: Assume also that there exsits a constant M > 0 such that : js(t) ? s(u)j  M jt ? ujs(t) 8(t; u) 2 [0 ; 1]  [0 ; 1]: P ?ks(x) sin(k x) is such that : Then, the function f (x) = k2IN

Proof : see appendix.

2 ? dimxB graph f = f (x) = s(x):

6. Construction using Iterated Function System(IFS) The third construction of continuous function with prescribed Holder function is based upon a generalization of the notion of IFS. This construction bears some analogy with the rst one, but here we directly manipulate the contraction ratios of ane functions instead of working on the coecients of the expansion in the Schauder basis. To begin with, we recall some basic facts about IFS. More details can be found in [13, 27, 16, 17, 15, 28] and others. 6.1. Recalls. Let K be a compact metric space whose distance is denoted by d(x; y) for x; y 2 K . Let H be the set of all non empty closed subsets of K . Then H is a compact metric space with the Hausdor metric [13] : h(A; B ) = maxfsup yinf d(x; y) ; sup yinf d(x; y)g 2B 2A x2A

x2B

which is de ned whenever A and B are subsets of K . Let wn : K ! K for n 2 f1; 2; : : : ; N g be N continuous functions. Then fK; wn : n = 1; 2; : : : ; N g is called an iterated function system (IFS). De ne W : H ! H by

W (A) = Any set G 2 H such that RR n2763

N [

n=1

wn(A) for A 2 H:

W (G) = G

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 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

is called an attractor of the IFS fK; wn : n = 1; 2; : : : ; N g. An IFS always admits at least one attractor. Indeed, start with any S 2 H , then the closure of the set om o(m?1)(S )), is an of all accumulation points of fW om(S )g1 m=1 , with W (S ) = W (W attractor of the IFS. If, for some s 2 [0; 1[ and all n 2 f1; : : : ; N g, d(wn (x); wn(y))  sd(x; y); 8(x; y) 2 K  K then the IFS is termed hyperbolic. In this case W is a contraction mapping, hence it admits a unique x point which is the unique attractor of the IFS. When the attractor G of an IFS is unique, it may be obtained as follows [14] :let p = (p1 ; : : : ; pN ) be a probability vector with each pn > 0 and P pn = 1. Start from n the x point x0 of w1 and de ne a sequence (xm ) by choosing successively xm 2 fw1(xm?1); : : : ; wN (xm?1)g for m 2 f1; 2; 3; : : : g, where probability pn is attached to the event xm = wn (xm?1). Then, the orbit fxmgm2N is dense in G. The pn's allow to generate a unique probability measure  on K which is stationary for the discrete time Markov process de ned as follows: The probability of transfer of x 2 K to a Borel subset B of K is :

p(x; B ) = where :

X n

pnwn(x) (B )

 1 if y 2 B y (B ) = 0 if y 2= B

We shall not developp here this aspect of IFS theory, and will now focus on the use of IFS for constructing graphs of continuous functions [14]. Given a set of points f(xn; yn) 2 [0 ; 1]  [u ; v]; n = 0; 1; :::; N g, with (u; v) 2 IR2 , consider the IFS given by the N contractions wn(n = 1; :::; N ) de ned on [0 ; 1]  [u ; v], by: wn(x; y) = (Ln (x) ; Fn(x; y)) where Ln is a contraction that maps [0 ; 1] to [xn?1 ; xn] and Fn : [0 ; 1][u ; v] ! [u ; v] is a function, contractive with respect to the second variable, such that : Fn(x0; y0) = yn?1 ; Fn(xN ; yN ) = yn: (6) The attractor of this IFS is the graph of a continuous function f which interpolates the points (xn; yn) [14] . If the Ln's are ane, Ln(x) = anx + hn, and if, for each n 2 f1; : : : ; N g, tnd(x; y)  d(wn(x); wn (y))  snd(x; y) for all x; y 2 K; INRIA

Construction of Continuous Functions with Prescribed Local Regularity

23

where 0 < tn  sn < 1, then : min(2; l)  dimH graph f  u where l and u are the positive solutions of : N X

n=1

tln = 1 and

and where the lower bound holds when :

N X

n=1

sun = 1

N !l X l

2

t1 tN  min(a1; aN )

n=1

tn :

Concerning the box dimension, if each Fn is ane with contraction ratio equal to cn , and if the interpolation points are equally spaced, then it is a classical result that [22] : + : : : + cN ) dimB graph f = 1 + log(c1 log N

6.2. Local behavior of self-ane functions. Under some conditions on the

Fn's, the function f de ned above is nowhere dierentiable. But here we want more, namely to control the regularity of f at each point. In this section we obtain the local Holder exponent of f at each point x 2 [0 ; 1] in the case where the Fn's are ane functions, and the interpolation points are equally spaced . We also derive the multifractal spectrum of f and recover the classical formula for the box dimension of the graph of f . Related results concerning the almost sure Holder exponent of f have already been obtained in [17]. Results concerning the multifractal spectrum were independently obtained in [18] and [19]. It is convenient to rewrite our setting in the following form : Let Si (0  i < m) be ane transformations represented in matrix notation by :  t   1=m 0  t   i=m  = S + i

x

a i ci x bi We suppose 0  t  1 and 1=m < ci < 1. Let f be the function whose graph is the attractor G of the IFS de ned by the Si's (with conditions on ai and bi corresponding to (6) to ensure the continuity of f ). Our result concerning the local regularity of f

is the following one : Proposition 8. Let 0:i1:::ik ::: be the terminating base-m expansion of a real t 2 [0 ; 1). Then the Holder exponent  of f at point t is :  log(ci :::cik ) ; lim inf log(cj :::cjk ) ; lim inf log(cl :::clk )   = min lim inf k!+1 log(m?k ) k!+1 log(m?k ) k!+1 log(m?k ) RR n2763 1

1

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 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

24

where, for any positive integer k, the k-uples (j1; :::; jk ) and (l1; :::; lk ) of non negative integers strictly smaller than m, are uniquely determined by : tk = m?k [mk t] if tk + m?k < 1 then t+k = tk + m?k = if tk ? m?k > 0 then t?k = tk ? m?k =

k X

p=1 k X p=1

jpm?p else t+k = tk lp m?p else t?k = tk

Proof : The proof is an adaptation of the classical computation of the box dimension of the graph of self ane curves [22]. Let k be a positive integer and (n1; :::; nk ) be a k-uple of integers such that 0  np < m, for every p = 1; :::; k. Let In :::nk be the interval of reals in [0 ; 1) whose base-m expansion begins with n1 :::nk . Then graph fj In :::nk = Sn  :::  Snk (G), which is a translation of Tn  :::  Tnk (G), where Ti is the linear part of Si . It is easily seen that the matrix representing Tn  :::  Tnk is :   m?k 0 m1?k an + m2?k cn an + ::: + cn cn :::cnk? ank cn cn :::cnk a Note a = max jai j; c = min(ci); r = c(1?(mc )? ) . We have : 1

1

1

1

1

1

1

2

1

2

1

1

2

1

jm1?k an + m2?k cn an + ::: + cn cn :::cnk? ank j  rcn :::cnk ; 1

1

2

1

2

1

1

so that if s is the height of the rectangle containing G, then graph fj In :::nk is contained in the rectangle whose height is (r + s)cn :::cnk . Consider now a real <  ; there exists a positive integer k0 such that, for every integer k > k0 we have : (ik ) > ; (jk ) > and (lk ) > where cn :::cnk ) : (nk ) = log( log(m?k ) 1

1

1

Let h be a real small enough so that the integer k, de ned by m?k?1  jhj < m?k , veri es k > k0 . Then either (i), (ii) or (iii) is true : (i) (t; t + h)  Ii :::ik (ii) (t; t + h)  Ii :::ik [ Ij :::jk (iii) (t; t + h)  Ii :::ik [ Il :::lk . INRIA 1

1

1

1

1

Construction of Continuous Functions with Prescribed Local Regularity

Denote r1 = r + s ; case (i) : we have :

25

jf (t + h) ? f (t)j  r1ci :::cik : 1

case (ii) : since f is continuous, we have :

jf (t + h) ? f (t)j  r1 ci :::cik + r1cj :::cjk : 1

1

case (iii) : using again the continuity of f , we have :

jf (t + h) ? f (t)j  r1ci :::cik + r1cl :::clk : 1

1

Hence, we always have :

jf (t + h) ? f (t)j  2r1jhj :

This implies that f 2 C ?(t), for every  > 0. On the other hand, consider now a real > . Assume w.l.o.g. that : log(cj :::cjk ) ;  = lim inf k!+1 log(m?k ) (the other cases are treated by simply changing j to i or l). Then, there exists a subsequence (k) such that, for every k, we have : log(cj :::cj k ) log(m?(k)) < : If q1 ; q2 and q3 are three non-colinear points in G, then Sj  :::  Sj k (G) contains the points (xn; f (xn )) = Sj  :::  Sj k (qn )(n = 1; 2; 3). The height d(k) of the triangle with these vertices is at least dcj :::cj k where d is the vertical distance from q2 to [q1 ; q3 ]. Thus, for every k, there exists a real hk such that jhk j < 2m?(k) and 1

1

( )

1

1

1

( )

jf (t + hk ) ? f (t)j  d2 cj :::cj k ; 1

which implies that :

( )

( )

( )

jf (t + hk ) ? f (t)j  d2 jhk j :

This shows that f 2= C +(t), for every  > 0, and the proof is complete. Using this proposition, it is easy to deduce the spectrum (; F ()) of singularity of f . The proof is analogous to the one for multinomial measures. RR n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

26

Corollary 3. With the same notations as above, and assuming that the proportion i(t) of (i ? 1)'s in the base-m expansion of t exists for each i, we have : mX ?1 mX ?1 mX ?1 f (t) = ? i (t) logm ci ; F () = ? i logm i ;  (q) = ? logm ci q i=0

i=0

i=0

(for de nition of F and  , see for instance [29].) Remark 1. Using the relation dimB graph f = 1 ?  (1) we recover the classical result [22] :

dimB graph f = 1 + logm

mX ?1 i=0

ci

It is now clear that, with this construction, we can not hope to control the local regularity at each point, since almost all points have the same Holder exponent (because the almost sure value of i(t) w.r.t the Lebesgue measure is m1 ) . We thus need to use some generalization, which will be presented in the next section. 6.3. Recursive construction. We set up here another way to construct fractals recursively, originally due to Anderssson [28]. We consider a collection of sets (F k )k2IN  , where each F k is a non-empty nite set of contractions Sik in K , for i = 0; : : : ; Nk ? 1, Nk  1 being an integer which denotes the cardinal of F k . We denote by cki the contraction ratio of Sik , for i = 0; : : : ; Nk ? 1, and k 2 IN . For n 2 IN  , let INmI n be the set of sequences of length n, de ned as follows: INmI n = f = (1; : : : ; n) : i 2 f0; : : : ; Ni ? 1g; i 2 IN  g and : INmI 1 = f = (1 ; 2; : : : ) : i 2 f0; : : : ; Ni ? 1g; i 2 IN  g De ne the operator W k : H ! H by :

W k (A) =

Nk [

n=1

Snk (A) for A 2 H

where Nk is the cardinal of Fk . De ne the conditions: (c) nlim !1

sup

(1 ;:::;n )2INmI n

) (Y n k=1

ckk = 0

9 81 j = < Y X k =0 j +1 x; x) c d ( S (c0 ) nlim sup k !1 ( ; ;::: )2INm 1 :j =n j k=1 ; I 1

2

+1

The proof of the following proposition can be found in [28].

INRIA

Construction of Continuous Functions with Prescribed Local Regularity

27

Proposition 9. If the conditions (c) and (c0 ) hold, then there exists a unique compact G such that :

lim W k  : : :  W 1(A) = G for every A 2 H:

k!1

We call G the attractor of the IFS (K; fF k gk2IN  ). We will use this generalized result to obtain more exibility in the construction of our functions. Let F k be the set of ane transformations Sik (0  i < m) represented in matrix notation by :      t   i=m  0 Sik xt = 1a=m k ck x + bk i

i

i

We suppose 0  t  1 and 1=m < cki < 1. We also assume that conditions (c) and (c0 ) hold to ensure that we have a unique and compact attractor. Then, if the aki 's and the bki 's satisfy some relations, analogous to those proposed in subsection 6.1, one can prove, using the same techniques as in [14], that the attractor of the IFS (K; fF k gk2N ) is the graph of a continuous function f . We, then, have the following result : Proposition 10. Let 0:i1:::ik ::: be the base-m expansion of a real t 2 [0 ; 1). Then: ! log(c1j :::ckjk ) log(c1l :::cklk ) log(c1i :::ckik ) ; lim inf ; lim inf f (t) = min lim inf k!+1 log(m?k ) k!+1 log(m?k ) k!+1 log(m?k ) where, for any integer k, if we denote tk = m?k [mk t], the k-uples (j1; :::; jk ) and (l1; :::; lk ) are given by : k X t+k = tk + m?k = jpm?p 1

1

t?k = tk ? m?k =

p=1 k X p=1

1

lpm?p

Proof : The proof uses the same techniques as in proposition 8. Although this generalization allows more exibility in the choice of f (t), it is still too much constrained. Indeed, it is easy to see that if two reals dier only at nite number of ranks in their base-m expansion, then they will have the same Holder exponent. Hence we can not control the regularity independently at each point. To so, let now F k be de ned as the set of ane transformations Sik (0  i  RR do n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

28

mk ? 1), each Sik operating only on [im?k+1 ; (i + 1)m?k+1] if i is even, and on [(i ? 1)m?k+1 ; im?k+1] if i is odd. Every Si maps to [im?k ; (i + 1)m?k]. Suppose, also, that we want to interpolate the points ( mi ; yi), for i = 0; : : : ; m, m  2 and yi 2 IR. Let the compact K be a rectangle containing the (xi ; yi)'s and write :  t   1=m 0   t   i=mk  k Si x = ak ck x + bki : i i We call (K; (F k )) a generalized ane IFS. De ne the following conditions, which allow the attractor to be the graph of a continuous function f (for sake of simplicity we will give conditions when m = 2, the general case being handled similarly): start with the graph of any non ane continuous function , and denote : (0) = u ; (1) = v: Then choose the contractions (or more precisely the aki and bki ) so that they verify

the following conditions: for i = 0; 1 : i + 1  i  1 1 Si (0; u) = m ; yi ; Si (1; v) = m ; yi+1

S02 (0; y0) = (0; y0) ; S02 (1=2; y1) = S12 (0; y0) ; S12 (1=2; y1) = (1=2; y1) S22 (1=2; y1) = (1=2; y1) ; S22 (1; y2) = S32 (1=2; y1) ; S32 (1; y2) = (1; y2): for k > 2 and for i = 0; : : : ; 2k ? 1 : if i is even then : if i < 2k?1 :

Sik  S ki ?1  S[k?i 2]  : : :  S[2 ki? ] (0; y0) = S ki ?1  S[k?i 2]  : : :  S[2 ki? ](0; y0) 2

22

2

2

2

22

2

2

Sik  S ki ?1  S[k?i 2]  : : :  S[2 ki? ] (1=2; y1) = Sik+1  S[ki?1 ]  S[ki?2 ]  : : :  S[2 ik? ](0; y0) if i  2k?1 : Sik  S ki ?1  S[k?i 2]  : : :  S[2 ki? ] (1=2; y1) = S ki ?1  S[k?i 2]  : : :  S[2 ki? ](1=2; y1) 2

22

2

2

22

2

+1 22

+1 2

2

2

2

22

2

2

+1 2

2

Sik  S ki ?1  S[k?i 2]  : : :  S[2 ki? ] (1; y2) = Sik+1  S[ki?1 ]  S[ki?2 ]  : : :  S[2 ik? ](1=2; y1) if i is odd, then : 2

22

2

2

+1 2

+1 22

2

+1 2

INRIA

Construction of Continuous Functions with Prescribed Local Regularity

29

if i < 2k?1 :

Sik  S[ki?]1  S[k?i 2]  : : :  S[2 ki? ] (1=2; y1) = S[ki?]1  S[k?i 2]  : : :  S[2 ki? ](1=2; y1) 22

2

2

2

2

22

2

2

if i  2k?1 :

Sik  S[ki?]1  S[k?i 2]  : : :  S[2 ki? ] (1; y2) = S[ki?]1  S[k?i 2]  : : :  S[2 ki? ] (1; y2): 22

2

2

2

2

22

2

2

Our main result is the following:

Proposition 11. Suppose that conditions (c) and (c0 ) hold. Then the attractor of the IFS de ned above is the graph of a continuous function f such that :   f mi = yi 8 i = 0; : : : ; m

and

f (t) = min(1 ; 2 ; 3)

where

8 log(ckmk? i > >  = lim inf 1 > k!+1 < log(ckmk? j  = lim inf 2 > k!+1 > log(ckmk? l > : 3 = lim inf k!+1

1 2 mk?2 i2 +:::+mik?1 +ik :::cmi1 +i2 ci1 ) ? k log(m ) c1 ) :::c2 1 +mk?2 j +:::+mj 1 2 k?1+jk mj1 +j2 j1 ? k log(m ) c1 ) :::c2 1 +mk?2 l +:::+ml 1 2 k?1 +lk ml1 +l2 l1 ? k log(m ) 1

1+

(7)

and where the ip's, jp's and the lp 's are de ned as in proposition 8.

Proof Let In :::nk be the interval of reals whose base-m expansion begins with n1 :::nk . De ne Gk to be the set obtained after k iterations in the process of generation of the attractor G, i.e : Gk = W k  : : :  W 1(G): Then, it easy to see that : 1 2 Gk j In :::nk = Smk k? n +mk? n +:::+mnk? +nk  : : :  Smn +n  Sn (G): 1

1

1

1

2

2

1

1

2

1

Using the same techniques as in the proof of proposition 8, the announced result follows.  RR n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

30

Remark 2. Given m reals r1; : : : ; rm 2] m1 ; 1[, de ne, for every integer k  1 and for every i 2 f0; : : : ; mk ? 1g, the cki 's as follows : cki = ri+1?m[ mi ]:

Then, we recover the original construction considered in proposition 8.

The following corollary allows to control the local singularity at each point, while interpolating the points ( mi ; yi ), for i = 0; : : : ; m. We rst need to state the following re nement of Lemma 2 : Lemma 3. Let s 2 H. Then there exists a sequence fRngn1 of piecewise polynomials such that : 8 s(t) = lim inf R (t) 8t 2 [0 ; 1] > < + n!+1 n ? (8) > jjRn0 jj1  n ; jjRn0 jj1  n 8 n  1

: jjRn jj1 

1 log n

where Rn0 + and Rn0 ? are respectively the right and the left derivative of Rn .

Proof : Let Qk be de ned as in Lemma 2 and de ne :

Rk = max(Qk ; log1 k ):

(9)

Corollary 4. Let s(t) be a function from [0 ; 1] to [0 ; 1], which is the lower limit of

a sequence of continuous functions. Then, there exists a generalized ane IFS whose attractor is the graph of a continuous function f which veri es : f (t) = s(t) Proof : Because of the continuity constraints, nding the generalized ane IFS amounts to determining the double sequence (cki )i;k . Let fRngn1 be a sequence of piecewise polynomials that veri es (8) and let M be the set of m-adic points of [0 ; 1]. Consider now the sequence frk gk1 of functions from M to IR de ned as follows. P For t 2 M, t = kp=1 ipm?p, let : 0

r1 (t) = R1 (i1m?1)

INRIA

Construction of Continuous Functions with Prescribed Local Regularity kX ?1

rk (t) = kRk (t) ? (k ? 1)Rk?1( and

p=1

31

ipm?p) for k = 2; : : : ; k0

rk (t) = kRk (t) ? (k ? 1)Rk?1(t) for k > k0: Now, for each k  1 and i = 0; : : : ; mk ? 1, set : cki = m?rk (im?k) : Using (9), one veri es that conditions (c) and (c0) are ful lled.

Using proposition 11, we get :

0 Pk r (t+) Pk r (t?) 1 Pk r (t ) j j j j C j j B j =1 j =1 j =1 C: ; lim inf ; lim inf lim inf f (t) = min B B k!+1 A @ k!+1 k k!+1 k k C

Since :

Pk r (t )

j =1

j j

k

= Rk (tk ) ;



Pk r (t+)

j =1

j j

k

= Rk (tk

+) ;

Pk r (t?)

j =1

j j

k

= Rk (t?k );

f (t) = min lim inf R (t ); lim inf R (t ; lim inf R k!+1 k k k!+1 k k k!+1 k +)



(t?) k

:

Using (8), we have : lim inf R (t) = lim inf R (t ) = lim inf R (t+) = lim inf R (t?): k!+1 k k!+1 k k k!+1 k k k!+1 k k We end up with :

f (t) = s(t): 

7. Concluding remarks 7.1. Non uniqueness of f . It is easy to see that, given a set of points f(xi; yi)gi=0;:::;N where xi = i=N , and a function s 2 H, there is an in nite number of continuous functions that interpolate the (xi; yi )'s and whose Holder function is s. Indeed, take the function f constructed in section 6:3 and consider the function g de ned by : f (x) + PL(x) g ( x ) =  RR n2763 1+

32

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

where PL(x) is the Legendre polynomial de ned by :

PL(x) =

N X i=0

QN (x ? x )

j 6=i yi Q N j 6=i

j

(xi ? xj )

and  is a real dierent from ?1. Then, since PL 2 C 1(IR), it is clear that g = s, and of course, the function g interpolates the (xi; yi )'s for every  2 IR n f?1g.

7.2. Size of Es. Let s 2 H and de ne : Es = fw 2 C 0([0 ; 1]) = w (x) = s(x) 8x 2 [0 ; 1]g: Proposition 12. Es is dense in C 0([0 ; 1]) for the uniform convergence norm jj : jj1 . Proof : Let IP be the set of polynomials de ned on [0 ; 1]. It is well known that IP is dense in C 0([0 ; 1]) for the uniform convergence norm. For f 2 C 0([0 ; 1]), let (Pn)n2IN be a sequence such that Pn 2 IP for every n 2 IN and

jjPn ? f jj1 ! 0 when n ! 1: Let now w be a function in Es, and consider the sequence (fn)n2IN  de ned by :

fn = Pn + wn for every n 2 IN :

Since Pn 2 C 1([0 ; 1]) for every n 2 IN , and w 2 Es, it is clear that fn is in Es for every n 2 IN . We have :

jjfn ? f jj1  jjPn ? f jj1 + jjwnjj1

w is a continuous function on a compact, and there exists a constant C > 0 such that jjwjj1  C , hence :

jjfn ? f jj1 ! 0 when n ! 1: 

INRIA

Construction of Continuous Functions with Prescribed Local Regularity

33

7.3. More re ned ways of characterizing the local regularity. The local regularity of the graphs of the functions constructed with the three methods we have presented above appears, in some cases, strikingly dierent (see section 8). Several improvements may be proposed in order to describe these discrepancies :  A well known method to measure more precisely the local structure would be to use ner scales of functions, as for instance functions of the form : n g(x) = x(log 1 ) (log log 1 ) : : : (log log : : : log 1 ) ; 1

x

2

x

x the Holder exponent at a point x0 would then be a vector (; 1; 2 ; : : : ; n).

 Another possibility is to characterize algebraic oscillations instead of taking the absolute values, i.e consider the two limits : lim sup g?h( h) and lim sup g+h( h) ;

where

h!0

h!0

g(x) = f (x0 + h) ? f (x0); g+(x) = max(g(x); 0); g?(x) = min(g(x); 0):

 Finally, especially for practical purposes, the speed of convergence to the local Holder exponent at x0 is of crucial importance. For instance, it is easy to show that, for the Schauder type function considered in section 4, if we take s(x) = x, then, for x0 > 0 and for some sequence hn ! 0, the best possible lower bound is : jf (x0 + hn ) ? f (x0)j  c1 jhn jx ?c jhnj where c1 and c2 are constants. But for the Weierstrass like functions of section 3, and also with s(x) = x, the best possible lower bound is : jf (x0 + hn ) ? f (x0)j  c0 hn x where c0 is a constant. When working with discrete data, this rst order dierence in h can make a big dierence (see gures in the next section). 8. Examples The following gures are graphs of continuous functions with prescribed local regularity. We have implemented the constructions described in section 4, 5, 6, and for each case, we show an example with s(t) = t and s(t) = j sin(5t)j. 0

2

0

RR n2763

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

34

Figure 1. Construction using the Schauder basis with s(t) = t 7 "f" 6

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2. Construction using the Schauder basis with s(t) = jsin(5t)j 8 "f" 7

6

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

INRIA

Construction of Continuous Functions with Prescribed Local Regularity Figure 3. Construction using the Weierstrass type function with

s(t) = t

8 "f" 6

4

2

0

-2

-4

-6

-8

-10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4. Construction using the Weierstrass type function with

s(t) = jsin(5t)j 12

"f" 10

8

6

4

2

0

-2

-4 0

RR n2763

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

35

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

36

Figure 5. Construction using generalized ane IFS with s(t) = t 5 "f" 4

3

2

1

0

-1

-2 0

0.2

0.4

0.6

0.8

1

Figure 6. Construction using generalized ane IFS with s(t) = jsin(5t)j 5 "f" 4

3

2

1

0

-1

-2

-3 0

0.2

0.4

0.6

0.8

1

INRIA

Construction of Continuous Functions with Prescribed Local Regularity

37

Appendix Proof of proposition 7

Recall that s is the Holder function of s. We begin by proving that f (t)  s(t): Let t be xed, 2 be a real such that 0 < 2  1, and h be a real such that 0 < jhj < 2 . Then we have : +1  X

f (t + h) ? f (t) =

k=1

?ks(t+h) sin(k (t + h)) ? ?ks(t) sin(k t)



= A + A0 where

A=

and

+1  X

k=1

A0 =

+1 X

k=1



?ks(t+h) ? ?ks(t) sin(k (t + h));

?




?ks(t) sin(k (t + h)) ? sin(k t) :

Let us give an upper bound for jAj. We have :

jAj  but :

+1 X

k=1

j?ks(t+h) ? ?ks(t)j

?ks(t+h) ? ?ks(t) = ?(log )  [s(t + h) ? s(t)]  (k?k ) where  2 [min(s(t); s(t + h)) ; max(s(t); s(t + h))]. Thus :

jAj  (log )js(t + h) ? s(t)j

+1 X

k?k

k=1 P +1 ? k Let C = k (0 < C < +1 because this series converges), then, since there k=1

exists a constant M > 0 such that : js(t + h) ? s(t)j  M jhjs(t); we have : where RR n2763

jAj  c1 jhjs(t)  c1 jhjs(t); c1 = CM log :

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

38

Let us now give an upper bound for jA0 j. For this purpose, we consider the integer N such that :

?(N +1)  jhj  ?N :

We have, using the main value theorem :

jA0 j  jhjX + 2Y

where

X=

and

Y=

but :

N X k=1

?k(s(t)?1)

+1 X

k=N +1

?ks(t)

X  1 ? 1s(t)?1 jhjs(t)?1

Y  1 ? 1?s(t) jhjs(t): Since s(t) is bounded, there exists a constant c2 > 0 such that : jA0 j  c2 jhjs(t): Finally, if c = c1 + c2 , we have : jf (t + h) ? f (t)j  cjhjs(t) which gives :

jf (t + h) ? f (t)j = 0: ( < s(t)) ) hlim !0 jhj

Now we will prove that f (t)  s(t). Let t be a real in [0 ; 1] and  a real in ]0 ; 2 [. Then, consider the integer N such that ?(N +1) <   ?N , and let h be a real such that ?(N +1) < jhj  . We have : X = jf (t + h) ? f (t) ? ?Ns(t)(sin(N (t + h) ? sin(N t))j

 B+2

+1 X

k=N

?ks(t) + jAj

P where B = N ?1 ?ks(t)j sin(k (t + h) ? sin(k t)j. We have :

k=1

s(t)?1 B  ?Ns(t) 1 ?(s(t)?1) :

INRIA

Construction of Continuous Functions with Prescribed Local Regularity

Since we have seen that :

39

jAj  c1 jhjs(t)  c1 ?Ns(t)

then :

X  ?Ns(t) (c1 + c3 ) ;

with

s(t)?1 ?s(t) c3 = 1 ?(s(t)?1) + 2 1 ? ?s(t) : Provided that  is large enough, we may choose c1 and c3 such that :

1 and c  1 c1  40 3 40

thus : but : and :

1 ?Ns(t) X  20





X  jf (t + h) ? f (t)j ? ?Ns(t)j sin(N (t + h)) ? sin(N t)j

jf (t + h) ? f (t)j  ?Ns(t)j sin(N (t + h)) ? sin(N t)j ? X: There exists a sequence [22] (hn), with ?(N +1) < jhn j    ?N for every n, such that :

j sin(N (t + hn)) ? sin(N t)j  101 8n

because 1  jhn jN  1 8n. We deduce : which gives :

jf (t + hn) ? f (t)j  201 ?Ns(t)  201 s(t)  201 jhn js(t) ( > s(t)) ) lim sup jf (t + jhh)j ? f (t)j = +1: h!0

Let us now check that f veri es conditions (c1 ) and (c2 ) of proposition 1. Let x be a real in [0 ; 1] and  be a real such that 0 <  < min(1 ; 2 ). For every  <  and t 2 B (x; ), we have seen that :   jRR f (t)n2763 ?f (u)j  c(t)M log  + 1 ? 1s(t)?1 + 1 ? 2?s(t) jt ? ujs(t) for every u 2 B (t; );

 VEHEL  K. DAOUDI, J. LEVY and Y. MEYER

40 where :

c(t) =

1 X k=1

k?k with  2 [min(s(t); s(u)) ; max(s(t); s(u))] :

This implies that : c(t; )  AM + B for every t 2 [0 ; 1] and  <  where : 1 X A = log  k?ka and B = 1 ? 1b?1 + 1 ?2?b :

Hence :

k=1

C (x; ) < +1 8  < ;

and the condition (c1 ) holds. Condition (c2 ) is easy to verify. Indeed, we have seen that there exists a real u 2 B (t; ),such that : jf (t) ? f (u)j  201 s(t); hence : 1 8 < ; c(t; )  20 which implies that : C (x; ) 6= 0: Now, since s is continuous, and conditions (c1 ) and (c2 ) hold, we get using proposition 1: 2 ? dimxB graph f = s(x) for every x 2 [0 ; 1]:  References 1. Gerald A. Edgar, editor. Classics On Fractals. Addison-Wesley Publishing Company,

1993. 2. Helge von Koch. On a Continuous Curve Tangent Constructible from Elementary Geometry. Arkiv for Matematik, Astronomi och Fysik, pages 681{702, 1904. 3. E.W. Hobson. The Theory of Functions of Real Variable and the Theory of Fourier's Series. Cambridge University Press, third edition, 1950. 4. Karl Weierstrass. On Continuous Function of a Real Argument that do not have a Well-De ned Dierential Quotient. Mathematische Werke, pages 71{74, 1895. 5. G.H. Hardy. On Weierstrass's Non-Dierentiable Function. Trans. Am. Math. Soc., 17:301{325, 1916. 6. T. Takagi. A Simple Example of a Continuous Function Without Derivative. Proceedings of the Physico-Mathematical Society of Japan, 1(2):176{177, 1903. INRIA

Construction of Continuous Functions with Prescribed Local Regularity

41

7. Masayoshi Hata. Singularities of the Weierstrass Type Functions. Journal d'Analyse Mathematique, 51, 1988. 8. T-Y. Hu and K-S. Lau. Fractal Dimensions and Singularities of the Weierstrass Type Functions. Trans. of the American Mathematical Society, 335(2):649{665, 1993. 9. R.D. Mauldin and S.C. Williams. On the Hausdor Dimension of Some Graphs. Trans. of the American Mathematical Society, 298(2):793{803, 1986. 10. S. Jaard. Construction de Fonctions Multifractales ayant un Spectre de Singularites Prescrit. C.R. Acad. Sci. Paris, pages 19{24, 1992. T. 315, Serie I. 11. W. Doeblin and R. Fortet. Sur des Cha^nes a Liaisons Completes. Bull. Soc. Math. de France, 65:132{148, 1937. 12. L. Dubins and F. Freedman. Invariant Probabilities for Certain Markov Process. Ann. Math. Statist., 37(1966):837{848. 13. J. Hutchinson. Fractals and Self-Similarity. Indiana University Journal of Mathematics, 30:713{747, 1981. 14. M. Barnsley. Fractal Functions and Interpolation. Constructive Approximation, 1985. 15. E.R. Vrcsay. Iterated Function Systems: Theory, Applications and the Inverse Problem. Fractal Geometry and Analysis, pages 405{468, 1991. 16. D.P. Hardin. Hyperbolic Iterated Function Systems and Applications. PhD thesis, Georgia Institute of Technology, 1985. 17. Tim Bedford. Holder Exponents and Box Dimension for Self-Ane Fractal Functions. Constructive Approximation, 5:33{48, 1989. 18. S. Jaard. Multifractal Formalism for Functions, Part 1 and Part 2. SIAM Journal of Mathematical Analysis. to appear. 19. Jacques Levy Vehel, Khalid Daoudi, and Evelyne Lutton. Fractal Modeling of Speech Signals. Fractals, 2(6), 1994. 20. Khalid Daoudi and Jacques Levy Vehel. Speech Modeling Based on Local Regularity Analysis. In Proceedings of the IASTED/IEEE International Conference on Signal and Image Processing, Las Vegas, USA, 20-23 November 1995. 21. C.A. Rogers. Hausdor Measures. Cambridge University Press, 1970. 22. K. J. Falconer. Fractal Geometry : Mathematical Foundation and Applications. John Wiley & Sons, 1990. 23. Claude Tricot. Mesures et Dimensions. PhD thesis, Univ. Paris 11, 1983. 24. Y. Meyer. Ondelettes. Hermann, 1990. 25. S. Jaard. Functions with Prescribed Holder Exponent. Applied and Computational Harmonic Analysis, 2(4):400{401, October 1995. 26. Claude Tricot. Courbes et Dimension Fractale. Springer-Verlag, 1993. 27. Michael F. Barnsley. Fractals Everywhere. AK Peters, 1993. 28. Leif M. Andersson. Recursive Construction of Fractals. Annales Academiae Scientiarum Fennicae, 1992. 29. A. Arneodo, J.-F. Muzy, and E. Bacry. Multifractal Formalism for Fractal Signals. Technical report, Paul-Pascal Research Center, France, 1992. RR n2763

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