Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions. P. Bouboulis, L. Dalla and M. Kostaki-Kosta

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1 BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume (79 95) Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions P. Bouboulis L. Dalla and M. Kostaki-Kosta Received 3 December 26 Accepted 3 September 27 Abstract This paper approaches the Hermite interpolation problem using fractal interpolation procedures. We generalise some theorems provided by Barnsley and others regarding the differentiability of fractal interpolation functions when recurrent iterated function systems are used. In addition we generalise the construction given by Navascués and Sebastián in [] and we provide a construction of smooth (C ) fractal surfaces using C Hermite Fractal Interpolation Functions. Keywords: dimension Fractals recurrent iterated function systems fractal interpolation fractal. Introduction The most commonly used technique for the construction of fractal sets defined in a complete metric space X is the use of Iterated Function Systems (IFS). IFS are sets of contractive mappings that take effect on H(X) the space containing all the non-empty compact subsets of X equipped with the Hausdorff metric h. This method was introduced by Barnsley and Demko in [3] and generalised by Barnsley Elton and Hardin in [4] to include Recurrent Iterated Function Systems (RIFS) using stochastic matrices with probabilities. IFS and RIFS are able to produce very complicated sets using only a handful of mappings. A special construction of IFS presented in [5] and [4] produces continuous functions whose fractal dimension is greater than. These are called Fractal Interpolation Functions (FIF) and were used especially in signal processing. Recently it was shown that FIF generalise Hermite interpolation polynomials (see []). In spite of the fact that this construction was extensively studied (see for example [] [2] [9]) it has remained restricted in the case of -dimensional data points. Several attempts were made to generalise the notion on R 2 but only in very specific cases (see []). The 79

2 8 P. Bouboulis L. Dalla and M. Kostaki-Kosta most general construction involving Fractal Interpolation Surfaces was published recently (see [7]) but still it involves interpolation points that are confined in some manner. In [8] FIF were used to construct Fractal Interpolation Surfaces on arbitrary points defined on grids. In this paper our intention is to provide a similar construction that gives rise to smooth (i.e. C ) surfaces. 2. Iterated Function Systems An Iterated Function System {X; w N } is defined as a pair of a complete metric space (X ρ) together with a finite set of continuous contractive mappings w i : X X with respective contraction factors s i for i = 2...N (N 2). The attractor of an IFS is the unique set E for which E = lim k W k (A ) for every starting compact set A where N W(A) = w i (A) for all A H(X) i= and H(X) is the complete metric space of all nonempty compact subsets of X with respect to the Hausdorff metric h (for the definition of the Hausdorff metric and properties of H(X) h see [6]). A simple example of an IFS defined on R 2 is the one that produces the well-known Sierpinski s Triangle (see figure (b)) which consists of the three mappings: ( ) ( ( ) ( ) x.5 x x w = w y.5) y 2 = y ( ) ( ) ( ) ( ) x.5 x 5 w 3 = +. y.5 y ( ( ) (.5 x +.5) y 5) The attractor of the following IFS looks like a natural fern (see (a)). ( ) ( ( ) x x w = y.6) y ( ) ( x.2.26 w 3 = y ( + ) ) ( x y) + ( ) ( ( ) x.85.4 x w 2 = y.4.85) y ( ) ( ) ( ( x.5.28 x w.6 3 = y.26.24) y ( +.6 ) + ) (.44 As we mentioned earlier a more general concept that allows the construction of even more complicated sets is that of the Recurrent Iterated Function System or RIFS for short which consists of the IFS {X; w N } together with an irreducible row-stochastic matrix P = (p νµ ) N (p νµ [ ] : ν µ =...N) such that N p νµ = ν =...N. () µ= ).

3 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions (a) (b) Figure : Two known attractors that arise from IFS. (a) a fern (b) Sierpinski s Triangle. The recurrent structure is given by the (irreducible) connection matrix C = (cνµ )N which is defined by if pµν > cνµ = if pµν = where ν µ = 2... N. The transition probability for a certain discrete time Markov process is pνµ which gives the probability of transfer into state µ given that the process is in state ν. Condition () says that whichever state the system is in (say ν) a set of probabilities is available that sum to one and describe the possible states to which the system transits at the next step. In this case the construction of the contractive map W needs a little more effort. First we define the mappings Wij wi (A) : H(X) H(X) with Wij (A) = pji > pji = for all A H(X) and the metric space H (X) = H(X)N = H(X) H(X) H(X) equipped with the metric B A A2 B2 h.. = max{h(ai Bi ); i = 2... N }..... AN BN (2)

4 82 P. Bouboulis L. Dalla and M. Kostaki-Kosta Then H(X) h is a complete metric space. The map W is now defined by A W W 2... W N A A 2 W : H(X) H(X) : W. = W 2 W W 2N... A 2. A N W N W N2... W NN A N w (A j ) j I() w 2 (A j ) = j I(2). w N (A j ) j I(N) where I(i) = {j : p ji > } for i = 2...N. If w i are contractions then W is a contraction and there is E = (E E 2...E N ) t H(X) such that W(E) = E and E i = j I(i) w i(e j ) for i = 2... N. Let A H(X). We define the sequences {A n } n N in H(X) and {A n } n N in H(X) as follows: A = (A A... A) t A n = W(A n ) and A n = N i= (A n) i for n N where A n = ((A n ) (A n ) 2...(A n ) N ). Then the set G = N i= E i is called the attractor of the RIFS {X; w N P }. Evidently G = lim n A n. We emphasize that the attractor of a RIFS depends not only from the corresponding IFS but also from the stochastic matrix. For example the following IFS equipped with different stochastic matrices produces different attractors (see figure 2). ( ) ( ( ) ( ( ) ( ( ) ( ) x.5 x x.5 x w = + w y.5) y ) 2 = + (3) y.5) y 5 ( ) ( ( ) ( ) ( ) ( ( ) ( ) x.5 x 5 x.5 x 5 w 3 = + w y.5) y 4 = +. (4) y.5) y 5 3. Fractal Interpolation Functions In this section we briefly describe the construction of fractal interpolation functions based on RIFSs as we will use it in our method. Let X = [ ] R and = {(x i y i ) : i =...N} be an interpolation set with N+ interpolation points such that = x < x < < x N =. The interpolation points divide [ ] into N intervals I i = [x i x i ] i =...N which we call domains. In addition let ˆ = {(ˆx j ŷ j ) : j =... M} be a subset of such that = ˆx < ˆx < < ˆx M =.

5 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions 83 P = 2 (a) (b) (c) P = P = Figure 2: The attractors of the IFS defined by equations 3-4 equipped with various stochastic matrices. We also assume that for every j =...M there is at least one i such that ˆx j < x i < ˆx j+. Thus the points of ˆ divide [ ] into M intervals J j = [ ˆx j ] j =...M which we call regions. Finally let J be a labelling map such that J: { 2... N} { 2... M} with J(i) = j. Let x i x i = δ i i = 2...N and ˆx j = ψ j j = 2...M. It is evident that each region contains an integer number of domains. In the special case where the interpolation points are equidistant (that is x i x i = δ i = 2... N and ˆx j = ψ j = 2...M) each region contains exactly α = ψ/δ N domains. We define N mappings of the form: ( ) ( ) x Li (x) w i = for i = 2...N (5) y F i (x y) where L i (x) = a i x+b i F i (x y) = s i y+q i (x) and q i (x) is a polynomial. Each map w i is constrained to map the endpoints of the region J J(i) to the endpoints of the domain I i (see figure 3). That is ) (ˆxj w i = ŷ j ( ) ) xi (ˆxj w y i = i ŷ j ( xi y i ) for i = 2...N. (6) Vertical segments are mapped to vertical segments scaled by the factor s i. The parameter s i is called the vertical scaling factor or the contraction factor of the map w i. It is easy to show that if s i < then there is a metric d equivalent to the Euclidean metric such that w i is a contraction (i.e. there is ŝ i : ŝ i < such that

6 84 P. Bouboulis L. Dalla and M. Kostaki-Kosta (p nm) = ¼ /2 /2 /2 /2 /2 /2 /2 /2 ½ ¼ (c nm) = ½ V=( 2 2 ) Figure 3: In the above figure the set consists of five interpolation points while the set ˆ consists of three points. The stochastic matrix the connection matrix and the connection vector are shown below the figure. d (w i ( x) w i ( y)) ŝ i d ( x y) x y R 2 see [6]). The N N stochastic matrix (p nm ) N is defined by the labelling function J as follows: { p nm = γ n if I n J J(m) otherwise. where γ n is the number of positive entries of the line n n = 2...N. This means that p nm is positive if the transformation L m maps the region containing the nth domain (i.e. I n ) to the mth domain (i.e. I m ). If we take a point in I n R n =...N we say that we are in state n. The number p nm shows the probability of applying the map w m to that point so that the system transits to state m. Sometimes it is more efficient to describe the matrix P through the connection matrix C or the connection vector V which are defined as follows: { pmn > c nm = otherwise V = (J() J(2)...J(N)).

7 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions Figure 4: The two FIFs shown above interpolate the points of the same set (consisting of six points). The difference is due to the selection of two distinct stochastic matrices. Next we consider C([x x N ]) where φ = max{ φ(x) x [x x N ]} and the complete metric subspace F = {g C([x x N ]) : g(x i ) = y i for i =... N}. The Read-Bajraktarevic operator T ˆ : F F is defined as follows ( (T ˆ g)(x) = F i L i (x) g ( L i (x) )) for x I i i = 2...N. It is easy to verify that T ˆ g is well defined and that T ˆ is a contraction with respect to the metric. According to the Banach fixed-point theorem there exists a unique f F such that T ˆ f = f. If f is any interpolation function and f n = T n ˆ f where T n ˆ = T ˆ T ˆ T ˆ then (f n) n N converges uniformly to f. The graph of the function f is the attractor of the RIFS {X w N (p ij ) N } associated with the interpolation points (see [6]). Note that f interpolates the points of for any selection of the parameters of the polynomials p i that satisfies (6). We will refer to a function of this nature as Recurrent Fractal Interpolation Function (RFIF). In the case where all the elements of the stochastic matrix are equal to (i.e. we have an IFS instead of a RIFS) the function will be simply referred to as Fractal Interpolation Function (FIF). We emphasize that a RFIF satisfies f(l i (x)) = s i f(x) + q i (x) (7) for all x I i i = 2...N. The most extensively studied case is that where q i (x) = c i x + f i x I i which means that w i are affine: ( ) x w i = y ( Li (x) F i (x y) ) = ( ) ( ( ) ai x ei + for i = 2...N. (8) c i s i y) f i

8 86 P. Bouboulis L. Dalla and M. Kostaki-Kosta The FIF that corresponds to the above RIFS is called affine FIF. From eq. (6) four linear equations arise which can always be solved for a i c i e i f i in terms of the coordinates of the interpolation points and the vertical scaling factor s i. Thus once the vertical scaling factor s i for each map has been chosen the remaining parameters may be easily computed (see [6]). 3.. The integral of a RFIF In [2] it is shown that the integral of a FIF defined from an IFS is also a FIF defined from a different IFS. Here we will show that the same is true in the case of the RFIF. The following theorems are extensions of the ones presented in [2]. Proposition 3. Let f be a RFIF constructed from the RIFS {R 2 ; w N P } where ( ) x w i = y ( ) Li (x) s i y + q i (x) with s i < a i i = 2...N and associated with the interpolation points = {(x i y i ) : i =...N} the points ˆ = {(ˆx j ŷ j ) : j =...M} and the labelling map J as given above. If f(x) = ỹ + f(t)dt x then the function f is the RFIF constructed from the RIFS {R 2 ; w N P } with where s i = a i s i ( ) x w i = y ( ) Li (x) s i y + q i (x) q i (x) = ỹ i a i s iˆỹ j + a i q i (t)dt and P is defined by the function J as mentioned above. for i = 2... N j = J(i) and ˆỹ j = ỹ I(j) where I(j) = i iff x i = ˆx j. The function f is associated with the interpolation points = {(x i ỹ i ) : i =...N} the points ˆ = {(ˆxj ˆỹ j ) : j =...M} and the labelling map J. The value ỹ is arbitrary while the values ỹ... ỹ N are computed as solutions of the linear system i = 2...N j = J(i). ỹ i = ỹ i + a i s i (ˆỹ j ˆỹ j ) + a i ˆxj q i (t)

9 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions 87 Proof. Li(x) f(l i (x)) = ỹ + f(t)dt = ỹ + x = ỹ i + Li(x) x i f(t)dt. i Substituting with t = L i (u) dt = a i du we have that f(l i (x)) = ỹ i + a i In addition if we use the relation (7) we have that f(l i (x)) = ỹ i + a i s i x f(t)dt + Li(x) x i f(l i (u))du with j = J(i). f(t)dt + a i q i (t)dt ˆxj f(t)dt = ỹ i + a i s i f(t)dt a i s i f(t)dt + a i q i (t)dt x x = ỹ i + a i s i ( f(x) ỹ ) a i s i (ˆỹ j ỹ ) + a i q i (t)dt f(l i (x)) = a i s i f(x) + ỹi a i s iˆỹ j + a i q i (t)dt (9) for all x I i. Thus f is a RFIF as stated by the theorem. Substituting x = ˆx j in (9) we obtain the linear system ỹ i = ỹ i + a i s i (ˆỹ j ˆỹ j ) + a i ˆxj q i (t) for i = 2... N j = J(i). Corollary 3. Let f f be two RFIF produced by the RIFS {R 2 ; w N P } and {R 2 ; w N P } respectively where ( ) ( ) ( ) ( ) x Li (x) x Li (x) w i = w y s i y + q i (x) i = y s i y + q i (x) with s i < s i < a i for i = 2... N and associated with the points of the sets ˆ and ˆ and the labelling map J as described above. Then f = f if and only if s i = a i s i and q i (x) = a iq i (x) for all x I i i = 2...N. ()

10 88 P. Bouboulis L. Dalla and M. Kostaki-Kosta Proof. The if part is immediate from the proposition 3.. For the converse we have: f(x i ) f(x i ) = ỹ i ỹ i = F i (ˆx j ˆỹ j ) F i ( ˆỹ j ) = a i s iˆỹ j + q i (ˆx j ) a i s iˆỹ j q i ( ) ˆxj = a i s i (ˆỹ j ˆỹ j ) + a i q i (t)dt. Thus ỹ i ỹ i i = 2...N satisfy the relations of proposition 3.. Therefore f is the RFIF constructed from the RFIS {R 2 w N P }. Due to () f = f. For a function g we symbolize g () = g and g (k) as its k-th order derivative. In addition we consider C n ([ ]) as the space of the functions that have continuous n-th order derivative equipped with the norm f = f + f + + f (n). The following theorem may be easily deduced from proposition 3.. Theorem 3. Consider the RIFS {R 2 w N P } whose attractor is the graph of a RFIF associated with the data points ˆ and the labelling map J where ( ) ( ) x Li (x) w i = y s i y + q i (x) s i < a n i q i C n ([ ]) for i = 2...N. We define F ki (x y) = s iy + q (k) i a k i (x). If for any k =... n each one of the 2N 2N linear systems y ki = F ki ( ŷ kj ) = a iŷ kj + q (k) i ( ) y ki = F ki (ˆx j ŷ kj ) = a iŷ kj + q (k) i (ˆx j ) with ŷ kj = y ki(j) I(j) = i iff ˆx j = x i j = J(i) i = 2...N has a unique solution for y ki then the RFIF f C n ([ ]) and f (k) is the RFIF defined by the RIFS {R 2 ; w k N P } where ( ) ( ) x Li (x) w ki = y F ki (x y) for k =...n. a k i a k i

11 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions Hermite Fractal Interpolation Functions Consider an interval [a b]. The Hermite interpolation problem is the search for a polynomial q of degree 2p + which satisfy q (k) (a) = a k q (k) (b) = b k where a k b k are given real numbers for k =...p. It has been proven that this problem has always a unique solution. The corresponding polynomial q is called Hermite interpolation polynomial. A more general definition is that of the Hermite interpolation function. Let D = {x < x < < x N } be a given partition of an interval [x x N ] where as usual I i = [x i x i ]. Then the space of the Hermite interpolation functions of order p denoted as H p+ D (p N) is defined as follows H p+ D = {φ : [x x N ] R; φ C p ([x x N ]) φ Ii P 2p+ i = 2... N} where the space P 2p+ is composed from the polynomials of degree at most 2p +. In order to approximate a given function y C p ([x x N ]) with a function φ H p+ D it is sufficient to choose the polynomials q i = φ Ii such that q (k) i (x i ) = y (k) (x i ) p (k) i (x i ) = y (k) (x i ). The following theorem generalizes the concept of Hermite interpolation using RFIF. We note that in [] Navascués and Sebastián gave a similar theorem using FIFs. Their work may be considered as a special case of theorem 3.2. Theorem 3.2 Consider x < x < < x N a partition of the interval [x x N ] (N N N 2) and ˆx < ˆx < < ˆx M (M N) a subset of it such that ˆx = x ˆx M = ˆx N. Consider also a mapping J : {... N} {... M} a set of numbers D y = {y ki : k =...p i =...N} and a i = (x i x i )/(ˆx j ) a i <. If the real numbers s s 2...s N satisfy s i < a p i i = 2... N then there is a unique RFIF f C p such that f (k) (x i ) = y ki for all k =... p i =...N. The function f is constructed as the attractor of a RIFS {R 2 ; w N P } with ( ) ( ) x Li (x) w i = y s i y + q i (x) where q i are polynomials of degree at most 2p +. Proof. Again we define the mapping I : {... M} {... N} as follows: I(j) = i iff ˆx j = x i. Using this mapping we construct the set ˆD y = {ŷ kj ; k =

12 9 P. Bouboulis L. Dalla and M. Kostaki-Kosta...p j =... M} such that ŷ kj = y ki(j). The coefficients of the polynomial q i are computed as solutions of the linear system F ki ( ŷ kj ) = s iŷ kj + q (k) i ( ) a k = y ki i F ki (ˆx j ŷ kj ) = s iŷ kj + q (k) i (ˆx j ) a k = y ki i for k =... p i = 2...N. We may rewrite this linear system as (q i L i (q i L i ) (k) (x i ) = y ki s iŷ kj a k i ) (k) (x i ) = y ki s iŷ kj k =... p i = 2... N. Thus (q i L i ) is an Hermite polynomial of degree p for all i = 2...N. This fact ensures that the linear system has a unique solution. In addition the conditions of theorem 3. are satisfied therefore f C p ([x x N ] n ). We note that in the special case where s i = i = 2...N the RFIF f is identical to the classical Hermite interpolation function (the proof is straightforward). a k i (a) (b) Figure 5: Two Hermite RFIF that interpolate the same set of data using (a) non zero contraction factors (b) zero contractions factors. 4. Fractal Interpolation Surfaces derived from RFIF The construction of fractal surfaces has drawn the attention of many researchers. Massopust was the first who gave a valid construction in [] using data points

13 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions 9 placed on a triangular domain. A lot of other attempts followed however most of them need strongly restricted data (for example the interpolation points that are placed on the boundary of the domain need to be colinear). The most general case was given in [7]. Here we describe a method that uses Hermite Fractal Interpolation Functions of order 3 to construct C Fractal Interpolation Surfaces on a rectangular grid of arbitrary interpolation points. Consider the interpolation points = {(x i y j z ij ) : i =...N; j =...M} [ ] [ ] R with = x < x < < x N = = y < y < < y M = and x i x i = δ i i =... N y j y j = δ j j =...M. Let S = {s s 2... s N } S = { s s 2... s M } be two sets of contraction factors and let P = (p nm ) N P = ( pnm ) M be two stochastic matrices with dimensions N N and M M respectively. Also let ˆ = {(ˆx k ŷ l ẑ kl ) : k =...K; l =...L} be a subset of such that ˆx = ˆx K = ŷ = ŷ L = and ˆx k ˆx k = ψ k ŷ l ŷ l = ψ l k =... K l =... L. Let J and J be defined as in section 3 associated with the matrices P and P respectively with J(i) = k J(j) = l. The points {x x... x N } divide [ ] into N domains I I 2... I N while the points {y y... y M } divide [ ] into M domains Ĩ Ĩ2... ĨM. Consequently the points {ˆx ˆx... ˆx K } divide [ ] into K regions J J 2... J K while the points {ŷ ŷ...ŷ L } divide [ ] into L regions J J 2... J L. In addition we define the mappings I : {... K} {... N} Ĩ : {... L} {... M} such that ˆx k = x I(k) and ŷ l = yĩ(l). We consider arbitrary continuous functions u i that interpolate the sets xi = {(x i y j z ij ) : j =...M} for i =... N. Then for y [ ] we construct an affine RIFS associated with the interpolation points y = {(x i y u i (y)) : i =... N} ˆ y = {ˆx k y u I(k) (y)) k =... K} the set of contraction factors S together with the matrix P which produce an affine RFIF f y : [ ] R (see figure 6). We define the function F : [ ] [ ] R such that F(x y) = f y (x). Similarly we consider arbitrary continuous functions v j that interpolate the sets yj = {(x i y j z ij ) : i =...N} for j =...M. As before for x [ ] we construct an affine RIFS associated with the interpolation points x = {(x y j v j (x)) : j =...M} ˆ x = {(x ŷ l vĩ(l) (x)) l =...L} the set of contraction factors S together with the matrix P which produce an affine FIF f x : [ ] R. Thus we define the function F : [ ] [ p] R such that F(x y) = f x (y). The functions F F are continuous functions that interpolate the data set (see [8]).

14 92 P. Bouboulis L. Dalla and M. Kostaki-Kosta (a) (b) (c) (d) Figure 6: An example of the construction of the function F is shown. (a) The points of where N = M = 8 p =. (b) The nine interpolation functions u u...u 8 (c) One of the FIFs f y (green line) (d) The graph of the function F. Using a similar method we may proceed in a construction of fractal interpolation surfaces of class C. In this case we consider arbitrary C functions u i that interpolate the sets xi = {(x i y j z ij ) : j =... M} for i =... N. We also consider some other arbitrary continuous functions u i defined on [ ] that satisfy the Lipschitz condition. These functions will be used as the x-partial derivative of the constructed surface. Then for y [ ] we construct a Hermite RFIF of order associated with the interpolation points y = {(x i y u i (y) u i (y)) : i =...N} ˆ y = {(ˆx k y u I(k) (y) u I(k) (y)) : k =... K} the set of contraction factors S together with the matrix P. The corresponding RIFS is {R 2 ; w y N P } where w yj are of the form ( ) x w yj = z ( ) ai x + b i s i z + q yi (x) for i = 2... N as defined in section 3.2. We define the function F : [ ] [ ] R such that F(x y) = f y (x).

15 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions 93 We can easily show that F is a C function. To this end we construct a RIFS whose attractor is identical to F. We begin with the set = {(x i y j z ij ) : i =...N; j =...M} [ ] [ ] R with = x < x < < x N = = y < y < < y M =. We define a subset ˆ of as follows ˆ = {(ˆx k y j z I(k)j ) : k =...K; j =...M}. The corresponding domains are D i = [x i x i ] [y y M ] i = 2...N and the regions are R k = [ˆx k ˆx k ] [y y M ]. The mappings w i are constructed as follows w i x y = a ix + b i L i (x) y = y z s i z + Q i (x y) s i z + Q i (x y) for i = 2...N (x y) D i such that Q i (x y) = q yi (x) where q yi are the polynomials that are used in the RIFS whose attractor is the Hermite RFIF f y. The fact that q yi are Hermite type polynomials of order 3 such that q yi (x i ) = u i (y) s i u k (y) q yi (x i ) = u i (y) s i u k (y) q yi (x i ) = u i (y) s iu k (y) a i q yi (x i) = u i (y) s iu k (y) a i where k = J(i) for all y [ ] ensures that Q i is a C function for all i = 2...N. We define the set F = {f C ([ ] 2 ) : f xi [] = u i f x x i [] = u i i =... N} and the operator T : F F such that Tf(x y) = s i f(l i (x) y) + Q i (L i (x) y) for all (x y) D i i = 2...N. We may easily show that T is well defined (based on the construction of f y and the fact that Q i are C ) and that it is a contraction. Therefore it has a unique fixed point which is identical to F. Hence F is a C function. In figure 7 we used sets of the form = {(x i y j z ij zij z ij ) : i =...N; j =... M} where the values zij z ij are the values of the x and y partial derivatives. We constructed the functions u i as Hermite RFIF that are associated with the data sets {(x i y j z ij zij ) : j =...M} for i =...N. The functions u i were constructed as the linear interpolation functions passing through the points {(x i y j zij ) : j =... M} for i =...N.

16 94 P. Bouboulis L. Dalla and M. Kostaki-Kosta (a) (b) (c) (d) Figure 7: An example of the construction of the smooth function F is shown. (a) The nine interpolation functions u u... u 8 (b) One of the FIFs f y (green line) (c) The graph of the C surface (d) Another example of a C surface using a different set of data. References. B. F. Barnsley J. Elton D. Hardin P. Massopust Hidden Variable Fractal Interpolation Functions SIAM J. Math. Anal. (2) B. F. Barnsley A. N. Harrington The calculus of Fractal Interpolation Functions J. Approx. Theory (57) M. F. Barnsley S. Demko Iterated function systems and the global construction of fractals Proc. Roy. Soc. London Ser A (399) M. F. Barnsley J. H. Elton D. P. Hardin Recurrent Iterated Function Systems Constr. Approx. (5) M. F. Barnsley Fractal functions and interpolation Constr. Approx. (2) M. F. Barnsley Fractals everywhere 2nd ed. Academic Press Professional P. Bouboulis L. Dalla V. Drakopoulos Construction of Recurrent Bivariate Fractal Interpolation Surfaces and Computation of their Box-Counting Dimension J. Approx. Theory (4) P. Bouboulis L. Dalla Fractal interpolation surfaces derived from fractal interpolation functions J. Math. Anal. Appl. (to appear).

17 Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions G. C. Donovan J.S. Geronimo D. P. Hardin P. R. Massopust Construction of Orthogonal Wavelets Using Fractal Interpolation Functions SIAM J. Math. Anal. (27) Peter R. Massopust Fractal Functions Fractal Surfaces and Wavelets Academic Press994.. M. A. Navascues and M. V. Sebastian Generalization of Hermite functions by fractal interpolation J. Approx. Theory (3) P. Bouboulis Department of Informatics and Telecommunications University of Athens Panepistimiopolis Athens Greece bouboulis@di.uoa.gr L. Dalla Department of Mathematics University of Athens Panepistimiopolis Athens Greece ldalla@math.uoa.gr M. Kostaki-Kosta Department of Mathematics University of Athens Panepistimiopolis Athens Greece mkostakikosta@yahoo.gr