Báo cáo hóa học: "SMOOTH FRACTAL INTERPOLATION" ppt

SMOOTH FRACTAL INTERPOLATION M. A. NAVASCU ´ ES AND M. V. SEBASTI ´ AN Received 12 December 2005; Revised 5 May 2006; Accepted 14 June 2006 Fractal methodology provides a general frame for the understanding of real-world phe- nomena. In particular, the classical methods of real-data interpolation can be generalized by means of fr actal techniques. In this paper, we describe a procedure for the construc- tion of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a fam- ily of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpo- lation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in Ᏼ 2 [a,b] is proven. Copyright © 2006 M. A. Navascu ´ es and M. V. Sebasti ´ an. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. 1. Introduction Fractal interpolation techniques provide good deterministic representations of complex phenomena. Barnsley [2, 3] and Hutchinson [8] were pioneers in the use of fractal func- tions to interpolate sets of data. Fractal interpolants can be defined for any continuous function defined on a real compact interval. This method constitutes an advance in the techniques of approximation, since all the classical methods of real-data interpolation can be generalized by means of fractal techniques (see, e.g., [5, 10, 12]). Fractal interpolation functions are defined as fixed points of maps between spaces of functions using iterated function systems. The theorem of Barnsley and Harr ington (see [4]) proves the existence of differentiable fractal interpolation functions. However, in some cases, it is difficult to find an iterated funcion system satisfying the hypothe- ses of the theorem, mainly whenever some specific boundary conditions are required (see [4]). In this paper, we describe a very general way of constructing smooth fractal Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 78734, Pages 1–20 DOI 10.1155/JIA/2006/78734 2 Smooth fractal interpolation functions with the help of Hermite osculatory polynomials. The proposed method solves the problem with the help of a classical interpolant. The fractal solution is unique and the constructed interpolant preserves the prefixed boundar y conditions. The procedure has a computational cost similar to that of the classical method. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. Each element of the class preserves the smoothness and the boundary conditions of the original. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. Assuming some a dditional conditions on the scaling factors, the convergence is also preserved. In the last section, a set of interpolating mappings associated to a cubic spline is de- fined, in the general frame of functions whose second derivative has an integrable square. In particular, the density of fractal cubic splines in Ᏼ 2 [a,b]isproven. 2. Construction of smooth fractal interpolants 2.1. Fractal interpolation functions. Let t 0 <t 1 < ··· <t N be real numbers, and I = [t 0 ,t N ] ⊂ R the closed interval that contains them. Let a set of data points {(t i ,x i ) ∈ I × R : i = 0,1,2, ,N} be given. Set I n = [t n−1 ,t n ]andletL n : I → I n , n ∈{1,2, ,N}, be contractive homeomorphisms such that L n  t 0  = t n−1 , L n  t N  = t n , (2.1)   L n  c 1  − L n  c 2    ≤ l   c 1 − c 2   ∀ c 1 ,c 2 ∈ I (2.2) for some 0 ≤ l<1. Let −1 <s n < 1, n = 1,2, ,N,andF = I × R,letN be continuous mappings, let F n : F → R be given satisfying F n  t 0 ,x 0  = x n−1 , F n  t N ,x N  = x n , n = 1,2, ,N, (2.3)   F n (t,x) − F n (t, y)   ≤   s n   | x − y|, t ∈ I, x, y ∈ R. (2.4) Now define functions w n (t,x) =  L n (t),F n (t,x)  ∀ n = 1,2, ,N, (2.5) and consider the following theorem. Theorem 2.1 [2, 3]. The iterated function system (IFS) {F,w n : n = 1,2, ,N} defined above admits a unique attractor G. G is the graph of a continuous function f : I → R which obeys f (t i ) = x i for i = 0,1,2, ,N. The prev ious function f is called a fractal interpolation function (FIF) corresponding to {(L n (t),F n (t,x))} N n =1 . f : I → R is the unique function satisfying the functional equa- tion f  L n (t)  = F n  t, f (t)  , n = 1,2, ,N, t ∈ I, (2.6) M. A. Navascu ´ es and M. V. Sebasti ´ an 3 or f (t) = F n  L −1 n (t), f ◦ L −1 n (t)  , n = 1,2, ,N, t ∈ I n =  t n−1 ,t n  . (2.7) Let Ᏺ be the set of continuous functions f :[t 0 ,t N ] → R such that f (t 0 ) = x 0 , f (t N ) = x N .DefineametriconᏲ by d( f ,g) =f − g ∞ = max    f (t) − g(t)   ; t ∈  t 0 ,t N  ∀ f ,g ∈ Ᏺ. (2.8) Then (Ᏺ,d) is a complete metric space. Define a mapping T : Ᏺ → Ᏺ by (Tf)(t) = F n  L −1 n (t), f ◦ L −1 n (t)  ∀ t ∈  t n−1 ,t n  , n = 1,2, ,N. (2.9) Using (2.1)–(2.4), it can be proved that (Tf)(t) is continuous on the interval [t n−1 ,t n ] for n = 1,2, ,N and at each of the points t 1 ,t 2 , ,t N−1 . T is a contractive mapping on the metric space (Ᏺ,d), Tf− Tg ∞ ≤|s| ∞  f − g ∞ , (2.10) where |s| ∞ = max{|s n |; n = 1,2, ,N}.Since|s| ∞ < 1, T possesses a unique fixed point on Ᏺ, that is to say, there is f ∈ Ᏺ such that (Tf)(t) = f (t)forallt ∈ [t 0 ,t N ]. This func- tion is the FIF corresponding to w n . The most widely studied fractal interpolation functions so far are defined by the fol- lowing IFS: L n (t) = a n t + b n , F n (t,x) = s n x + q n (t) (2.11) with a n =  t n − t n−1   t N − t 0  , b n =  t N t n−1 − t 0 t n   t N − t 0  . (2.12) s n is called the vertical scaling factor of the iterated function system and s = (s 1 , s 2 , ,s N ) is the scale vector of the transformation. If q n (t) are linear for t ∈ [t 0 ,t N ]then the FIF is called affine (AFIF) (see [2, 11]). The cubic FIF (see [10, 13]) is constructed using q n (t)asacubicpolynomial. In many cases, the data are evenly sampled, then h = t n − t n−1 , t N − t 0 = Nh. (2.13) In the particular case, s n = 0foralln = 1,2, ,N,then F n (t,x) = q n (t) (2.14) and f (t) = q n ◦ L −1 n (t)forallt ∈ I n . 4 Smooth fractal interpolation 2.2. Differentiable fractal interpolation functions. In this section, we study the con- struction of smooth fractal interpolation functions. The theorem of Barnsley and Har- rington [4] proves the existence of di fferentiable FIFs and gives the conditions for their existence. We look for IFS satisfying the hypotheses of this theorem. Theorem 2.2 (Barnsley and Harrington [4]). Let t 0 <t 1 <t 2 < ··· <t N and L n (t), n = 1,2, ,N, the affine function L n (t) = a n t + b n satisfying the expressions (2.1)-(2.2). Let a n = L  n (t) = (t n − t n−1 )/(t N − t 0 ) and F n (t,x) = s n x + q n (t), n = 1,2, ,N,verifying(2.3)-(2.4). Suppose for some integer p ≥ 0 , |s n | <a p n ,andq n ∈ Ꮿ p [t 0 ,t N ]; n = 1,2, ,N.Let F nk (t,x) = s n x + q (k) n (t) a k n , k = 1, 2, , p, (2.15) x 0,k = q (k) 1  t 0  a k 1 − s 1 , x N,k = q (k) N  t N  a k N − s N , k = 1, 2, , p. (2.16) If F n−1,k (t N ,x N,k ) = F nk (t 0 ,x 0,k ) with n = 2,3, ,N and k = 1,2, , p, then  L n (t),F n (t,x)  N n =1 (2.17) determines a FIF f ∈ Ꮿ p [t 0 ,t N ] and f (k) is the FIF determined by  L n (t),F nk (t,x)  N n =1 (2.18) for k = 1, 2, , p. From here on, we consider a uniform partition in order to simplify the calculus. In this case, a n = 1 N . (2.19) If we consider a generic polynomial q n , for instance, the equality proposed in the the- orem implies the resolution of systems of equations. Sometimes the system has no solu- tion, mainly whenever some boundary conditions are imposed on the function (see [4]). We will proceed in a different way. In order to define an IFS satisfying Theorem 2.2,we consider the following mappings: L n (t) = a n t + b n , F n (t,x) = s n x + q n (t), (2.20) where q n (t) = g ◦ L n (t) − s n b(t), (2.21) g is a continuous function satisfying g  t i  = x i , i = 0,1, ,N, (2.22) M. A. Navascu ´ es and M. V. Sebasti ´ an 5 and b(t) is a real continuous function, b = g,suchthat b  t 0  = x 0 , b  t N  = x N . (2.23) The IFS satisfies the hypotheses (2.1)–(2.5) of Barnsley’s theorem (see [11]). In [11], we proved some properties of this fractal function. Definit ion 2.3. Consider g ∈ Ꮿ(I) and a partition of the closed interval I = [t 0 ,t N ], Δ : t 0 <t 1 < ···<t N .Letb be defined as before and let s = (s 1 , s N ) be the scaling vector of the IFS defined by (2.11)and(2.21). The corresponding FIF g s Δb , g s b , g s Δ or simply g s is called s-fractal function of g with respect to the partition Δ and the function b. Theorem 2.4 (see [11]). The s-fractal function g s b of g with respect to Δ and b satisfies the inequality   g s b − g   ∞ ≤ | s| ∞ 1 −|s| ∞ g − b ∞ , (2.24) where |s| ∞ = max 1≤n≤N {|s n |}. Besides, g s b interpolates to g, that is to say, g s b  t n  = g  t n  ∀ n = 0,1, ,N. (2.25) Consequence 2.5. If s = 0, then g s b = g. Remark 2.6. By (2.7), for all t ∈ I n , n = 1,2, ,N, g s b (t) = g(t)+s n  g s b − b  ◦ L −1 n (t). (2.26) The first step is to check which conditions should satisfy b(t) in order to fulfill the hypotheses of the theorem of Barnsley and Harrington. Let us consider p ≥ 0, |s n | < 1/N p ,andq n (t) ∈ Ꮿ p [t 0 ,t N ], n = 1,2, ,N. The prescribed conditions are F n−1,k  t N ,x N,k  = F nk  t 0 ,x 0,k  , (2.27) where n = 2, 3, , N, k = 1,2, , p. We have from the assumptions (2.15)ofthetheorem, F nk (t,x) = s n x + q (k) n (t) a k n . (2.28) In this particular case, q n (t) = g ◦ L n (t) − s n b(t) (2.29) as L n (t) = (1/N)t + b n and L  n (t) = 1/N = a n ,wehaveforallk = 0,1, , p, q (k) n (t) = 1 N k g (k)  L n (t)  − s n b (k) (t) (2.30) 6 Smooth fractal interpolation so that (2.27)becomes N k s n−1 g (k)  t N  − N k s N b (k)  t N  1 − N k s N − s n−1 N k b (k)  t N  = N k s n g (k)  t 0  − N k s 1 b (k)  t 0  1 − N k s 1 − s n N k b (k)  t 0  . (2.31) If we consider constant scale factors s n = s 1 for all n = 1, ,N, g (k)  t N  − b (k)  t N  = g (k)  t 0  − b (k)  t 0  . (2.32) Asufficient condition in order to satisfy this equality is b (k)  t 0  = g (k)  t 0  , b (k)  t N  = g (k)  t N  (2.33) for k = 0,1,2, , p. In this case, we look for a function b which agrees with g at the ex- tremes of the interval until the pth derivative. The conditions (2.33) will be satisfied if a Hermite interpolating polynomial b is con- sidered, with nodes t 0 , t N and p derivatives at the extremes. In this case, (see [14]), b(t) = H g (t) = p  k=0 g (k)  t 0  ᏸ 0k (t)+ p  k=0 g (k)  t N  ᏸ Nk (t). (2.34) The functions ᏸ ik are defined by means of intermediate l ik ,fori = 0,N and 0 ≤ k ≤ p, l 0k (t) =  t − t 0  k k!  t − t N t 0 − t N  p+1 , l Nk (t) =  t − t N  k k!  t − t 0 t N − t 0  p+1 (2.35) so that ᏸ 0p (t) = l 0p (t), ᏸ Np (t) = l Np (t), (2.36) and for k = p − 1, p − 2, ,0, ᏸ 0k (t) = l 0k (t) − p  ν=k+1 l (ν) 0k  t 0  ᏸ 0ν (t), ᏸ Nk (t) = l Nk (t) − p  ν=k+1 l (ν) Nk  t N  ᏸ Nν (t). (2.37) The mappings ᏸ ik satisfy ᏸ (σ) ik  t j  = ⎧ ⎨ ⎩ 1ifi = j, k = σ, 0 otherwise. (2.38) M. A. Navascu ´ es and M. V. Sebasti ´ an 7 The degree of H g (t)is2p + 1. The function g is an interpolant of the data such that g ∈ Ꮿ p . According to the theorem of Barnsley and Harrington, the IFS associated with the kth derivative of a FIF is expressed by L n (t) = 1 N t + b n , F nk (t,x) = N k s 1 x + N k q (k) n (t), k = 0,1,2, , p. (2.39) In our case, q n (t) = g ◦ L n (t) − s 1 b(t), (2.40) where b(t) is a Hermite interpolating polynomial of degree 2p +1ofg.Thederivativesof q n (t)become q (k) n (t) = 1 N k g (k)  L n (t)  − s 1 b (k) (t), k = 0,1,2, , p, (2.41) so that the IFS defining the kth derivative of g s b is (2.15), L n (t) = 1 N t + b n , F nk (t,x) = N k s 1 x + g (k) ◦ L n (t) − N k s 1 b (k) (t), k = 0,1,2, , p, (2.42) that is to say, the map q nk corresponding to F nk is q nk (t) = g (k) ◦ L n (t) − N k s 1 b (k) (t), k = 0,1,2, , p, (2.43) so that the kth derivative of the s-fractal function of g with respect to s and b, g s b ,agrees with the fractal function of g (k) with respect to the scaling vector N k s and b (k) (Definition 2.3):  g s b  (k) =  g (k)  N k s b (k) , k = 0, 1, 2, , p. (2.44) Proposition 2.7. (g s b ) (k) interpolates to g (k) at the nodes of Δ,for0 ≤ k ≤ p. Proof. The ordinates of (g s b ) (k) at the extremes of the interval are given in the theorem of Barnsley and Harrington. Applying (2.16), (2.33), and (2.41),  g s b  (k)  t 0  = x 0,k = q (k) 1  t 0  a k 1 − s 1 = 1 a k 1 − s 1  1 N k g (k)  L 1  t 0  − s 1 b (k)  t 0   = 1 1 − s 1 N k  g (k)  t 0  − s 1 N k b (k)  t 0  = g (k)  t 0  . (2.45) Inthesameway,  g s b  (k)  t N  = g (k)  t N  . (2.46) 8 Smooth fractal interpolation Now, applying the fixed point equation (2.26) corresponding to kth IFS at t n ,  g s b  (k)  t n  = F nk  L −1 n  t n  ,  g s b  (k) ◦ L −1 n  t n   = N k s 1  g s b  (k) ◦ L −1 n  t n  + g (k)  t n  − N k s 1 b (k) ◦ L −1 n  t n  = g (k)  t n  (2.47) since L −1 n  t n  = t N ,  g s b  (k)  t N  = g (k)  t N  = b (k)  t N  . (2.48)  The properties of g s b are as the following. (i) (g s b ) (k) interpolates to g (k) at the nodes of the partition Δ,for0≤ k ≤ p. (ii) g s b may be close to g (choosing suitably the scale vector according to (2.24)). (iii) g s b preserves the p-smoothness of g. (iv) g s b preserves the boundary conditions of g. (v) If s = 0, g s b = g,thatistosay,g is a particular case of g s b . Note. Despite the similarity between g s b and g, in general, they do not agree. In fact, if s = 0andb = g,theng s b = g. Let us assume that g s b = g.Ifs 1 = 0, applying (2.26)forL n (t) ∈ I n , g ◦ L n (t) = g ◦ L n (t)+s 1 (g − b)(t), g(t) = b(t) (2.49) for all t ∈ I. 2.3. Uniform bounds. In order to bound the distance between g and g s b , we consider a theorem of Ciarlet et al. concerning Hermite interpolation. Given a partition Δ : t 0 <t 1 < ···<t N of an interval [t 0 ,t N ], I n = [t n−1 ,t n ]for1≤ n ≤ N, the Hermite function space (see [14]) H p+1 Δ (p ∈ N)isdefinedby H p+1 Δ =  ϕ :  t 0 ,t N  −→ R ; ϕ ∈ Ꮿ p  t 0 ,t N  , ϕ| I n ∈ ᏼ 2p+1  , (2.50) where ᏼ 2p+1 is the space consisting of all polynomials of degree at most 2p +1. Theorem 2.8 (Ciarlet et al. [6]). Let g ∈ Ꮿ r [t 0 ,t N ] with r ≥ 2p +2,letΔ be any partition of [t 0 ,t N ],letΔ : t 0 <t 1 < ···<t N ,andletϕ(t) be the unique interpolation of g(t) in H p+1 Δ , that is, g (l) (t n ) = ϕ (l) (t n ),forall0 ≤ n ≤ N, 0 ≤ l ≤ p. Then   g (k) − ϕ (k)   ∞ ≤  Δ 2p+2−k 2 2p+2−2k k!(2p +2− 2k)!   g (2p+2)   ∞ (2.51) for all k = 0, 1, , p +1. In the case in study, we consider a single subinterval of length T = b − a. To bound the difference between the kth derivative of g and the kth derivative of g s b , we can use M. A. Navascu ´ es and M. V. Sebasti ´ an 9 Theorem 2.4,    g s b  (k) − g (k)   ∞ =    g (k)  N k s b (k) − g (k)   ∞ ≤ N k   s 1   1 − N k   s 1     g (k) − b (k)   ∞ . (2.52) Considering that b(t) = ϕ(t) is the Hermite interpolating polynomial of degree 2p + 1ofg, theorem of Ciarlet et al. can be used in order to bound g (k) − b (k)  ∞ ,sothat applying (2.51), (2.52) and considering g ∈ Ꮿ (2p+2) ,    g s b  (k) − g (k)   ∞ ≤ N k   s 1   1 − N k   s 1     g (k) − b (k)   ∞ ≤ N k   s 1   1 − N k   s 1   T 2p+2−k 2 2p+2−2k k!(2p +2− 2k)!   g (2p+2)   ∞ , k = 0, 1, , p. (2.53) 2.4. An operator of Ꮿ p (I). From here on, we denote by g s the s-fractal function of g ∈ Ꮿ p (I) with respect to a fixed partition Δ of the interval, a scaling vector s with constant coordinates s n = s 1 for all n = 1,2, N and b(t) = H g (t) defined in the preceding sections. For fixed Δ, let us consider the operator of Ꮿ p (I) which assigns g s to the function g, Ᏸ s p (g) = g s . (2.54) Theorem 2.9. Ᏸ s p is a linear, injective, and bounded operator of Ꮿ p (I). Proof. The operator is linear as by (2.26)forallt ∈ I n , f s (t) = f (t)+s n  f s − H f  ◦ L −1 n (t), g s (t) = g(t)+s n  g s − H g  ◦ L −1 n (t). (2.55) Multiplying the first equation by λ and the second by μ and considering that λH f + μH g = H λf+μg , (2.56) the function λf s + μg s (2.57) satisfies the equation corresponding to (λf + μg) s . (2.58) By the uniqueness of the solution, the linearity is proved. To prove the injectivity, let us consider that g s = 0. In this case, for all t ∈ I n by (2.26), 0 = g(t) − s 1 H g ◦ L −1 n (t) (2.59) but this equation is satisfied by g(t) = 0 and due to the uniqueness of the solution g = 0. 10 Smooth fractal interpolation We consider Ꮿ p (I) endowed with the norm  f  Ꮿ p (I) =  p k =0  f (k)  ∞ . Using the defi- nition of H g (t)(2.34),   H ( j) g (t)   ∞ = sup      p  k=0  g (k)  t 0  ᏸ ( j) 0k (t)+g (k)  t N  ᏸ ( j) Nk (t)       ,   H ( j) g (t)   ∞ ≤ sup t∈I g Ꮿ p (I) p  k=0     ᏸ ( j) 0k (t)    +    ᏸ ( j) Nk (t)     . (2.60) Let us consider λ p = sup t∈I 0 ≤ j≤p p  k=0     ᏸ ( j) 0k (t)    +    ᏸ ( j) Nk (t)     (2.61) then   H g   Ꮿ p (I) ≤g Ꮿ p (I) λ p ,   g − H g   Ꮿ p (I) ≤  λ p +1   g Ꮿ p (I) . (2.62) On the other hand, using Theorem 2.4,(2.52), and N j   s 1   1 − N j   s 1   ≤ N p   s 1   1 − N p   s 1   (2.63) for 0 ≤ j ≤ p,onehas   g s − g   Ꮿ p (I) ≤ N p   s 1   1 − N p   s 1     g − H g   Ꮿ p (I) , (2.64) by (2.64)and(2.62),   g s   Ꮿ p (I) −g Ꮿ p (I) ≤ N p   s 1   1 − N p   s 1    λ p +1   g Ꮿ p (I) ,   g s   Ꮿ p (I) ≤ 1+λ p N p   s 1   1 − N p   s 1    g Ꮿ p (I) . (2.65) As a consequence, Ᏸ s p is bounded and   Ᏸ s p   ≤ 1+λ p N p   s 1   1 − N p   s 1   . (2.66)  2.5. Convergence in Ꮿ p (I). Let x ∈ Ꮿ p (I) be an original function providing the data and let g Δ N ∈ Ꮿ p (I) be an interpolant of x on the partition Δ N . We consider the fractal function g s N Δ N of g Δ N with respect to the partition Δ N , the scale vector s N with constant coordinates, and the function b defined by the equality (2.34). [...]... J E Hutchinson, Fractals and self-similarity, Indiana University Mathematics Journal 30 (1981), no 5, 713–747 20 Smooth fractal interpolation [9] P.-J Laurent, Approximation et optimisation, Hermann, Paris, 1972 [10] M A Navascu´ s and M V Sebasti´ n, Some results of convergence of cubic spline fractal interpolae a tion functions, Fractals 11 (2003), no 1, 1–7 , Fitting curves by fractal interpolation:... 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Ᏼ2 [a,b] (3.30), σΔN − b 16 Smooth fractal interpolation Since |sN | < 1/N 2+r , taking limits in (3.34) as N → ∞, 1 N s σΔN − σΔN Ᏼ2 [a,b] − 0 → (3.36) Finally, by (3.30) and (3.31), N s σΔN −→ x (3.37) in Ᏼ2 [a,b] Consequence 3.10 The fractal cubic splines with non-null scale vector are dense in Ᏼ2 [a,b] Note For sN = 0, we retrieve the standard spline case 3.2 Fractal cubic spline operator Let ΔN... application to the quantification of cognitive [11] brain processes, Thinking in Patterns: Fractals and Related Phenomena in Nature (M M Novak, ed.), World Scientific, New Jersey, 2004, pp 143–154 , Generalization of Hermite functions by fractal interpolation, Journal of Approximation [12] Theory 131 (2004), no 1, 19–29 , A fractal version of Schultz’s theorem, Mathematical Inequalities & Applications 8 [13]... (S) = Ss is the space 2 of s -fractal functions of S with respect to the partition ΔN : a = t0 < t1 < t2 < · · · < tN+1 = b (3.45) Since Ᏸs is linear, 2 s Ᏸs (S) = Ss = span σkN 2 N k =1 (3.46) s The family σkN is orthonormal respect to (3.43) because s σkN t j = σkN t j = δ jk (3.47) s for k, j = 1,2, ,N As a consequence, {σkN }N=1 is a basis of Ss k Let us define the fractal cubic spline operator... (ti ,xi ), then b a (property of minimum · 2 σΔ (t) dt ≤ b 2 x (t) dt a (3.14) 2 -norm) Theorem 3.9 states the convergence of fractal cubic splines towards the original function if an additional condition is imposed on the scaling factors We need a previous result concerning fractal interpolation functions Lemma 3.8 For a uniform partition and constant scale factors sn = s1 , for all n = 1, ,N, fs−... = s2 1 s 2 L2 (a,b) f −f b an n =1 = s2 1 a 2 f s − b (t) dt, (3.17) s f −b 2 L2 (a,b) 14 Smooth fractal interpolation As a consequence, fs− f L2 (a,b) = s1 fs− f L2 (a,b) ≤ f s −b s1 f −b 1 − s1 fs− f ≤ s1 L2 (a,b) L2 (a,b) + f −b L2 (a,b) , (3.18) L2 (a,b) The next theorem states the convergence of fractal cubic splines towards any original function x ∈ Ᏼ2 [a,b] when the partition is refined A part... obtained Consequence 2.11 If one considers a scaling vector such that |sN | < 1/N p+r , for r > 0 1 sN fixed, the fractal interpolant gΔN converges in Ꮿ p (I) towards the original x if gΔN does (as N tends to ∞) Note The constant λ p does not depend on ΔN but only on the extremes of the interval 3 Fractal cubic splines In this section, we study the particular case p = 2, considering the following IFS: Ln... function f at the points t0 ,t1 , ,tN ∈ ΔN , being σΔN type I or II Then (r) f (r) − σΔN ∞ ≤ Cr Lh4−r (r = 0,1,2) with C0 = 5/384, C1 = 1/24, C2 = 3/8 The constants C0 and C1 are optimum (3.3) 12 Smooth fractal interpolation Remark 3.2 A spline is type I if its first derivatives at a and b are known A spline is type II if it can be explicitly represented by its second derivatives at a and b Theorem 3.3... j,k = 1,2, ,N, s σkN t j = σkN t j = δ jk s Proposition 3.12 ᏿N is linear Proof It is an immediate consequence of (3.48) Without loss of generality, we consider I = [0,1] and tk = k/N (3.50) 18 Smooth fractal interpolation Lemma 3.13 Let f and g be defined in I such that f (tk ) = g(tk ) for k = 1,2, ,N for some N, then s s ᏿N ( f ) = ᏿N (g) (3.51) Proof For all t ∈ I, N s ᏿N ( f )(t) = N s s g tk σkN . integrable square. In particular, the density of fractal cubic splines in Ᏼ 2 [a,b]isproven. 2. Construction of smooth fractal interpolants 2.1. Fractal interpolation functions. Let t 0 <t 1 <. L −1 n (t)forallt ∈ I n . 4 Smooth fractal interpolation 2.2. Differentiable fractal interpolation functions. In this section, we study the con- struction of smooth fractal interpolation functions Sebasti ´ an, Some results of convergence of cubic spline fractal interpola- tion functions,Fractals11 (2003), no. 1, 1–7. [11] , Fitting curves by fractal interpolation: an applicat ion to the quantification