FRACTAL COUNTABLE INTERPOLATION SCHEME: EXISTENCE AND AFFINE INVARIANCE

EXISTENCE AND AFFINE INVARIANCE

NICOLAE ADRIAN SECELEAN

We extend some properties of the fractal interpolation from the finite case to the case of countable set of data. The main result is that, given an countable system of data ∆ in [a, b]×Y, where [a, b] is a real interval andY a compact metric space andψa proper continuous function which transforms ∆ in some other countable system of data, thenψ(A) is the attractor corresponding toψ(∆), whereAis the attractor associated with ∆. The particular case ∆⊂R² space is described.

AMS 2010 Subject Classification: Primary 28A80, Secondary 65D15.

Key words: countable iterated function system, interpolatory scheme, attractor, countable system of data, fractal interpolation function

1. INTRODUCTION

In his famous paper [4], J.E. Hutchinson proves that, given a set of con- tractions (IFS) (ω_n)^N_n=1 in a complete metric space X, there exists a unique nonempty compact set A ⊂X, named the attractor of IFS which is its fixed point. This attractor is, generally, a fractal set. These ideas have been ex- tended to infinitely many contractions, such a generalization can be found in [5] for Countable Iterated Function Systems (CIFS) on a compact metric space.

Barnsley ([1], [2]) shows that, for a given finite system of data ∆_N = (x_n, y_n)^N_n=0, there exists a unique fractal interpolation function f, i.e., a con- tinuous function such that f(x_n) =y_n for all 0 ≤n≤N and its graph is the attractor of a proper IFS.

In [7] the problem of fractal interpolation has been extended to the case of a countable system of data ∆ = (xn, yn)n≥0 in the compact metric spaceX= [a, b]×Y,Y being a compact metric space. One constructs a CIFS onXwhose attractor is the graph of a proper countable fractal interpolation function f (namely, a continuous functionf withf(x_n) =y_n for everyn= 0,1, . . .).

MATH. REPORTS13(63),1 (2011), 75–87

2. PRELIMINARIES

2.1. Iterated function systems and countable iterated function systems

In this subsection we recall some well known aspects on Fractal Theory used in the sequel (more complete treatments may be found in [4], [2], [5], [7]).

Let (X,d) be a complete metric space andK(X) the class of all compact non-empty subsets of X.

The functionh:K(X)×K(X)→R+,h(A, B) = max{d(A, B),d(B, A)}, where d(A, B) = sup

x∈A

y∈Binf d(x, y)

, for all A, B ∈ K(X), is a metric, called the Hausdorff metric. The setK(X) is a complete metric space with respect to this metric h.

Suppose that (X,d) is a complete metric space. A set of contractionsω_n: X →X, 1≤ n≤N, is called aniterated function system (IFS). Such a sys- tem of maps induces a set function S_N :K(X)→ K(X),S_N(E) =

N

S

n=1

ω_n(E), which is a contraction onK(X) with contraction ratior≤ max

1≤n≤Nr_n,r_n being the contraction ratio of ωn, n= 1, . . . , N. According to the Banach contrac- tion principle, there exists a unique set A_N ∈ K(X) which is invariant with respect to S_N, that is, A_N = S_N(A_N) =

N

S

n=1

ω_n(A_N). We say that the set AN ∈ K(X) is theattractor of the IFS (ωn)^N_n=1 .

Now, we suppose further that (X,d) is a compact metric space.

A sequence of contractions (ωn)n≥1 on X whose contraction ratios are, respectively,r_n>0, such that sup

n

r_n<1 is called acountable iterated function system, for short CIFS. For each N ≥1, (ω_n)^N_n=1 will be a partial IFS of the considered CIFS and S_N defined as below is the associated set function.

If we consider the CIFS (ω_n)n≥1, then the set functionS :K(X)→ K(X), given by

(1) S(E) = [

n≥1

ω_n(E)

(the bar means the closure of the respective set), is a contraction map on (K(X), h) with contraction ratior ≤sup

n

rn. Thus, there exists a unique non- empty compact set A ⊂ X invariant for the family (ω_n)n≥1, that is, A = S(A) = S

n≥1

ωn(A).The set A is calledthe attractor of the CIFS (ωn)n≥1 and it can be obtained as limit in the Hausdorff metric of the sequence of attractors (AN)N≥1 of the partial IFS (ωn)^N_n=1,N = 1,2, . . . (see [5]).

Theorem1. [8]Assume that(Bk)kis a convergent sequence of nonempty compact subsets ofX. ThenAcan be approximated by the sequence of compact nonempty sets (S_k^p(Bk))p,k. More precisely, we have lim

p lim

k S_k^p(Bk) = A, the limiting processes being taken in the Hausdorff metric, andS_k^p:=S_k◦ · · · ◦ S_k

| {z }

p times

.

2.2. Countable fractal interpolation

A.Now, we describe an extension of the fractal interpolation to the case of the countable system of data (more details can be found in [7]).

Let (Y,dY) be a compact and arcwise connected metric space. A coun- table system of data (abbreviated CSD) is a set of points of the form

∆ = (x_n, y_n)n≥0={(x_n, y_n)∈R×Y, n= 0,1, . . .},

where the sequence (x_n)n≥0 is strictly increasing and bounded and (y_n)n≥0Y is convergent.

In the sequel ∆ = (x_n, y_n)n≥0 will be a CSD and we put a = x₀, b = limn xn,M = lim

n yn andX = [a, b]×Y. Clearly, (b, M)∈X.

Aninterpolation functioncorresponding to this CSD is a continuous map f : [a, b]→Y such that f(xn) =yn forn= 0,1, . . .. The points (xn, yn)∈X, n ≥ 0, are called the interpolation points. Note that f(b) = M, since f is continuous. Define a metric δ onX by

(2) δ((x, y),(x⁰, y⁰)) =|x−x⁰|+θdY(y, y⁰),

for all (x, y),(x⁰, y⁰) ∈ X, where θ is a positive real number to be specified below. The metric space (X, δ) is compact.

Let c and s be real numbers with 0 ≤ s < 1 and c > 0. For each n= 1,2, . . ., letϕ_n:X →Y be a function which satisfies the inequalities (3) d_Y(ϕ_n(α, y), ϕ_n(β, y))≤c|α−β|, ∀α, β ∈[a, b], ∀y∈Y, and

(4) d_Y(ϕn(α, y), ϕn(α, z))≤sd_Y(y, z), ∀α∈[a, b], ∀y, z ∈Y.

Define a transformation ω_n:X→X by

ωn(x, y) = (ln(x), ϕn(x, y)) for all (x, y)∈X, n= 1,2, . . . ,

whereln: [a, b]→[xn−1, xn] is the invertible transformationln(x) =anx+en, with

(5) an= xn−xn−1

b−a and en= bxn−1−axn

b−a .

Notice that

(6) an>0 for all n≥1 and sup

n≥1

an<1.

Theorem2. [7]Let the family(ω_n)n≥1 be defined as above. Assume that there exist real constants c and s such that c >0, 0 ≤s <1, and conditions (3) and(4) are fulfilled. Letθ be the constant in the definition of the metricδ in equation (2) given by θ= inf

n≥1(1−an)/2c. Then (ωn)n≥1 is a CIFS in the metric δ. Consequently, there exists a unique nonempty compact set A ⊂X such that A= S

n≥1

ω_n(A).

We refer to the CIFS (ω_n)_nfrom the preceding theorem associated to ∆ and Φ:= (ϕn)n as a countable interpolatory schemeσ(∆,Φ).

We now constrain the CIFS (ω_n)n≥1 onX defined above to ensure that the attractor includes the CSD. Namely, we assume that

(7) ϕn(x0, y0) =yn−1 and ϕn(b, M) =yn forn= 1,2, . . . . Then it follows that

(8) ω_n(x₀, y₀) = (xn−1, yn−1) and ω_n(b, M) = (x_n, y_n) forn= 1,2, . . . . Theorem 3. [7] Let σ(∆,Φ) be a countable interpolatory scheme. As- sume also that there exist real constants c and s such that 0≤s < 1, c > 0, and conditions (3),(4)and(7)are fulfilled. Then there exists an interpolation functionf corresponding to the CSD such that the graphAoff is the attractor of the CIFS (ω_n)n≥1. That is, A=

(x, f(x)); x∈[a, b] . B.We consider now the particular case when ∆ =

(xn, yn)∈R² : n= 0,1,2, . . . , a=x0 < x1 <· · ·, b= lim

n xn, (yn)n≥0 is a convergent sequence, M = lim

n y_n, Y is a compact interval and X := [a, b]×Y. We will obtain a generalization for countable sets of data of Barnsley’s construction [2].

The following lemma proves that there exists an interpolation function for such CSD.

Lemma1. There exists aninfinite polygonal linewhich is an interpolation function corresponding to the CSD considered above.

Proof. For each n = 1,2, . . ., let be the affine function fn : [a, b] → Y given by

f_n(x) =yn−1+ y_n−yn−1

x_n−xn−1

(x−xn−1) and f : [a, b]→Y given by

(9) f(x) =

f_n(x) forx∈[xn−1, x_n), n= 1,2, . . . , M forx=b.

The functionf is obviously continuous on [a, b). To prove thatf is continuous atb, let be ε >0. Then there existn¹_ε ≥1 andn²_ε ≥1 such that

n≥n¹_ε ⇒ |M−y_n|< ε 2 and, respectively,

n, m≥n²_ε ⇒ |y_m−y_n|< ε 2.

Let Nε = max{n¹_ε, n²_ε}. For any x ∈ (xNε, b), there exists a unique n > Nε

such that xn−1≤x≤x_n, hencef(x) =f_n(x) and therefore

|M −f(x)|=

M−yn−1− y_n−yn−1

x_n−xn−1

·(x−xn−1) ≤

≤ |M−yn−1|+|y_n−yn−1| · x−xn−1

x_n−xn−1

≤

≤ |M −yn−1|+|y_n−yn−1| ≤ ε 2 +ε

2 =ε.

Thus f is continuous and, clearly, f(x_n) = f_n(x_n) = y_n for all n= 0,1, . . ., therefore f is an interpolation function for the CSD (xn, yn)n≥0.

The maps ϕ_n, n ≥1, in the construction of (ω_n)n≥1 can be the affine transformations

ϕ_n(x, y) =c_nx+d_ny+g_n, c_n, d_n, g_n∈R, n= 1,2, . . . .

If d_n is any real number, we obtain, using the above notationsω_n:X→R², (10) ω_n(x, y) = (a_nx+e_n, c_nx+d_ny+g_n) for any n= 1,2, . . . ,

where the constants used will be a_n= x_n−xn−1

b−a , en= bxn−1−axn

b−a , c_n= y_n−yn−1

b−a −d_n(M −y₀) b−a , g_n= byn−1−ay_n

b−a −d_nby₀−aM b−a . (11)

We deduce immediately

(12) en=xn−anb, gn=yn−cnb−dnM, ∀n= 1,2, . . . .

Standard calculus proves the existence, for any n ≥ 1, of the vertical scaling factor d_n satisfying the conditionsd_n≥0, sup

n≥1

d_n<1 andω_n(X)⊂X for all n. The metric δ on R² from (2) is equivalent to the Euclidean metric and (ωn)n≥1 is a CIFS on the compact metric space (X, δ) which isassociated

to the considered CSD. We refer to the CIFS (ωn)nassociated to CSD ∆ with d= (d_n)n≥1 as acountable interpolatory scheme σ(∆,d).

From Theorem 3 we obviously obtain

Theorem 4. Under the above conditions, there exists an interpolation function corresponding to the countable set of data such that its graph is the attractor of the associated CIFS (ωn)n≥1.

The interpolation function whose graph is the attractor of CIFS described above is called the countable fractal interpolation function.

3. AFFINE INVARIANCE OF COUNTABLE INTERPOLATORY SCHEME

We now extend some aspects concerning the affine invariance of an in- terpolatory scheme described by Koci´c and Simoncelli [3] from the finite case to the case of countable data. The particular case when the countable sys- tem of data is contained in the R² Euclidean space is treated. Also we will give some results concerning the approximation in the Hausdorff metric of the attractor corresponding to a transformed countable system of data under a feasible mapping.

Let ∆ = (xn, yn)n≥0 be a CSD andX= [a, b]×Y defined in Section 2.2 and ψ:X →X be a continuous function.

Definition 1. We say that the mappingψon Xisoutstandingfor a given CSD ∆ = (x_n, y_n)n≥0 on X if ψ(∆) = (x⁰_n, y⁰_n)n≥0 is also a CSD.

Let σ(∆,Φ) be a countable interpolatory scheme and ψ an affine out- standing mapping for it. There exists a new CIFS (ω_n⁰)n≥1 corresponding to the transformed CSD ψ(∆) on the compact space ψ(X)⊂X.

Lemma 2. Let X, Y be two metric spaces and f :X → Y a continuous function. Then

f(A) =f(A), for any relatively compact set A⊂X.

Proof.Because f is continuous, we have f(A)⊂f(A). Conversely, y∈f(A)⇒ ∃(yn)n⊂f(A), yn→y⇒ ∃(xn)n⊂A, f(xn) =yn→y.

Next, A being compact, by Weierstrass-Bolzano Theorem there exists a con- vergent subsequence (xnk)k of (xn)n. Hencexnk →

k x∈A.

Thusf(x_nk)→

k f(x) =y, consequently y∈f(A).

Remark. If in the above lemma the metric spaceX is compact, then the equality from statement holds for any subset A of X.

Lemma 3. Assume that

(13) ω⁰_n◦ψ=ψ◦ω_n, ∀n≥1.

Then ψ(A) is the attractor of the associated CIFS of CSD ψ(∆) (A being the attractor of (ω_n)_n).

Proof. We must show that ψ(A) = S⁰ ψ(A)

=

∞

S

n=1

ω⁰_n ψ(A)

. Since A=S(A), it remain to prove the equalityψ S(A)

=S⁰ ψ(A)

. By Lemma 2 and the hypothesis, we have

ψ S(A)

=ψ [^∞

n=1

ωn(A)

=ψ [^∞

n=1

ωn(A)

=

∞

[

n=1

(ψ◦ωn)(A) =

=

∞

[

n=1

(ω_n⁰ ◦ψ)(A) =S⁰ ψ(A)

.

We consider now the particular case described in Section 2.2 A (using the notations from there). Let a nonsingular affine mapping ψ : R² → R² given by

(14) ψ(x, y) = (px+qy+α, rx+sy+β), p, q, r, s, α, β∈R, ps6=qr.

Recall that a mapping is affine if and only if it is a composition of a regular linear transformation (α = β = 0) and a translation (q = r = 0, p=s= 1,|α|+|β|>0).

Let us consider a countable interpolatory scheme σ(∆,d) and suppose that ψis an outstanding mapping for it. The CIFS corresponding to the new interpolatory scheme σ ∆⁰ =ψ(∆),d

on the spaceψ(X)⊂R² will be (15) ω_n⁰(x, y) = (a⁰_nx+e⁰_n, c⁰_nx+dny+g⁰_n), n= 1,2, . . . , with

a⁰_n= p(x_n−xn−1) +q(y_n−yn−1) p(b−a) +q(M−y0) , (16)

c⁰_n= r(x_n−xn−1) +s(y_n−yn−1)

p(b−a) +q(M−y0) −d_n r(b−a) +s(b−a) p(b−a) +q(M −y0), e⁰_n= (pb+qM+α)(pxn−1+qyn−1+α)−(pa+qy₀+α)(px_n+qy_n+α)

p(b−a) +q(M −y0) ,

g_n⁰ =rx_n+sy_n+β−c⁰_n(px_n+qy_n+β)−d_n(rx_n+sy_n+β).

We get obviously

(17) e⁰_n=pxn+qyn+α−a⁰_n(pb+qM+α), ∀n= 1,2, . . . .

Notice that (ω_n⁰)n is indeed a CIFS because the denominators in (16) cannot vanish because of the regularity properties of ψ . . ..

The next theorem establishes the relationship between the attractors of the two CIFSs (ωn)n and (ω⁰_n)n.

Lemma4. Under the hypothesis of the present section the condition(13) is fulfilled if and only if q = 0.

Proof.I:The regular linear case (α=β = 0) By using (10), (14), (12), one has

(ψ◦ωn)(x, y) = p(anx+en)+q(cnx+dny+gn), r(anx+en)+s(cnx+dny+gn))

= pa_nx+px_n−pa_nb+qc_nx+qd_ny+qy_n−qc_nb−qd_nM, ra_nx+rx_n−ra_nb+sc_nx+sd_ny+sy_n−sc_nb−sd_nM

. for every (x, y)∈X andn≥1. Next, by the formulas (14), (15), (17), we have

ω_n⁰ ψ(x, y)

= a⁰_n(px+qy) +e⁰_n, c⁰_n(px+qy) +dn(rx+sy) +g⁰_n

=

= pa⁰_nx+qa⁰_ny+x_n−a⁰_nb, pc⁰_nxqc⁰_ny+rd_nx+sd_ny+y_n−c⁰_nb−d_nM . for all (x, y)∈X and n= 1,2, . . ., it follows from the last two relations that

(ψ◦ωn)(x, y)−(ω_n⁰ ◦ψ)(x, y) = (0,0), ∀(x, y)∈X, if and only if

p(a_n−a⁰_n) +qc_n

(b−x) +q(d_n−a⁰_n)(M −y) = 0 and

r(an−dn) +scn−pc⁰_n

(b−x)−qc⁰_n(M −y) = 0

for any (x, y) ∈ X and any n = 1,2, . . .. The last two equalities hold if and only if

(18) p(a_n−a⁰_n) +qc_n= 0, q(d_n−a⁰_n) = 0, ∀n and

(19) r(an−dn) +scn−pc⁰_n= 0, qc⁰_n= 0, ∀n.

Since, in (16),q = 0 impliesa⁰_n=an andc⁰_n= ^r(an−d_p^n)+scn and, consequently, the equalities (18) and (19) are valid.

We need to show thatq cannot be null. By the second equations in (18) and (19), c⁰_n = 0, a⁰_n = dn and hence the first equations by (18) and (19) will be

p(a_n−d_n) +qc_n= 0, r(a_n−d_n) +sc_n= 0, ∀n.

The above linear equations system has nontrivial solution iffps−rq = 0 which, by assumption in (14), never occurs. So, one must have q = 0.

II:The translation case (q =r= 0,p=s= 1,|α|+|β|>0)

We have ψ(x, y) = (x+α, y+β), |α|+|β|>0,a⁰_n=a_n,c⁰_n=c_n hence, by (17) and (12),

e⁰_n=en+ (1−an)α, g_n⁰ =gn+ (1−dn)β−cnα, for any n= 1,2, . . ..

In view of the preceding facts, it is simple to verify that, for each n= 1,2, . . . and any (x, y)∈X, the equality (ψ◦ωn)(x, y) = (ω_n⁰ ◦ψ)(x, y) holds, thus completing the proof.

Theorem 5. If q = 0 and A is the attractor of the associated CIFS of CSD∆, thenψ(A)represents the attractor of the associated CIFS of CSDψ(∆).

Proof.The assertion follows from Lemmas 3 and 4.

Remark. We notice that, if ψ is nonsingular, then q = 0 implies p6= 0, assuring that ψis outstanding.

InK(R²), define the scalar multiplicationby

λM =

(x, λy); (x, y)∈M , M ∈ K(R²), λ∈R.

Taking p = 1, q = r = α = β = 0 and s 6= 0 in (14), we obtain an outstanding affine transformation for a given CSD ∆ = (xn, yn)n≥0.

Applying Theorem 5 we obtain

Corollary 1. (Homogeneity) Let ∆ be a CSD and λ ∈ R. Assume that the corresponding countable interpolatory schemes of ∆and λ∆have the same vertical scaling factor d. Then λA is the attractor corresponding to the CSD λ∆ (A being the attractor of the associated CIFS of∆). In other words, the countable interpolating scheme given by σ(∆,d) is homogeneous.

In the sequel we give some methods for the approximation of the attractor of ψ(∆), ψbeing defined in (14) with q= 0. First, we need

Lemma 5. Let (X,d), (Y, δ) be two metric spaces and f : X → Y a uniformly continuous map. If (An)n, A ∈ K(X) are such that An →A, then f(A_n)→f(A), the limits being taken in the Hausdorff metrics.

Proof. Assume by reductio ad absurdum that An → A while f(An) does not converge to f(A). Then there exists ε₀ >0 such that n

∀n∈N, ∃k_n≥nsuch that h f(A_kn), f(A)

≥ε₀

(we will denote by h the Hausdorff metric in both spacesX andY). That is, for each n= 1,2, . . . ,one has

sup

y⁰∈f(A)

inf

y∈f(A_nk)δ(y⁰, y)

≥ε₀ or sup

y∈f(A_nk)

inf

y⁰∈f(A)δ(y, y⁰)

≥ε₀.

CaseI: By considering, if necessary, a subsequence, we can suppose that sup

y⁰∈f(A)

y∈f(Ainf_nk)δ(y⁰, y)

≥ ε₀ for any n ≥ 1, hence, for each n = 1,2, . . ., there exists yn ∈f(Ank) such that for any y⁰ ∈f(A) we have δ(yn, y⁰) ≥ε0. So, there exists x_n∈A_n such that

(20) δ f(xn), f(x⁰)

=δ(yn, y⁰)≥ε0,

for every x⁰ ∈ A. Now, let ε > 0, ε < ε0. By uniform continuity of f, there exists η >0 such that

∀x, x⁰ ∈X with d(x, x⁰)< η ⇒ δ f(x), f(x⁰)

< ε.

Next, A_n → A implies A_kn → A, hence there exists n_η ≥ 1 such that h(A_kn, A) < η for all n ≥ n_η. It follows that sup

x∈A_kn

xinf⁰∈Ad(x, x⁰)

< η and therefore, for anyn≥nη and anyx∈Aknthere existsx⁰ ∈Awith d(x, x⁰)< η.

Consequently, d(xn, x⁰_n) < η and hence δ f(xn), f(x⁰_n)

< ε < ε0, con- tradicting (20).

CaseII: We proceed in a similar way as in the preceding case. Suppose that sup

y∈f(A_nk)

inf

y⁰∈f(A)δ(y, y⁰)

≥ε₀ for alln≥1. Then, for everyn= 1,2, . . ., there exists x⁰₀ ∈A such that, for any x∈A_kn, one has

(21) δ f(x⁰₀), f(x)

≥ε₀.

At the same time, for an arbitrary ε >0,ε < ε0, there existsη >0 such that (22) ∀x⁰, x∈X with d(x⁰, x)< η ⇒ δ f(x⁰), f(x)

< ε.

Next,

A_n→A ⇒ A_kn →A ⇒ ∃n_η ∈N such that sup

x⁰∈A

x∈Ainf_knd(x⁰, x)

< η, ∀n≥n_η,

namely, for any x⁰ ∈ A, there exists x_kn ∈ A_kn such that d(x⁰, x_kn) < η.

In particular, taking x⁰ = x⁰₀ ∈ A and x = x_kn ∈ A_kn, we have, by (22), d(x⁰, x)< ηand henceδ f(x⁰₀), f(x)

< ε, which contradicts relation (21).

Remark. Iff is an uniformly continuous map between two metric spaces X and Y, then the corresponding set function F : K(X) → K(Y), F(A) = f(A) for anyA∈ K(X), is continuous in the Hausdorff metrics. In particular, ifX is compact, it is sufficient forf to be continuous.

For the rest of paper we suppose that ψ is an outstanding mapping with respect to the CSD ∆. The associated CIFS of ∆ will be (ω_n)n≥1 whose attractor is denoted byA and (ω_n⁰)n≥1 will be the CIFS corresponding to the

CSD ψ(∆). Assume thatω_n⁰ ◦ψ=ψ◦ωn for any n= 1,2, . . .. In the case of

∆⊂R², we consider q= 0.

In the sequel some results concerning the approximation of the attractor of the CIFS associated to the transformed CSD ψ(∆) will be formulated.

Proposition 1. The sequence of sets ψ(∆_p)

p converges in the Haus- dorff metric to the attractor of ψ(∆), where ∆1 :=S(∆), ∆p :=S(∆_p−1), for p≥2.

Proof. For any B ∈ K(X), S^p(B) = S ◦ · · · ◦ S

| {z }

p times

(B) converges (in the Hausdorff metric) to the fixed point A of S, namely the attractor of the CIFS associated to CSD ∆. Next, by the uniform continuity of ψ it follows from Lemma 5 that ψ(S^p) →

p ψ(A). The assertion comes now by applying

Lemma 3.

Proposition 2. For each N ≥1, let A_N be the attractor of the partial IFS(ω_n)^N_n=1 of CIFS associated to the countable interpolatory schemeσ(∆,Φ).

Then ψ(A_N)

N converges in the Hausdorff metric to the attractor correspond- ing to the transformed by ψ of the considered countable interpolatory scheme.

Proof.From Section 2.1 we deduce that AN →A,Abeing the attractor of (ω_n)n≥1. Then, using Lemma 5, one has ψ(A_N)→ψ(A) and the assertion comes clearly from Lemma 3.

As a consequence of Theorems 5 and Proposition 1, we get obviously Proposition 3. Let ∆ be a CSD on R² and ψ an outstanding affine transformation on X given by (14) with q = 0. If A is the attractor of the associated CIFS of σ(∆,d), then ψ(Bp)

p converges to the attractor of the associated CIFS of σ ψ(∆),d

, where, for p ≥ 1, B_p := S(B_p−1), B₀ being the compact set ∆ or B₀ :=L₀, the infinite polygonal line defined in (9).

The next proposition is a simple consequence of Theorem 1 and Lemma 3 (respectively 5). It gives a sequence of finite nonempty sets which approximate

“from inside” the attractor of the transformed CSD. This is very useful for computer graphical representation.

We will use the notation ∆_N :=

(x_n, y_n); n = 0,1, . . . , N , N ≥ 1, for the partial CSD. For each N = 1,2, . . . , we define S_N⁰ (E) :=

SN

n=1

ω_n⁰(E), S_N^0p(E) :=S_N⁰ S_N^0p−1(E)

.

Proposition 4. The sequence of sets S_N^0p(ψ(∆_N))

N,p converges with respect to the Hausdorff metric to the attractorψ(A)corresponding to CSDψ(∆).

We shall conclude with an example of application of Theorem 5.

Example. Take [a, b] = [0,1] and ∆ = (x_n, y_n)n≥0 ⊂R², where x_n:=

√n

√n+ 1, y_n:= sinπ(√ n+ 1)

√n+ 1 , n= 0,1, . . . .

We apply Theorem 5 with the vertical scaling factor d_n = 0.4 for any n ≥1 and the affine mapping

Ψ(x, y) = (x+ 1,0.1x−8.5y−1.8).

The attractor A corresponding to the countable interpolatory scheme σ =

∆,d = (dn)n

is represented in Figure 1 and the attractor ψ(A) associated with σ⁰= ψ(∆),d= (dn)n

appears in Figure 2.

Fig. 1

Fig. 2

REFERENCES

[1] M.F. Barnsley, Fractal functions and interpolations. Constructive Approximation 2 (1986), 303–329.

[2] M.F. Barnsley,Fractals Everywhere. Academic Press, 1988.

[3] Lj.M. Koci´c and A.C. Simoncelli, Notes on fractal interpolation. Novi Sad J. Math.30 (2000),3, 59–68.

[4] J. Hutchinson,Fractals and self-similarity. Indiana Univ. J. Math.30(1981), 713–747.

[5] N.A. Secelean,Countable iterated function systems. Far East J. Dynamical Systems3(2) (2001), 149–167.

[6] N.A. Secelean,Measure and Fractals. University “Lucian Blaga” of Sibiu Press, 2002.

[7] N.A. Secelean,The fractal interpolation for countable systems of data. Beograd University Publikacije Electrotehn. Fak. Ser. Matematika14(2003), 11–19.

[8] N.A. Secelean, Approximation of the attractor of a countable iterated function system.

General Mathematics3(2009).

Received 3 June 2010 “Lucian Blaga” University of Sibiu Department of Mathematics

Sibiu, Romania nicolae.secelean@ulbsibiu.ro