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Nonlinear bivariate fractal interpolation function on grids
Chaos, Solitons and Fractals 81 (2015) 351–358
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Nonlinear bivariate fractal interpolation function on grids✩ SongIl Ri∗ Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 March 2015 Accepted 15 October 2015
a b s t r a c t In this paper we introduce a new construction of the fractal interpolation surface (FIS) using an even more general IFS which can generate self-affine and non self-affine fractal surfaces. Here we present the general types of fractal surfaces that are based on nonlinear IFSs. © 2015 Elsevier Ltd. All rights reserved.
MSC: 37C45 28A80 Keywords: Function system (IFS) Attractor Nonlinear fractal interpolation surface
1. Introduction The concept of fractal interpolation functions (FIFs) was introduced by Barnsley [1] on the basis of the theory of iterated function systems (IFSs). Barnsley defined a fractal interpolation function (FIF) in 1986 and presented a construction of fractal curves by fractal interpolation. In the developments of theory of FIFs, many researchers have generalized the notion of FIFs in different ways. The fractal interpolated curves have been discussed in detail in the literature [2,3] and Massopust [4] was the first to put forward the construction of fractal surfaces via IFS. In recent years, many authors studied construction methods of FISs with the data set given on rectangular grids (see [5–12]). Xie et al. [8] introduced the construction of an attractor that contains the interpolation points of a rectangular data set, but generally is not a graph of a continuous function. In order to ensure the continuity of the surface, Dalla [9] assumed that the interpolation nodes on the boundary are collinear. But the vertical scaling factors used in the iterated function system must all be constant in this case. Vertical scaling factors in
✩ ∗
Construction was presented on D.P.R.K. 2015. Tel.: +86015543562662. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.chaos.2015.10.020 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
IFS are important one that characterizes the constructed fractal set . Feng et al. [10] presented a construction of fractal surfaces on a new condition that the function vertical scaling factors are 0 on the grid of rectangular domain. In this article we introduce a new construction of a bivariate fractal interpolation surface (BFIS) of the set of data and generalize the construction of fractal surfaces in [9,10]. We show that the graph of a nonlinear fractal interpolation function is the attractor of an underlying contractive nonlinear IFS. 2. Fixed point of operator and attractor of iterated function system Let a set {(xi , yi , zi, j ) : i = 1, 2, . . . , M; j = 1, 2, . . . , N} be given. The following theorem devises a general construction of the nonlinear FIS. Let I = [a, b], J = [c, d], a = x0 < x1 < · · · < xM = b and c = y0 < y1 < · · · < yN = d. Let each of the sets
{(x0 , y j , z0, j ) : j = 1, 2, . . . , N}, {(xM , y j , zM, j ) : j = 0, 1, . . . , N}, {(xi , y0 , zi,0 ) : i = 0, 1, . . . , M} and {(xi , yN , zi,N ) : i = 0, 1, . . . , M} be collinear. Let Ii , Jj , D and Di, j denote [xi−1 , xi ], [y j−1 , y j ], I × J and Ii × Jj . Let ai :=
xi −xi−1 xM −x0 , bi
:= xi−1 −
xi −xi−1 xM −x0 x0 , c j
:=
352
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
y j −y j−1 yN −y0
y −y
and d j := y j−1 − yj −yj−1 y0 . For i = 1, 2, . . . , M and j = N 0 1, 2, . . . , N, we define mappings li, j : D → Di, j , where li, j is a continuous function such that
li, j (x, y) : = (ai x + bi , c j x + d j ) f or i = 1, 2, . . . , M; j = 1, 2, . . . , N. Let L: D → D be a function such that L(x, y) := li,−1j (x, y) for (x, y) ∈ Di, j . That is,
⎧ −1 l1,1 (x, y), ⎪ ⎪ ⎪ ⎨ ··· L(x, y) := li,−1j (x, y), ⎪ ⎪ ⎪ ⎩ −1· · · lM,N (x, y),
(x, y) ∈ [x0 , x1 ] × [y0 , y1 ] (x, y) ∈ [xi−1 , xi ] × [y j−1 , y j ] (x, y) ∈ [xM−1 , xM ] × [yN−1 , yN ].
Similarly, Tg(xi , y j−1 ) = zi, j−1 , Tg(xi−1 , y j ) = zi−1, j , Tg(xi−1 , y j−1 ) = zi−1, j−1 . Hence Tg(xi , y j ) = zi, j for all i = 1, 2, . . . , M; j = 1, 2, . . . , N. Then Tg is not always a continuous surface over
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}. But T is well defined. It is obvious that Tg(x, y) is continuous when (x, y) ∈ Di, j . For (xi , y) ∈ Di, j ∩ Di+1, j , let (x , y ) := li,−1j (xi , y) and
−1 (x
, y
) := li+1, (x , y). j i Since g(x , y ) = b(x , y ), g(x
, y
) = b(x
, y
), L(xi , y) = li,−1j (xi , y) = (x , y ) = (xM , y ) for (x, y) ∈ Di, j and L(x, y) = −1 (x, y) = (x
, y
) = (x0 , y
) for (x, y) ∈ Di+1, j , li+1, j
Definition 2.1. Let for some constants L1 > 0, L2 > 0, | f (x1 , y1 ) − f (x2 , y2 )| L1 |x1 − x2 | + L2 |y1 − y2 |(∀(x1 , y1 ), ∀(x2 , y2 ) ∈ D ⊂ R2 ). Then we call f: D → R Lipschitz continuous function on the region D ⊂ R2 .
Tg(xi , y) =
Let S: D → R be Lipschitz continuous function such that s := max(x,y)∈D |S(x, y)| < 1. We denote the set of Lipschitz continuous functions f: D → R by C 0 = C 0 (D). Let C∗ ⊂ C0 be the set of Lipschitz continuous functions defined on D, such that
=
i=1 j=1
−b(L(xi , y)))]χIi ×J j (xi , y)
−b(xM+1 , y ))]χIi ×J j (xi , y) =
f ((1 − λ)x0 + λxM , yN ) = (1 − λ)z0,N + λzM,N
N M [h(xi , y) + S(xi , y)(g(x0 , y
) i=1 j=1
−b(x0 , y
))]χIi ×J j (xi , y) = h(xi , y)
f (xM , (1 − λ)y0 + λyN ) = (1 − λ)zM,0 + λzM,N ,
for λ ∈ [0, 1]. Let C ∗∗ := { f ∈ C ∗ : f (xi , y j ) = zi, j , i = 0, 1, . . . , M; j = 0, 1, . . . , N}. It is well-known that (C∗ , d∞ ) and (C∗∗ , d∞ ) are complete metric spaces, where
N M [h(xi , y) + S(xi , y)(g(xM+1 , y ) i=1 j=1
f (x0 , (1 − λ)y0 + λyN ) = (1 − λ)z0,0 + λz0,N , f ((1 − λ)x0 + λxM , y0 ) = (1 − λ)z0,0 + λzM,0 ,
N M [h(xi , y) + S(xi , y)(g(L(xi , y))
for (xi , y) ∈ Di, j and (xi , y) ∈ Di+1, j . (xi−1 , y) ∈ Di−1, j ∩ Di, j , (x, y j−1 ) ∈ Di, j−1 ∩ Di, j and (x, y j ) ∈ Di, j+1 ∩ Di, j can be dealt similarly. Hence T: C∗ → C∗∗ and since C∗∗ ⊂ C∗ , Tn : C∗∗ → C∗∗ (n ≥ 2).
d∞ ( f, g) = max{| f (x, y) − g(x, y)| : (x, y) ∈ D}.
Theorem 2.3. If s < 1 then T is a contraction and it possesses a unique fixed point f ∈ C∗∗ . In particular, f: D → R is a continuous function that interpolates the given data
Let b ∈ C∗ and h ∈ C∗∗ . We define T: C∗ → C0 by
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
Tg(x, y) =
N M
Proof. To prove contractivity, we observe that
i=1 j=1
d∞ (Tg1 , Tg2 ) = max |S(x, y)(g1 (L(x, y)) − g2 (L(x, y)))|
[h(x, y) + S(x, y)(g(L(x, y))
− b(L(x, y)))]χIi ×J j (x, y) for (x, y) ∈ D, where χIi ×J j denotes the characteristic function of a set Ii × Jj . Lemma 2.2. Tg ∈ C∗∗ . That is, T: C∗ → C∗∗ and Tn : C∗∗ → C∗∗ (n ≥ 2). Proof. Since
li, j (x0 , y0 ) = (xi−1 , y j−1 ), li, j (x0 , yN ) = (xi−1 , y j ), li, j (xM , y0 ) = (xi , y j−1 ), li, j (xM , yN ) = (xi , y j ), L(xi−1 , y j−1 ) = (x0 , y0 ), L(xi−1 , y j ) = (x0 , yN ), L(xi , y j−1 ) = (xM , y0 ), L(xi , y j ) = (xM , yN ) and g, b ∈ C∗ , hence
Tg(xi , y j ) =
N M [h(xi , y j ) + S(xi , y j )(g(L(xi , y j )) i=1 j=1
− b(L(xi , y j )))]χIi ×J j (xi , y j ) = h(xi , y j ) = zi, j for all i = 1, 2, . . . , M; j = 1, 2, . . . , N.
(x,y)∈D
s max
(x,y)∈Di, j
{|g1 (li,−1j (x, y)) − g2 (li,−1j (x, y))| : i = 1, 2, . . . ,
M; j = 1, 2, . . . , N} = sd∞ (g1 , g2 ). By Lemma 2.2, the existence of an unique fixed point f ∈ C∗∗ (when s < 1) follows from the contraction mapping theorem and graph of f passes through the interpolation points. This completes the proof. For i = 1, 2, . . . , M and j = 1, 2, . . . , N, define mappings D × R → Di, j × R by
wi, j (x, y, z) := (li, j (x, y), Fi, j (x, y, z)), where functions li, j : D → Di, j and Fi, j : D × R → R are defined by li, j (x, y) := (ai x + bi , c j y + d j ) and
Fi, j (x, y, z) := h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y)). That is,
wi, j
x y z
=
ai x + bi c jy + d j , h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y))
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
353
where
Then for any (x1 , y1 , z1 ), (x2 , y2 , z2 ) ∈ D × [−k, k];
y j − y j−1 xi − xi−1 xi − xi−1 ai := , bi := xi−1 − x0 , c j := , x M − x0 x M − x0 y N − y0 y j − y j−1 d j := y j−1 − y0 y N − y0
|S(li, j (x1 , y1 ))z1 − S(li, j (x2 , y2 ))z2 | |S(li, j (x1 , y1 ))||z1 − z2 | + |z2 ||S(li, j (x1 , y1 )) −S(li, j (x2 , y2 ))| s|z1 − z2 | + k|S(li, j (x1 , y1 )) − S(li, j (x2 , y1 ))| +k|S(li, j (x2 , y1 )) − S(li, j (x2 , y2 ))| s|z1 − z2 | + kLS,x ai |x1 − x2 | + kLS,y c j |y1 − y2 |,
for i = 1, 2, . . . , M; j = 1, 2, . . . , N. Let Mb := max(x,y)∈D |b(x, y)|, S: D → R be Lipschitz continuous functions such that s := max(x,y)∈D |S(x, y)| < 1 and
Mh,S,b :=
max |h(li, j (x, y))
max
1iM,1 jN (x,y)∈D
for any (x1 , y1 ), (x2 , y2 ) ∈ D;
−S(li, j (x, y))b(x, y)|. Let (x, y, z) ∈ D × [−k, k]. Because s < 1, if k ≥ Mh,S,b (1 − s)k and
Mh,S,b 1−s
then
|Fi, j (x, y, z)| = |h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y))| |S(li, j (x, y))||z| + |h(li, j (x, y)) − S(li, j (x, y))b(x, y)| sk + Mh,S,b sk + (1 − s)k = k.
|h(li, j (x1 , y1 )) − h(li, j (x2 , y2 ))| |h(li, j (x1 , y1 )) − h(li, j (x2 , y1 ))| +|h(li, j (x2 , y1 )) − h(li, j (x2 , y2 ))| Lh,x ai |x1 − x2 | + Lh,y c j |y1 − y2 | and
has been constructed.
|S(li, j (x1 , y1 ))b(x1 , y1 ) − S(li, j (x2 , y2 ))b(x2 , y2 )| |S(li, j (x1 , y1 ))||b(x1 , y1 ) − b(x2 , y2 )| +|b(x2 , y2 )||S(li, j (x1 , y1 )) − S(li, j (x2 , y2 ))| s|b(x1 , y1 ) − b(x2 , y2 )| + |b(x2 , y2 )|(LS,x ai |x1 −x2 | + kLS,y c j |y1 − y2 |) sLb,x |x1 − x2 | + sLb,y |y1 − y2 | + Mb LS,x ai |x1 −x2 | + Mb LS,y c j |y1 − y2 |).
Theorem 2.4. We consider the IFS defined above, associated with the set of data.
So
M
h,S,b Hence for any k ≥ 1−s and (x, y, z) ∈ D × [−k, k], it follows that |Fi, j (x, y, z)| ≤ k. That is,
Fi, j : D × [−k, k] → D × [−k, k]. Then an iterated function system
{D × [−k, k]; wi, j : i = 1, 2, . . . , M; j = 1, 2, . . . , N}
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) − ai |x1 − x2 |
−c j |y1 − y2 | = θ [h(li, j (x1 , y1 )) + S(li, j (x1 , y1 ))
Let s := max{|S(x, y)|: (x, y) ∈ D} < 1, S: D → R be Lipschitz Mh,S,b 1−s .
continuous functions and k ≥ Then there exists some metric d∗ that is equivalent to the Euclidean metric d on [x0 , xM ] × [y0 , yN ] × k such that the wi, j are contractions for all i = 0, 1, . . . , M; j = 0, 1, . . . , N with respect to d∗ . In particular, there exists a unique nonempty compact set G ⊂ [x0 , xM ] × [y0 , yN ] × k, such that
G=
N M
wi, j (G).
i=1 j=1
(z1 − b(x1 , y1 ))] − [h(li, j (x2 , y2 )) + S(li, j (x2 , y2 ))
(z2 − b(x2 , y2 ))]
θ s|z1 − z2 | + kLS,x ai |x1 − x2 | + kLS,y c j |y1 − y2 | +Lh,x ai |x1 − x2 | + Lh,y c j |y1 − y2 | + sLb,x |x1 − x2 | sθ |z1 − z2 | + θ (kLS,x ai + Lh,x ai + sLb,x +Mb LS,x ai )|x1 − x2 | + θ (kLS,y c j + Lh,y c j
Proof. We define a metric d∗ on R3 by
d∗ ((x1 , y1 , z1 ), (x2 , y2 , z2 )) := |x1 − x2 | + |y1 − y2 | + θ |z1 − z2 |, where θ is a positive real number which is specified below. Then this metric is equivalent to the Euclidean metric on R3 . Let
|S(x1 , y) − S(x2 , y)| LS,x |x1 − x2 |(∀y ∈ J), |S(x, y1 ) − S(x, y2 )| LS,y |y1 − y2 |(∀x ∈ I), |b(x1 , y) − b(x2 , y)| Lb,x |x1 − x2 |(∀y ∈ J), |b(x, y1 ) − b(x, y2 )| Lb,y |y1 − y2 |(∀x ∈ I), |h(x1 , y) − h(x2 , y)| Lh,x |x1 − x2 |(∀y ∈ J),
+sLb,y + Mb LS,y c j )|y1 − y2 |. Hence
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) {ai + θ [kLS,x ai + Lh,x ai + sLb,x + Mb LS,x ai ]}|x1 − x2 | +{c j + θ [kLS,y c j + Lh,y c j + sLb,y + Mb LS,y c j ]}|y1 − y2 | +sθ |z1 − z2 |. Let
θ := min
1 − max1iM ai , 2 max1iM (kLS,x ai + Lh,x ai + sLb,x + Mb LS,x ai )
1 − max1 jN c j . 2 max1 jN (kLS,y c j + Lh,y c j + sLb,y + Mb LS,y c j )
and
|h(x, y1 ) − h(x, y2 )| Lh,y |y1 − y2 |(∀x ∈ I).
+sLb,y |y1 − y2 | + Mb LS,x ai |x1 − x2 | + Mb LS,y c j |y1 − y2 |
We have
354
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) 1 + cj 1 + ai |x1 − x2 | + |y1 − y2 | + sθ |z1 − z2 | 2 2
1 + max1iM ai 1 + max1 jN c j , ,s max 2 2
(|x1 − x2 | + |y1 − y2 | + θ |z1 − z2 |)
max d
∗
1 + max1iM ai 1 + max1 jN c j , ,s 2 2
((x1 , y1 , z1 ), (x2 , y2 , z2 )).
Since ai <1(i = 1, 2, . . . , M), c j <1( j = 1, 2, . . . , N) and s < 1, the wi, j are all contraction mappings on metric space ([x0 , xM ] × [y0 , yN ] × R, d∗ ). Because metric d∗ is equivalent to the Euclidean metric d, therefore there exists a unique nonempty compact set G ⊂ [x0 , xM ] × [y0 , yN ] × R on metric space ([x0 , xM ] × [y0 , yN ] × R, d) such that
G=
N M
wi, j (G).
Proof. Let f(x, y) be a fixed point of operator T and G denote the graph of f. Then
f (li, j (x, y)) = (T f )(li, j (x, y)) = h(li, j (x, y))
z +z
−z
−z
−s (z
+z
−z
0,0 0,N i, j−1 i, j M,N gi, j = i, j i−1, j−1 i−1,(j y −y N 0 )(xM −x0 ) = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 −si, j (zM,N + z0,0 − z0,N − zM,0 ),
z
−z
−s (z
−z
)−g y (x −x )
z
−z
−s (z
−z
)−g x (y −y )
−zM,0 )
ei, j = i−1, j−1 i, j−1 i, j x00,0−xMM,0 i, j 0 0 M = −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )], fi, j = i−1, j−1 i−1, j i, j y00,0−yN 0,N i, j 0 0 N = −[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )] and
+[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )] −si, j zM,N − [zi, j + zi−1, j−1 − zi−1, j − zi, j−1 −si, j (zM,N + z0,0 − z0,N − zM,0 )] = zi−1, j−1 − si, j z0,0 for all i = 1, 2, . . . , M; j = 1, 2, . . . , N(see [9], p.54–55). So we obtain that
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j
+S(li, j (x, y))( f (x, y) − b(x, y))
= −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )]x
= Fi, j (x, y, f (x, y)) for (x, y) ∈ D and for i = 1, 2, . . . , M; j = 1, 2, . . . , N. Hence we obtain that
wi, j (G) = {wi, j (x, y, f (x, y)) : (x, y) ∈ D}
{(ai x + bi , c j y + d j , h(li, j (x, y)) +S(li, j (x, y))( f (x, y) − b(x, y)) : (x, y) ∈ D} = {(ai x + bi , c j y + d j , (T f )(li, j (x, y)) : (x, y) ∈ D} = {(ai x + bi , c j y + d j , f (li, j (x, y)) : (x, y) ∈ D} = {(x, y, f (x, y)) : (x, y) ∈ Di, j }. =
Hence
i=1 j=1
where
= zi, j + [zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )]
Corollary 2.5. Let G be an attractor of the IFS {D × R; wi, j , i = 1, 2, . . . , M; j = 1, 2, . . . , N} defined above and f(x, y) be a fixed point of operator T : g → h + S · (g ◦ L − b ◦ L). Then the graph f is G.
{(x, y, f (x, y)) : (x, y) ∈ Di, j } =
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j ,
= zi, j − ei, j − fi, j − si, j zM,N − gi, j
This completes the proof.
M N
Remark 2.7. Let [x0 , xM ] × [y0 , yN ] = [0, 1] × [0, 1]. 1. In [9],
ki, j = zi, j − ei, j xM − fi, j yN − si, j zM,N − gi, j xM yN
i=1 j=1
G=
is the restriction of fˆ on [x0 , xM ] × [y0 , yN ], where fˆ : [x−1 , xM+1 ] × [y−1 , yN+1 ] → R is a fractal interpolation function according to Theorem 2.3 and Theorem 2.4.
M N
wi, j (G).
i=1 j=1
By Theorem 2.4, only one nonempty compact set G is an attractor of the IFS. Remark 2.6. General Construction (see [9], p.57) Now let’s consider new data
{(xi , y j , zˆi, j ) : i = −1, 0, . . . , M + 1; j = −1, 0, . . . , N + 1}, where zˆi, j = zi, j for i = 0, 1, . . . , M; j = 0, 1, . . . , N, x−1 < x0 , y−1 < y0 , xM < xM+1 , yN < yN+1 and zˆ−1, j = zˆM+1, j = zˆi,−1 = zˆi,N+1 ≡ 0 for all i = −1, 0, . . . , M + 1; j = −1, 0, . . . , N + 1. Then the function f which interpolates the given data
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}
−[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )]y +[zi, j + zi−1, j−1 − zi−1, j − zi, j−1 − si, j
(zM,N + z0,0 − z0,N − zM,0 )]xy +si, j z + [zi−1, j−1 − si, j z0,0 ] = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy] + si, j [z − (z0,0 (1 − x)(1 − y) +zM,0 x(1 − y) + z0,N (1 − x)y + zM,N xy)]. Let Si, j (x, y) := si, j = s(|si, j | < 1) for all i = 1, 2, . . . , M; j = 1, 2, . . . , N, b(x, y) := z0,0 (1 − x)(1 − y) + zM,0 x(1 − y) + z0,N (1 − x)y + zN,M xy and h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy)]. Then b ∈ C∗ , h ∈ C∗∗ and
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy)] + si, j [z − (z0,0 (1 − x)(1 − y) +zM,0 x(1 − y) + z0,N (1 − x)y + zM,N xy)] = h(li, j (x, y)) + si, j (z − b(x, y)). 2. In [10],
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ), where
si, j (x, y) = =
λi, j (x − x0 )(xM − x)(y − y0 )(yN − y) λi, j xy(1 − x)(1 − y),
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
355
zi, j + zi−1, j−1 − zi−1, j − zi, j−1 (xM − x0 )(yN − y0 ) = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 ,
3. The Minkowski dimension of the nonlinear bivariate fractal interpolation surfaces
−y0 (zi, j − zi−1, j ) + yN (zi, j−1 − zi−1, j−1 ) (xM − x0 )(yN − y0 ) = zi, j−1 − zi−1, j−1 ,
In this section we present the formula for the Minkowski dimension of a bivariate fractal interpolation surface. We use the following notation: I := [0, 1] × [0, 1] j−1 j i and Ii, j := [ i−1 N , N ] × [ N , N ] for i, j = 1, 2, . . . , N. Let E :=
g i, j =
e i, j =
xM (zi−1, j − zi−1, j−1 ) − x0 (zi, j − zi, j−1 ) (xM − x0 )(yN − y0 ) = zi−1, j − zi−1, j−1 ,
fi, j =
{( Ni , Nj , zi, j ); i, j = 0, 1, 2, . . . , N} be the set of data and Pk be a rectangle, that is,
Pk :=
and
x0 y0 zi, j − x0 yN zi, j−1 − xM y0 zi−1, j + xM yN zi−1, j−1 k i, j = (xM − x0 )(yN − y0 ) = zi−1, j−1 . (see [10], p.1898) Hence
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ) = λi, j xy(1 − x)(1 − y)z + [zi, j−1 − zi−1, j−1 ]x +[zi−1, j − zi−1, j−1 ]y + [zi, j + zi−1, j−1 −zi−1, j − zi, j−1 ]xy + zi−1, j−1 = λi, j xy(1 − x)(1 − y)z + [zi−1, j−1 (1 − x)(1 − y) +zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy].
i
+zi−1, j (1 − x)y + zi, j xy]. Then
λi, j xy(1 − x)(1 − y)z + [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy] = h(li, j (x, y)) + si, j (x, y)z = h(li, j (x, y)) +si, j (x, y)(z − b(x, y)).
R f [I] :=
sup
(x1 ,y1 ),(x2 ,y2 )∈I
| f (x1 , y1 ) − f (x2 , y2 )|.
Lemma 3.1. Let f be BFIF of the set of data E. If s := max {|S(x, y)|: (x, y) ∈ I} < 1 then
R f [Ii, j ]
maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x +Lb,y ) + (Lh,x + Lh,y ) . 1−s
Proof. Let for k ∈ N,
k := max{| f (x, y) − z0,0 |; (x, y) ∈ Pk }, λk := max max{| f (x, y) − zi−1, j−1 |; i, j=1,2,...,N (x,y)
(x, y) ∈ Pk ∩ Ii, j }. By Theorem 2.3, Theorem 2.4, for (x, y) ∈ Pk ∩ Ii, j ,
f (li, j (x, y)) = (T f )(li, j (x, y)) = h(li, j (x, y)) +S(li, j (x, y))( f (x, y) − b(x, y))
Fi, j (x, y, z) =
Example 2.8. Let [x0 , xM ] × [y0 , yM ] = [0, 1] × [0, 1]. Let b(x, y) := z0,0 (1 − x)(1 − y) + zM,0 x(1 − y) + z0,N (1 − x)y + zN,M xy. Then z0,0 − b(x0 , y0 ) = 0, z1,0 − b(x1 , y0 ) = 0, z0,1 − b(x0 , y1 ) = 0 and z1,1 − b(x1 , y1 ) = 0. Let h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy)] and S: [0, 1] × [0, 1] → R be Lipschitz continuous functions such that
j ; i, j = 0, 1, 2, . . . , Nk . Nk
We denote the maximum range of a function f by
Let b(x, y) ≡ 0 and
h(li, j (x, y)) = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y)
N
, k
= Fi, j (x, y, f (x, y)) and for (x, y) ∈ D,
f (x, y) := h(x, y) + S(x, y)( f (L(x, y)) − b(L(x, y))). Since li, j (0, 0) = xi−1, j−1 , h(xi−1, j−1 ) = zi−1, j−1 and b(0, 0) = z0,0 , we obtain
f (x, y) − zi−1, j−1 = h(x, y) + S(x, y)( f (L(x, y)) −b(L(x, y))) − zi−1, j−1 = h(x, y) + S(x, y)( f (L(x, y)) − b(L(x, y))) − h(li, j (0, 0)) = h(x, y) + S(x, y)( f (L(x, y)) − z0,0 +b(0, 0) −b(L(x, y))) −h(li, j (0, 0))
xy si, j (x, y) = . (1 + x)(1 + y)
|S(x, y)|| f (L(x, y)) − z0,0 | + |S(x, y)||b(0, 0)
Define Fi, j : [0, 1] × [0, 1] × R → R by Fi, j (x, y, z) = h(li, j (x, y)) + si, j (x, y)(z − b(x, y)). By Theorem 2.3 and Theorem 2.4, there exists a continuous function f: [0, 1] × [0, 1] → R that interpolates the given data {(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}. This result is a substantial generalization of [9], Proposition 2.2, p.56 and [10], Theorem 3.1, p.1898.
s| f (L(x, y)) − z0,0 | + s|b(0, 0) − b(L(x, y)))|
Remark 2.9. If b(x, y) ∈ C∗ and h(x, y) ∈ C∗∗ or si, j (x, y) = 0 and h(x, y) ∈ C∗∗ then there exists a continuous function f: [0, 1] × [0, 1] → R that interpolates the given data
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
−b(L(x, y)))| + |h(x, y) − h(li, j (0, 0))| +|h(x, y) − h(li, j (0, 0))| s| f (L(x, y)) − z0,0 | + s(Lb,x |x| +Lb,y |y|) + Lh,x |x| + Lh,y |y| s| f (L(x, y)) − z0,0 | + s(Lb,x +Lb,y ) + (Lh,x + Lh,y ). Hence
λk+1 =
max
max{| f (x, y) − zi−1, j−1 | : (x, y)
i, j=1,2,...,N (x,y)
∈ Pk+1 ∩ Ii, j }
356
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
max
max{s| f (L(x, y)) − z0,0 | + s(Lb,x + Lb,y )
and
i, j=1,2,...,N (x,y)
Fi, j (x, y, z) = h(li, j (x, y)) + si, j (z − b(x, y))
+(Lh,x + Lh,y ) : (x, y) ∈ Pk+1 ∩ Ii, j } s max{| f (x, y) − z0,0 |; (x, y) ∈ Pk }
= ei, j x + fi, j y + gi, j xy + si, j z + ki, j .
+s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) sk + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ). That is, λk+1 sk + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ). Let for some (xp , yq ) ∈ Pk , k = max{| f (x, y) − z0,0 |; (x, y) ∈ Pk } = | f (x p , yq ) − z0,0 | and (x p , yq ) ∈ Ii p , jq (i p , jq ∈ {1, 2, . . . , N}). Because | f (x p , yq ) − z0,0 | | f (x p , yq ) − zi, j | + |zi, j − z0,0 | for all i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }, | f (x p , yq ) − z0,0 | max{| f (x p , yq ) − zi, j | + |zi, j − z0,0 | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} max{| f (x p , yq ) − zi, j | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} + max{|zi, j − z0,0 | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} maxi, j=1,2,...,N max(x,y) {| f (x, y) − zi−1, j−1 |; (x, y) ∈ Pk ∩ Ii, j } + max{| f (x, y) − z0,0 |; (x, y) ∈ Pk } = λk + 1 . Since k 1 + λk , we obtain that
λk+1 s(1 + λk ) + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) s[1 + sk−1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] +[s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] = [s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] + s[sk−1 +s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] [s1 + s(Lb,x + Lb,y ) + (Lh,x +Lh,y )]+s[s(1 + λk−1 ) +s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] = [s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] +s[s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] + sλk−1 . It follows by induction that
λk+1 = max max{| f (x, y) −zi−1, j−1 | : (x, y) ∈ Pk+1 ∩ Ii, j } i, j=1,2,...,N (x,y)
s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) . = 1−s
Because k ∈ N is arbitrary,
R f [Ii, j ]
Remark 3.2. In [9], Let si, j (x, y) := s, |s| < 1 for all i, j = 1, 2, . . . , N,
+z0,N (1 − x)y + zN,N xy
| f (x, y) − zi−1, j−1 | = |Fi, j (L(x, y), f (L(x, y))) − zi−1, j−1 | |ei, j | + | fi, j | + |gi, j | + |si, j f (L(x, y)) − zi−1, j−1 + ki, j | = |ei, j | + | fi, j | + |gi, j | + |si, j f (L(x, y)) − si, j z0,0 | |s| max |zi, j − z0,0 | + 4|s| max |zi1 , j1 − zi2 , j2 | i1 , j1 ,i2 , j2 ∈{0,N}
i, j=0,1,...,N
+4 max max{|zi, j − zi, j−1 |, |zi, j − zi, j+1 |, |zi, j i, j=1,...,N
−zi−1, j |, |zi, j − zi+1, j |}. So
R f [Ii, j ]
1 1 − |s|
max
i1 , j1 ,i2 , j2 ∈{0,N}
|s| max |zi, j − z0,0 | + 4|s| i, j=0,1,...,N
|zi1 , j1 − zi2 , j2 |
+4 max max{|zi, j − zi, j−1 |, |zi, j i, j=1,...,N
−zi, j+1 |, |zi, j − zi−1, j |, |zi, j − zi+1, j |} . (see [11], p.1152, Theorem 1]) By Lemma 3.1, it also has
R f [Ii, j ]
1 1−|s|
maxi, j=0,1,...,N |zi, j −z0,0 ||s|+|s|(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−|s|
|s| max |zi, j − z0,0 |
|zi, j | + 8 max |zi, j | .
i, j=0,1,...,N
+8|s| max
i, j={0,N}
i, j={1,...,N}
In [10],
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ), where
maxi, j=0,1,...,N |zi, j −z0,0 |s+s(Lb,x +Lb,y )+(Lh,x +Lh,y ) . 1−s
b(x, y) := z0,0 (1 − x)(1 − y) + zN,0 x(1 − y)
Hence
si, j (x, y) =
λi, j xy(1 − x)(1 − y),
g i, j = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 , e i, j = zi, j−1 − zi−1, j−1 , fi, j = zi−1, j − zi−1, j−1 ,
k i, j = zi−1, j−1 . Hence
and
h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy]. Since
gi, j = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 − si, j
(zM,N + z0,0 − z0,N − zN,0 ), ei, j = −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zN,0 )]; fi, j = −[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )]; ki, j = zi−1, j−1 − si, j z0,0
| f (x, y) − zi−1, j−1 | = |Fi, j (L(x, y), f (L(x, y))) − zi−1, j−1 |, |e i, j | + | fi, j | + |g i, j | + |si, j (x, y) f (L(x, y)) −zi−1, j−1 + k i, j | = |e i, j | + | fi, j | + |g i, j | + |si, j (x, y) f (L(x, y))| max |si, j (x, y)||zi, j | i, j=0,1,...,N
+4 max max{|zi, j − zi, j−1 |, |zi, j i, j=1,...,N
−zi, j+1 |, |zi, j − zi−1, j |, |zi, j − zi+1, j |}.
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
357
Also, since b(x, y) ≡ 0 and
+[−b(x1 , y1 ) + b(x2 , y2 )]|
h(li, j (x, y)) = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy)],
s
R f [Ii, j ] 1 1−|s|
maxi, j=0,1,...,N |si, j (x,y)||zi, j −z0,0 |+|si, j (x,y)|(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−|s|
maxi, j=0,1,...,N |si, j (x, y)||zi, j − z0,0 |
+8 maxi, j={0,1,...,N} |zi, j | .
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
+(Lh,x aim |x1 − x2 | + Lh,y c jm |y1 − y2 |) + (Lb,x |x1 − x2 | +Lb,y |y1 − y2 |) sR f (Ii1 , j1 ;i2 , j2 ;...;im−1 , jm−1 ) +
Let I := [0, 1] × [0, 1], m ∈ N, k ∈ {1, 2, . . . , m} and
Ii1 , j1 ;i2 , j2 ;...;im , jm := lim , jm ◦ lim−1 , jm−1 ◦ · · · ◦ li1 , j1 (I) for all ik , jk ∈ {1, 2, . . . , N}.
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm ,
Lh,x aim + Lh,y c jm + Lb,x + Lb,y . Nm−1
So
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sR f (Ii1 , j1 ;i2 , j2 ;...;im−1 , jm−1 ) +
Lemma 3.3. Let f be fractal interpolation function of the data set E := {( Ni , Nj , zi, j ); i, j = 0, 1, 2, . . . , N} with function vertical scaling factor S(x, y). (1) If N1 < s, then
| f (x1 , y1 ) − f (x2 , y2 )|
Lh,x aim + Lh,y c jm + Lb,x + Lb,y . Nm−1
By induction,
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm R f (I) +(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m−1
where R is a constant such that
p=0
maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s Lh,x aim + Lh,y c jm + Lb,x + Lb,y + . s(1 − (Ns)−1 )
R := 2N2
1 N,
(2) If 0 < s
then
m Nm−1
.
f (li, j (x, y)) = h(li, j (x, y)) + S(li, j (x, y))( f (x, y) − b(x, y)) = Fi, j (x, y, f (x, y)) for (x, y) ∈ D and
| f (x1 , y1 ) − f (x2 , y2 )|,
we obtain
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) = R f (lim , jm ◦ lim−1 , jm−1 ◦ · · · ◦ li1 , j1 (I)) = = =
sup
| f (x1 , y1 ) − f (x2 , y2 )|
sup
| f (x1 , y1 ) − f (x2 , y2 )|
(x1 ,y1 ),(x2 ,y2 )∈lim , jm ◦lim−1 , jm−1 ◦···◦li1 , j1 (I)
(x1 ,y1 ),(x2 ,y2 )∈lim , jm (lim−1 , jm−1 ◦···◦li1 , j1 (I))
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
| f (lim , jm (x1 , y1 ))
− f (lim , jm (x2 , y2 ))| =
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
|Fim , jm (x1 , y1 , f (x1 , y1 ))
=
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
Lh,x aim + Lh,y c jm + Lb,x + Lb,y m−1 1 s (sN)m−1−p p=0
Lh,x aim + Lh,y c jm + Lb,x + Lb,y
s(1 − (Ns)−1 )
.
R := 2N2 maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s Lh,x aim + Lh,y c jm + Lb,x + Lb,y . + s(1 − (Ns)−1 ) Since
R f [Ii, j ]
maxi, j=0,1,...,N |zi, j −z0,0 |s+s(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−s
(by Lemma 3.1),
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm . (2) If s
1 N
then we obtain the estimation
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m−1
sp
p=0
|[h(lim , jm (x1 , y1 ))
+S(lim , jm (x1 , y1 ))( f (x1 , y1 ) −b(x1 , y1 ))]−[h(lim , jm (x2 , y2 ))
< 1 then
Let
−Fim , jm (x2 , y2 , f (x2 , y2 ))|
1 Ns
1 . Nm−1−p
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm
sm R f (I) +
Proof. Since
sup
< s, that is,
+(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
(x1 ,y1 ),(x2 ,y2 )∈I
1 N
R f (I) +
R f (I) R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Nm
R f [I] =
(1) If
sp
R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y ) Nm
m−1 p=0
1 Nm−1−p
1 1 N p Nm−1−p
R f (I) m + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y ) m−1 . Nm N
+S(lim , jm (x2 , y2 ))( f (x2 , y2 ) − b(x2 , y2 ))]| =
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
|S(lim , jm (x1 , y1 ))[ f (x1 , y1 )
− f (x2 , y2 )] + [h(lim , jm (x1 , y1 )) − h(lim , jm (x2 , y2 ))]
Theorem
3.4. (The
Minkowski
dimension)
Let
E :=
{( Ni , Nj , zi, j ) : i, j = 0, 1, . . . , N} be the data set such that the points of
358
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
i
j , zi , j : j = 0, 1, . . . , N N N 0 i j 0 , , zi, j0 : i = 0, 1, . . . , N or E j0 := N N Ei0 :=
0
,
are non-collinear for some i0 ∈ {0, 1, . . . , N}(or some j0 ∈ {0, 1, . . . , N}). Let f be the BFIF of the set of data E. (1) If N1 < minx,y∈[0,1] |S(x, y)| maxx,y∈[0,1] |S(x, y)| = s < 1, then
2 < 3 + logN min
x,y∈[0,1]
(2) If 0 < s
1 N,
|S(x, y)| dimM Gr( f ) 3 + logN s < 3.
then dimM Gr( f ) = 2 .
Proof. Without loss of generality, we may assume that j j i j i i p1 = ( N1 , N0 , zi1 , j0 ), p2 = ( N2 , N0 , zi2 , j0 ), p3 = ( N3 , N0 , zi3 , j0 ) are non-collinear for some i1 , i2 , i3 , j0 ∈ {0, 1, . . . , N}. Let m ∈ N, k ∈ {1, 2, . . . , m} and
wi1 , j1 ;i2 , j2 ;...;im , jm := wim , jm ◦ wim−1 , jm−1 ◦ · · · ◦ wi1 , j1
3 + logN min
x,y∈[0,1]
(2) Let 0 < s <
|S(x, y)| dimM Gr( f ) 3 + logN s.
1 N.
Since
R f (I) Nm
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm )
+(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m Nm−1
(by Lemma 3.3 (2)), we have
NN−m (Gr( f )) N2m (2 + R f (Ii1 , j1 ;i2 , j2 ;...;im , jm )Nm )
N2m 2 +
R (I) f
+Lb,x + Lb,y )
Nm
+ (Lh,x aim + Lh,y c jm
m Nm−1
Nm
= N2m [2 + R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y )mN].
for all ik , jk ∈ {1, 2, . . . , N}. Then
Thus, dimM Gr( f ) 2. Since the Minkowski dimension of any surface is at least 2, dimM Gr( f ) = 2.
wi1 , j1 ;i2 , j2 ;...;im , jm ( p1 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p2 ), wi1 , j1 ;i2 , j2 ;...;im , jm
Acknowledgment
( p3 ) ∈ wi1 , j1 ;i2 , j2 ;...;im , jm (Gr( f )). The height of the triangle with vertices
wi1 , j1 ;i2 , j2 ;...;im , jm ( p1 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p2 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p3 ) is at least h( minx,y∈[0,1] |S(x, y)|)m , where h is the vertical distance from p2 to the segment [p1 , p3 ]. (1) Let N1 < minx,y∈[0,1] |S(x, y)| maxx,y∈[0,1] |S(x, y)| = s < 1. By Lemma 3.3 (1) and the assumption N1 < minx,y∈[0,1] |S(x, y)| s < 1, we obtain
h
min
x,y∈[0,1]
m |S(x, y)| R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm .
We cover Gr(f) by boxes of side N−m . Let NN−m (Gr( f )) denote the minimal number of boxes of side N−m that cover the graph of f. Then
N2m h N
2m
min
x,y∈[0,1]
m |S(x, y)| Nm NN−m (Gr( f ))
(2 + Rsm Nm ).
Since
dimM Gr( f ) = lim
m→+∞
log NN−m (Gr( f )) m log N
(if this limit exists), we obtain that
The author would like to thank Maurice Courbage for help during preparation of this paper. References [1] Barnsley MF. Fractal functions and interpolation. Constr Approx 1986;2:303–29. [2] Barnsley MF. Fractals Everywhere. New York: Academic Press; 1988. [3] Barnsley MF, Massopust PR. Bilinear fractal interpolation and box dimension. J Approx Theory 2015;192:362–78. [4] Massopust PR. Fractal surfaces. J Math Anal Appl 1990;151:275–90. [5] Bouboulis P, Dalla L. Closed fractal interpolation surfaces. J Math Anal Appl 2007;327:116–26. [6] Bouboulis P, Dalla L. Fractal interpolation surfaces derived from fractal interpolation functions. J Math Anal Appl 2007;336:919–36. [7] Bouboulis P, Dalla L, Drakopoulos V. Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension. J Approx Theory 2006;141:99–117. [8] Xie H, Sun H. The study on bivariate fractal interpolation functions and creation of fractal interpolated surfaces. Fractals 1997;5(4):625–34. [9] Dalla L. Bivariate fractal interpolation functions on grids. Fractals 2002;10(1):53–8. [10] Feng ZG, Feng YZ, Yuan ZY. Fractal interpolation surfaces with function vertical scaling factors. Appl Math Lett 2012;25(11):1896–900. [11] Malysz R. The Minkowski dimension of the bivariate fractal interpolation surfaces. Chaos Solitons Fractals 2006;27(5):1147–56. [12] Zhao N. Construction and application of fractal interpolation surfaces. Vis Comput 1996;12:132–46.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Nonlinear bivariate fractal interpolation function on grids✩ SongIl Ri∗ Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 March 2015 Accepted 15 October 2015
a b s t r a c t In this paper we introduce a new construction of the fractal interpolation surface (FIS) using an even more general IFS which can generate self-affine and non self-affine fractal surfaces. Here we present the general types of fractal surfaces that are based on nonlinear IFSs. © 2015 Elsevier Ltd. All rights reserved.
MSC: 37C45 28A80 Keywords: Function system (IFS) Attractor Nonlinear fractal interpolation surface
1. Introduction The concept of fractal interpolation functions (FIFs) was introduced by Barnsley [1] on the basis of the theory of iterated function systems (IFSs). Barnsley defined a fractal interpolation function (FIF) in 1986 and presented a construction of fractal curves by fractal interpolation. In the developments of theory of FIFs, many researchers have generalized the notion of FIFs in different ways. The fractal interpolated curves have been discussed in detail in the literature [2,3] and Massopust [4] was the first to put forward the construction of fractal surfaces via IFS. In recent years, many authors studied construction methods of FISs with the data set given on rectangular grids (see [5–12]). Xie et al. [8] introduced the construction of an attractor that contains the interpolation points of a rectangular data set, but generally is not a graph of a continuous function. In order to ensure the continuity of the surface, Dalla [9] assumed that the interpolation nodes on the boundary are collinear. But the vertical scaling factors used in the iterated function system must all be constant in this case. Vertical scaling factors in
✩ ∗
Construction was presented on D.P.R.K. 2015. Tel.: +86015543562662. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.chaos.2015.10.020 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
IFS are important one that characterizes the constructed fractal set . Feng et al. [10] presented a construction of fractal surfaces on a new condition that the function vertical scaling factors are 0 on the grid of rectangular domain. In this article we introduce a new construction of a bivariate fractal interpolation surface (BFIS) of the set of data and generalize the construction of fractal surfaces in [9,10]. We show that the graph of a nonlinear fractal interpolation function is the attractor of an underlying contractive nonlinear IFS. 2. Fixed point of operator and attractor of iterated function system Let a set {(xi , yi , zi, j ) : i = 1, 2, . . . , M; j = 1, 2, . . . , N} be given. The following theorem devises a general construction of the nonlinear FIS. Let I = [a, b], J = [c, d], a = x0 < x1 < · · · < xM = b and c = y0 < y1 < · · · < yN = d. Let each of the sets
{(x0 , y j , z0, j ) : j = 1, 2, . . . , N}, {(xM , y j , zM, j ) : j = 0, 1, . . . , N}, {(xi , y0 , zi,0 ) : i = 0, 1, . . . , M} and {(xi , yN , zi,N ) : i = 0, 1, . . . , M} be collinear. Let Ii , Jj , D and Di, j denote [xi−1 , xi ], [y j−1 , y j ], I × J and Ii × Jj . Let ai :=
xi −xi−1 xM −x0 , bi
:= xi−1 −
xi −xi−1 xM −x0 x0 , c j
:=
352
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
y j −y j−1 yN −y0
y −y
and d j := y j−1 − yj −yj−1 y0 . For i = 1, 2, . . . , M and j = N 0 1, 2, . . . , N, we define mappings li, j : D → Di, j , where li, j is a continuous function such that
li, j (x, y) : = (ai x + bi , c j x + d j ) f or i = 1, 2, . . . , M; j = 1, 2, . . . , N. Let L: D → D be a function such that L(x, y) := li,−1j (x, y) for (x, y) ∈ Di, j . That is,
⎧ −1 l1,1 (x, y), ⎪ ⎪ ⎪ ⎨ ··· L(x, y) := li,−1j (x, y), ⎪ ⎪ ⎪ ⎩ −1· · · lM,N (x, y),
(x, y) ∈ [x0 , x1 ] × [y0 , y1 ] (x, y) ∈ [xi−1 , xi ] × [y j−1 , y j ] (x, y) ∈ [xM−1 , xM ] × [yN−1 , yN ].
Similarly, Tg(xi , y j−1 ) = zi, j−1 , Tg(xi−1 , y j ) = zi−1, j , Tg(xi−1 , y j−1 ) = zi−1, j−1 . Hence Tg(xi , y j ) = zi, j for all i = 1, 2, . . . , M; j = 1, 2, . . . , N. Then Tg is not always a continuous surface over
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}. But T is well defined. It is obvious that Tg(x, y) is continuous when (x, y) ∈ Di, j . For (xi , y) ∈ Di, j ∩ Di+1, j , let (x , y ) := li,−1j (xi , y) and
−1 (x
, y
) := li+1, (x , y). j i Since g(x , y ) = b(x , y ), g(x
, y
) = b(x
, y
), L(xi , y) = li,−1j (xi , y) = (x , y ) = (xM , y ) for (x, y) ∈ Di, j and L(x, y) = −1 (x, y) = (x
, y
) = (x0 , y
) for (x, y) ∈ Di+1, j , li+1, j
Definition 2.1. Let for some constants L1 > 0, L2 > 0, | f (x1 , y1 ) − f (x2 , y2 )| L1 |x1 − x2 | + L2 |y1 − y2 |(∀(x1 , y1 ), ∀(x2 , y2 ) ∈ D ⊂ R2 ). Then we call f: D → R Lipschitz continuous function on the region D ⊂ R2 .
Tg(xi , y) =
Let S: D → R be Lipschitz continuous function such that s := max(x,y)∈D |S(x, y)| < 1. We denote the set of Lipschitz continuous functions f: D → R by C 0 = C 0 (D). Let C∗ ⊂ C0 be the set of Lipschitz continuous functions defined on D, such that
=
i=1 j=1
−b(L(xi , y)))]χIi ×J j (xi , y)
−b(xM+1 , y ))]χIi ×J j (xi , y) =
f ((1 − λ)x0 + λxM , yN ) = (1 − λ)z0,N + λzM,N
N M [h(xi , y) + S(xi , y)(g(x0 , y
) i=1 j=1
−b(x0 , y
))]χIi ×J j (xi , y) = h(xi , y)
f (xM , (1 − λ)y0 + λyN ) = (1 − λ)zM,0 + λzM,N ,
for λ ∈ [0, 1]. Let C ∗∗ := { f ∈ C ∗ : f (xi , y j ) = zi, j , i = 0, 1, . . . , M; j = 0, 1, . . . , N}. It is well-known that (C∗ , d∞ ) and (C∗∗ , d∞ ) are complete metric spaces, where
N M [h(xi , y) + S(xi , y)(g(xM+1 , y ) i=1 j=1
f (x0 , (1 − λ)y0 + λyN ) = (1 − λ)z0,0 + λz0,N , f ((1 − λ)x0 + λxM , y0 ) = (1 − λ)z0,0 + λzM,0 ,
N M [h(xi , y) + S(xi , y)(g(L(xi , y))
for (xi , y) ∈ Di, j and (xi , y) ∈ Di+1, j . (xi−1 , y) ∈ Di−1, j ∩ Di, j , (x, y j−1 ) ∈ Di, j−1 ∩ Di, j and (x, y j ) ∈ Di, j+1 ∩ Di, j can be dealt similarly. Hence T: C∗ → C∗∗ and since C∗∗ ⊂ C∗ , Tn : C∗∗ → C∗∗ (n ≥ 2).
d∞ ( f, g) = max{| f (x, y) − g(x, y)| : (x, y) ∈ D}.
Theorem 2.3. If s < 1 then T is a contraction and it possesses a unique fixed point f ∈ C∗∗ . In particular, f: D → R is a continuous function that interpolates the given data
Let b ∈ C∗ and h ∈ C∗∗ . We define T: C∗ → C0 by
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
Tg(x, y) =
N M
Proof. To prove contractivity, we observe that
i=1 j=1
d∞ (Tg1 , Tg2 ) = max |S(x, y)(g1 (L(x, y)) − g2 (L(x, y)))|
[h(x, y) + S(x, y)(g(L(x, y))
− b(L(x, y)))]χIi ×J j (x, y) for (x, y) ∈ D, where χIi ×J j denotes the characteristic function of a set Ii × Jj . Lemma 2.2. Tg ∈ C∗∗ . That is, T: C∗ → C∗∗ and Tn : C∗∗ → C∗∗ (n ≥ 2). Proof. Since
li, j (x0 , y0 ) = (xi−1 , y j−1 ), li, j (x0 , yN ) = (xi−1 , y j ), li, j (xM , y0 ) = (xi , y j−1 ), li, j (xM , yN ) = (xi , y j ), L(xi−1 , y j−1 ) = (x0 , y0 ), L(xi−1 , y j ) = (x0 , yN ), L(xi , y j−1 ) = (xM , y0 ), L(xi , y j ) = (xM , yN ) and g, b ∈ C∗ , hence
Tg(xi , y j ) =
N M [h(xi , y j ) + S(xi , y j )(g(L(xi , y j )) i=1 j=1
− b(L(xi , y j )))]χIi ×J j (xi , y j ) = h(xi , y j ) = zi, j for all i = 1, 2, . . . , M; j = 1, 2, . . . , N.
(x,y)∈D
s max
(x,y)∈Di, j
{|g1 (li,−1j (x, y)) − g2 (li,−1j (x, y))| : i = 1, 2, . . . ,
M; j = 1, 2, . . . , N} = sd∞ (g1 , g2 ). By Lemma 2.2, the existence of an unique fixed point f ∈ C∗∗ (when s < 1) follows from the contraction mapping theorem and graph of f passes through the interpolation points. This completes the proof. For i = 1, 2, . . . , M and j = 1, 2, . . . , N, define mappings D × R → Di, j × R by
wi, j (x, y, z) := (li, j (x, y), Fi, j (x, y, z)), where functions li, j : D → Di, j and Fi, j : D × R → R are defined by li, j (x, y) := (ai x + bi , c j y + d j ) and
Fi, j (x, y, z) := h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y)). That is,
wi, j
x y z
=
ai x + bi c jy + d j , h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y))
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
353
where
Then for any (x1 , y1 , z1 ), (x2 , y2 , z2 ) ∈ D × [−k, k];
y j − y j−1 xi − xi−1 xi − xi−1 ai := , bi := xi−1 − x0 , c j := , x M − x0 x M − x0 y N − y0 y j − y j−1 d j := y j−1 − y0 y N − y0
|S(li, j (x1 , y1 ))z1 − S(li, j (x2 , y2 ))z2 | |S(li, j (x1 , y1 ))||z1 − z2 | + |z2 ||S(li, j (x1 , y1 )) −S(li, j (x2 , y2 ))| s|z1 − z2 | + k|S(li, j (x1 , y1 )) − S(li, j (x2 , y1 ))| +k|S(li, j (x2 , y1 )) − S(li, j (x2 , y2 ))| s|z1 − z2 | + kLS,x ai |x1 − x2 | + kLS,y c j |y1 − y2 |,
for i = 1, 2, . . . , M; j = 1, 2, . . . , N. Let Mb := max(x,y)∈D |b(x, y)|, S: D → R be Lipschitz continuous functions such that s := max(x,y)∈D |S(x, y)| < 1 and
Mh,S,b :=
max |h(li, j (x, y))
max
1iM,1 jN (x,y)∈D
for any (x1 , y1 ), (x2 , y2 ) ∈ D;
−S(li, j (x, y))b(x, y)|. Let (x, y, z) ∈ D × [−k, k]. Because s < 1, if k ≥ Mh,S,b (1 − s)k and
Mh,S,b 1−s
then
|Fi, j (x, y, z)| = |h(li, j (x, y)) + S(li, j (x, y))(z − b(x, y))| |S(li, j (x, y))||z| + |h(li, j (x, y)) − S(li, j (x, y))b(x, y)| sk + Mh,S,b sk + (1 − s)k = k.
|h(li, j (x1 , y1 )) − h(li, j (x2 , y2 ))| |h(li, j (x1 , y1 )) − h(li, j (x2 , y1 ))| +|h(li, j (x2 , y1 )) − h(li, j (x2 , y2 ))| Lh,x ai |x1 − x2 | + Lh,y c j |y1 − y2 | and
has been constructed.
|S(li, j (x1 , y1 ))b(x1 , y1 ) − S(li, j (x2 , y2 ))b(x2 , y2 )| |S(li, j (x1 , y1 ))||b(x1 , y1 ) − b(x2 , y2 )| +|b(x2 , y2 )||S(li, j (x1 , y1 )) − S(li, j (x2 , y2 ))| s|b(x1 , y1 ) − b(x2 , y2 )| + |b(x2 , y2 )|(LS,x ai |x1 −x2 | + kLS,y c j |y1 − y2 |) sLb,x |x1 − x2 | + sLb,y |y1 − y2 | + Mb LS,x ai |x1 −x2 | + Mb LS,y c j |y1 − y2 |).
Theorem 2.4. We consider the IFS defined above, associated with the set of data.
So
M
h,S,b Hence for any k ≥ 1−s and (x, y, z) ∈ D × [−k, k], it follows that |Fi, j (x, y, z)| ≤ k. That is,
Fi, j : D × [−k, k] → D × [−k, k]. Then an iterated function system
{D × [−k, k]; wi, j : i = 1, 2, . . . , M; j = 1, 2, . . . , N}
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) − ai |x1 − x2 |
−c j |y1 − y2 | = θ [h(li, j (x1 , y1 )) + S(li, j (x1 , y1 ))
Let s := max{|S(x, y)|: (x, y) ∈ D} < 1, S: D → R be Lipschitz Mh,S,b 1−s .
continuous functions and k ≥ Then there exists some metric d∗ that is equivalent to the Euclidean metric d on [x0 , xM ] × [y0 , yN ] × k such that the wi, j are contractions for all i = 0, 1, . . . , M; j = 0, 1, . . . , N with respect to d∗ . In particular, there exists a unique nonempty compact set G ⊂ [x0 , xM ] × [y0 , yN ] × k, such that
G=
N M
wi, j (G).
i=1 j=1
(z1 − b(x1 , y1 ))] − [h(li, j (x2 , y2 )) + S(li, j (x2 , y2 ))
(z2 − b(x2 , y2 ))]
θ s|z1 − z2 | + kLS,x ai |x1 − x2 | + kLS,y c j |y1 − y2 | +Lh,x ai |x1 − x2 | + Lh,y c j |y1 − y2 | + sLb,x |x1 − x2 | sθ |z1 − z2 | + θ (kLS,x ai + Lh,x ai + sLb,x +Mb LS,x ai )|x1 − x2 | + θ (kLS,y c j + Lh,y c j
Proof. We define a metric d∗ on R3 by
d∗ ((x1 , y1 , z1 ), (x2 , y2 , z2 )) := |x1 − x2 | + |y1 − y2 | + θ |z1 − z2 |, where θ is a positive real number which is specified below. Then this metric is equivalent to the Euclidean metric on R3 . Let
|S(x1 , y) − S(x2 , y)| LS,x |x1 − x2 |(∀y ∈ J), |S(x, y1 ) − S(x, y2 )| LS,y |y1 − y2 |(∀x ∈ I), |b(x1 , y) − b(x2 , y)| Lb,x |x1 − x2 |(∀y ∈ J), |b(x, y1 ) − b(x, y2 )| Lb,y |y1 − y2 |(∀x ∈ I), |h(x1 , y) − h(x2 , y)| Lh,x |x1 − x2 |(∀y ∈ J),
+sLb,y + Mb LS,y c j )|y1 − y2 |. Hence
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) {ai + θ [kLS,x ai + Lh,x ai + sLb,x + Mb LS,x ai ]}|x1 − x2 | +{c j + θ [kLS,y c j + Lh,y c j + sLb,y + Mb LS,y c j ]}|y1 − y2 | +sθ |z1 − z2 |. Let
θ := min
1 − max1iM ai , 2 max1iM (kLS,x ai + Lh,x ai + sLb,x + Mb LS,x ai )
1 − max1 jN c j . 2 max1 jN (kLS,y c j + Lh,y c j + sLb,y + Mb LS,y c j )
and
|h(x, y1 ) − h(x, y2 )| Lh,y |y1 − y2 |(∀x ∈ I).
+sLb,y |y1 − y2 | + Mb LS,x ai |x1 − x2 | + Mb LS,y c j |y1 − y2 |
We have
354
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
d∗ (wi, j (x1 , y1 , z1 ), wi, j (x2 , y2 , z2 )) 1 + cj 1 + ai |x1 − x2 | + |y1 − y2 | + sθ |z1 − z2 | 2 2
1 + max1iM ai 1 + max1 jN c j , ,s max 2 2
(|x1 − x2 | + |y1 − y2 | + θ |z1 − z2 |)
max d
∗
1 + max1iM ai 1 + max1 jN c j , ,s 2 2
((x1 , y1 , z1 ), (x2 , y2 , z2 )).
Since ai <1(i = 1, 2, . . . , M), c j <1( j = 1, 2, . . . , N) and s < 1, the wi, j are all contraction mappings on metric space ([x0 , xM ] × [y0 , yN ] × R, d∗ ). Because metric d∗ is equivalent to the Euclidean metric d, therefore there exists a unique nonempty compact set G ⊂ [x0 , xM ] × [y0 , yN ] × R on metric space ([x0 , xM ] × [y0 , yN ] × R, d) such that
G=
N M
wi, j (G).
Proof. Let f(x, y) be a fixed point of operator T and G denote the graph of f. Then
f (li, j (x, y)) = (T f )(li, j (x, y)) = h(li, j (x, y))
z +z
−z
−z
−s (z
+z
−z
0,0 0,N i, j−1 i, j M,N gi, j = i, j i−1, j−1 i−1,(j y −y N 0 )(xM −x0 ) = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 −si, j (zM,N + z0,0 − z0,N − zM,0 ),
z
−z
−s (z
−z
)−g y (x −x )
z
−z
−s (z
−z
)−g x (y −y )
−zM,0 )
ei, j = i−1, j−1 i, j−1 i, j x00,0−xMM,0 i, j 0 0 M = −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )], fi, j = i−1, j−1 i−1, j i, j y00,0−yN 0,N i, j 0 0 N = −[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )] and
+[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )] −si, j zM,N − [zi, j + zi−1, j−1 − zi−1, j − zi, j−1 −si, j (zM,N + z0,0 − z0,N − zM,0 )] = zi−1, j−1 − si, j z0,0 for all i = 1, 2, . . . , M; j = 1, 2, . . . , N(see [9], p.54–55). So we obtain that
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j
+S(li, j (x, y))( f (x, y) − b(x, y))
= −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )]x
= Fi, j (x, y, f (x, y)) for (x, y) ∈ D and for i = 1, 2, . . . , M; j = 1, 2, . . . , N. Hence we obtain that
wi, j (G) = {wi, j (x, y, f (x, y)) : (x, y) ∈ D}
{(ai x + bi , c j y + d j , h(li, j (x, y)) +S(li, j (x, y))( f (x, y) − b(x, y)) : (x, y) ∈ D} = {(ai x + bi , c j y + d j , (T f )(li, j (x, y)) : (x, y) ∈ D} = {(ai x + bi , c j y + d j , f (li, j (x, y)) : (x, y) ∈ D} = {(x, y, f (x, y)) : (x, y) ∈ Di, j }. =
Hence
i=1 j=1
where
= zi, j + [zi−1, j−1 − zi, j−1 − si, j (z0,0 − zM,0 )]
Corollary 2.5. Let G be an attractor of the IFS {D × R; wi, j , i = 1, 2, . . . , M; j = 1, 2, . . . , N} defined above and f(x, y) be a fixed point of operator T : g → h + S · (g ◦ L − b ◦ L). Then the graph f is G.
{(x, y, f (x, y)) : (x, y) ∈ Di, j } =
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j ,
= zi, j − ei, j − fi, j − si, j zM,N − gi, j
This completes the proof.
M N
Remark 2.7. Let [x0 , xM ] × [y0 , yN ] = [0, 1] × [0, 1]. 1. In [9],
ki, j = zi, j − ei, j xM − fi, j yN − si, j zM,N − gi, j xM yN
i=1 j=1
G=
is the restriction of fˆ on [x0 , xM ] × [y0 , yN ], where fˆ : [x−1 , xM+1 ] × [y−1 , yN+1 ] → R is a fractal interpolation function according to Theorem 2.3 and Theorem 2.4.
M N
wi, j (G).
i=1 j=1
By Theorem 2.4, only one nonempty compact set G is an attractor of the IFS. Remark 2.6. General Construction (see [9], p.57) Now let’s consider new data
{(xi , y j , zˆi, j ) : i = −1, 0, . . . , M + 1; j = −1, 0, . . . , N + 1}, where zˆi, j = zi, j for i = 0, 1, . . . , M; j = 0, 1, . . . , N, x−1 < x0 , y−1 < y0 , xM < xM+1 , yN < yN+1 and zˆ−1, j = zˆM+1, j = zˆi,−1 = zˆi,N+1 ≡ 0 for all i = −1, 0, . . . , M + 1; j = −1, 0, . . . , N + 1. Then the function f which interpolates the given data
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}
−[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )]y +[zi, j + zi−1, j−1 − zi−1, j − zi, j−1 − si, j
(zM,N + z0,0 − z0,N − zM,0 )]xy +si, j z + [zi−1, j−1 − si, j z0,0 ] = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy] + si, j [z − (z0,0 (1 − x)(1 − y) +zM,0 x(1 − y) + z0,N (1 − x)y + zM,N xy)]. Let Si, j (x, y) := si, j = s(|si, j | < 1) for all i = 1, 2, . . . , M; j = 1, 2, . . . , N, b(x, y) := z0,0 (1 − x)(1 − y) + zM,0 x(1 − y) + z0,N (1 − x)y + zN,M xy and h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy)]. Then b ∈ C∗ , h ∈ C∗∗ and
Fi, j (x, y, z) = ei, j x + fi, j y + gi, j xy + si, j z + ki, j = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy)] + si, j [z − (z0,0 (1 − x)(1 − y) +zM,0 x(1 − y) + z0,N (1 − x)y + zM,N xy)] = h(li, j (x, y)) + si, j (z − b(x, y)). 2. In [10],
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ), where
si, j (x, y) = =
λi, j (x − x0 )(xM − x)(y − y0 )(yN − y) λi, j xy(1 − x)(1 − y),
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
355
zi, j + zi−1, j−1 − zi−1, j − zi, j−1 (xM − x0 )(yN − y0 ) = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 ,
3. The Minkowski dimension of the nonlinear bivariate fractal interpolation surfaces
−y0 (zi, j − zi−1, j ) + yN (zi, j−1 − zi−1, j−1 ) (xM − x0 )(yN − y0 ) = zi, j−1 − zi−1, j−1 ,
In this section we present the formula for the Minkowski dimension of a bivariate fractal interpolation surface. We use the following notation: I := [0, 1] × [0, 1] j−1 j i and Ii, j := [ i−1 N , N ] × [ N , N ] for i, j = 1, 2, . . . , N. Let E :=
g i, j =
e i, j =
xM (zi−1, j − zi−1, j−1 ) − x0 (zi, j − zi, j−1 ) (xM − x0 )(yN − y0 ) = zi−1, j − zi−1, j−1 ,
fi, j =
{( Ni , Nj , zi, j ); i, j = 0, 1, 2, . . . , N} be the set of data and Pk be a rectangle, that is,
Pk :=
and
x0 y0 zi, j − x0 yN zi, j−1 − xM y0 zi−1, j + xM yN zi−1, j−1 k i, j = (xM − x0 )(yN − y0 ) = zi−1, j−1 . (see [10], p.1898) Hence
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ) = λi, j xy(1 − x)(1 − y)z + [zi, j−1 − zi−1, j−1 ]x +[zi−1, j − zi−1, j−1 ]y + [zi, j + zi−1, j−1 −zi−1, j − zi, j−1 ]xy + zi−1, j−1 = λi, j xy(1 − x)(1 − y)z + [zi−1, j−1 (1 − x)(1 − y) +zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy].
i
+zi−1, j (1 − x)y + zi, j xy]. Then
λi, j xy(1 − x)(1 − y)z + [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy] = h(li, j (x, y)) + si, j (x, y)z = h(li, j (x, y)) +si, j (x, y)(z − b(x, y)).
R f [I] :=
sup
(x1 ,y1 ),(x2 ,y2 )∈I
| f (x1 , y1 ) − f (x2 , y2 )|.
Lemma 3.1. Let f be BFIF of the set of data E. If s := max {|S(x, y)|: (x, y) ∈ I} < 1 then
R f [Ii, j ]
maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x +Lb,y ) + (Lh,x + Lh,y ) . 1−s
Proof. Let for k ∈ N,
k := max{| f (x, y) − z0,0 |; (x, y) ∈ Pk }, λk := max max{| f (x, y) − zi−1, j−1 |; i, j=1,2,...,N (x,y)
(x, y) ∈ Pk ∩ Ii, j }. By Theorem 2.3, Theorem 2.4, for (x, y) ∈ Pk ∩ Ii, j ,
f (li, j (x, y)) = (T f )(li, j (x, y)) = h(li, j (x, y)) +S(li, j (x, y))( f (x, y) − b(x, y))
Fi, j (x, y, z) =
Example 2.8. Let [x0 , xM ] × [y0 , yM ] = [0, 1] × [0, 1]. Let b(x, y) := z0,0 (1 − x)(1 − y) + zM,0 x(1 − y) + z0,N (1 − x)y + zN,M xy. Then z0,0 − b(x0 , y0 ) = 0, z1,0 − b(x1 , y0 ) = 0, z0,1 − b(x0 , y1 ) = 0 and z1,1 − b(x1 , y1 ) = 0. Let h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) + zi−1, j (1 − x)y + zi, j xy)] and S: [0, 1] × [0, 1] → R be Lipschitz continuous functions such that
j ; i, j = 0, 1, 2, . . . , Nk . Nk
We denote the maximum range of a function f by
Let b(x, y) ≡ 0 and
h(li, j (x, y)) = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y)
N
, k
= Fi, j (x, y, f (x, y)) and for (x, y) ∈ D,
f (x, y) := h(x, y) + S(x, y)( f (L(x, y)) − b(L(x, y))). Since li, j (0, 0) = xi−1, j−1 , h(xi−1, j−1 ) = zi−1, j−1 and b(0, 0) = z0,0 , we obtain
f (x, y) − zi−1, j−1 = h(x, y) + S(x, y)( f (L(x, y)) −b(L(x, y))) − zi−1, j−1 = h(x, y) + S(x, y)( f (L(x, y)) − b(L(x, y))) − h(li, j (0, 0)) = h(x, y) + S(x, y)( f (L(x, y)) − z0,0 +b(0, 0) −b(L(x, y))) −h(li, j (0, 0))
xy si, j (x, y) = . (1 + x)(1 + y)
|S(x, y)|| f (L(x, y)) − z0,0 | + |S(x, y)||b(0, 0)
Define Fi, j : [0, 1] × [0, 1] × R → R by Fi, j (x, y, z) = h(li, j (x, y)) + si, j (x, y)(z − b(x, y)). By Theorem 2.3 and Theorem 2.4, there exists a continuous function f: [0, 1] × [0, 1] → R that interpolates the given data {(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}. This result is a substantial generalization of [9], Proposition 2.2, p.56 and [10], Theorem 3.1, p.1898.
s| f (L(x, y)) − z0,0 | + s|b(0, 0) − b(L(x, y)))|
Remark 2.9. If b(x, y) ∈ C∗ and h(x, y) ∈ C∗∗ or si, j (x, y) = 0 and h(x, y) ∈ C∗∗ then there exists a continuous function f: [0, 1] × [0, 1] → R that interpolates the given data
{(xi , y j , zi, j ) : i = 0, 1, . . . , M; j = 0, 1, . . . , N}.
−b(L(x, y)))| + |h(x, y) − h(li, j (0, 0))| +|h(x, y) − h(li, j (0, 0))| s| f (L(x, y)) − z0,0 | + s(Lb,x |x| +Lb,y |y|) + Lh,x |x| + Lh,y |y| s| f (L(x, y)) − z0,0 | + s(Lb,x +Lb,y ) + (Lh,x + Lh,y ). Hence
λk+1 =
max
max{| f (x, y) − zi−1, j−1 | : (x, y)
i, j=1,2,...,N (x,y)
∈ Pk+1 ∩ Ii, j }
356
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
max
max{s| f (L(x, y)) − z0,0 | + s(Lb,x + Lb,y )
and
i, j=1,2,...,N (x,y)
Fi, j (x, y, z) = h(li, j (x, y)) + si, j (z − b(x, y))
+(Lh,x + Lh,y ) : (x, y) ∈ Pk+1 ∩ Ii, j } s max{| f (x, y) − z0,0 |; (x, y) ∈ Pk }
= ei, j x + fi, j y + gi, j xy + si, j z + ki, j .
+s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) sk + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ). That is, λk+1 sk + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ). Let for some (xp , yq ) ∈ Pk , k = max{| f (x, y) − z0,0 |; (x, y) ∈ Pk } = | f (x p , yq ) − z0,0 | and (x p , yq ) ∈ Ii p , jq (i p , jq ∈ {1, 2, . . . , N}). Because | f (x p , yq ) − z0,0 | | f (x p , yq ) − zi, j | + |zi, j − z0,0 | for all i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }, | f (x p , yq ) − z0,0 | max{| f (x p , yq ) − zi, j | + |zi, j − z0,0 | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} max{| f (x p , yq ) − zi, j | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} + max{|zi, j − z0,0 | : i ∈ {i p − 1, i p }, j ∈ { jq − 1, jq }} maxi, j=1,2,...,N max(x,y) {| f (x, y) − zi−1, j−1 |; (x, y) ∈ Pk ∩ Ii, j } + max{| f (x, y) − z0,0 |; (x, y) ∈ Pk } = λk + 1 . Since k 1 + λk , we obtain that
λk+1 s(1 + λk ) + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) s[1 + sk−1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] +[s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] = [s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] + s[sk−1 +s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] [s1 + s(Lb,x + Lb,y ) + (Lh,x +Lh,y )]+s[s(1 + λk−1 ) +s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] = [s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] +s[s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y )] + sλk−1 . It follows by induction that
λk+1 = max max{| f (x, y) −zi−1, j−1 | : (x, y) ∈ Pk+1 ∩ Ii, j } i, j=1,2,...,N (x,y)
s1 + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) . = 1−s
Because k ∈ N is arbitrary,
R f [Ii, j ]
Remark 3.2. In [9], Let si, j (x, y) := s, |s| < 1 for all i, j = 1, 2, . . . , N,
+z0,N (1 − x)y + zN,N xy
| f (x, y) − zi−1, j−1 | = |Fi, j (L(x, y), f (L(x, y))) − zi−1, j−1 | |ei, j | + | fi, j | + |gi, j | + |si, j f (L(x, y)) − zi−1, j−1 + ki, j | = |ei, j | + | fi, j | + |gi, j | + |si, j f (L(x, y)) − si, j z0,0 | |s| max |zi, j − z0,0 | + 4|s| max |zi1 , j1 − zi2 , j2 | i1 , j1 ,i2 , j2 ∈{0,N}
i, j=0,1,...,N
+4 max max{|zi, j − zi, j−1 |, |zi, j − zi, j+1 |, |zi, j i, j=1,...,N
−zi−1, j |, |zi, j − zi+1, j |}. So
R f [Ii, j ]
1 1 − |s|
max
i1 , j1 ,i2 , j2 ∈{0,N}
|s| max |zi, j − z0,0 | + 4|s| i, j=0,1,...,N
|zi1 , j1 − zi2 , j2 |
+4 max max{|zi, j − zi, j−1 |, |zi, j i, j=1,...,N
−zi, j+1 |, |zi, j − zi−1, j |, |zi, j − zi+1, j |} . (see [11], p.1152, Theorem 1]) By Lemma 3.1, it also has
R f [Ii, j ]
1 1−|s|
maxi, j=0,1,...,N |zi, j −z0,0 ||s|+|s|(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−|s|
|s| max |zi, j − z0,0 |
|zi, j | + 8 max |zi, j | .
i, j=0,1,...,N
+8|s| max
i, j={0,N}
i, j={1,...,N}
In [10],
Fi, j (x, y, z) = si, j (x, y)z + (e i, j x + fi, j y + g i, j xy + k i, j ), where
maxi, j=0,1,...,N |zi, j −z0,0 |s+s(Lb,x +Lb,y )+(Lh,x +Lh,y ) . 1−s
b(x, y) := z0,0 (1 − x)(1 − y) + zN,0 x(1 − y)
Hence
si, j (x, y) =
λi, j xy(1 − x)(1 − y),
g i, j = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 , e i, j = zi, j−1 − zi−1, j−1 , fi, j = zi−1, j − zi−1, j−1 ,
k i, j = zi−1, j−1 . Hence
and
h(li, j (x, y)) := [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy]. Since
gi, j = zi, j + zi−1, j−1 − zi−1, j − zi, j−1 − si, j
(zM,N + z0,0 − z0,N − zN,0 ), ei, j = −[zi−1, j−1 − zi, j−1 − si, j (z0,0 − zN,0 )]; fi, j = −[zi−1, j−1 − zi−1, j − si, j (z0,0 − z0,N )]; ki, j = zi−1, j−1 − si, j z0,0
| f (x, y) − zi−1, j−1 | = |Fi, j (L(x, y), f (L(x, y))) − zi−1, j−1 |, |e i, j | + | fi, j | + |g i, j | + |si, j (x, y) f (L(x, y)) −zi−1, j−1 + k i, j | = |e i, j | + | fi, j | + |g i, j | + |si, j (x, y) f (L(x, y))| max |si, j (x, y)||zi, j | i, j=0,1,...,N
+4 max max{|zi, j − zi, j−1 |, |zi, j i, j=1,...,N
−zi, j+1 |, |zi, j − zi−1, j |, |zi, j − zi+1, j |}.
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
357
Also, since b(x, y) ≡ 0 and
+[−b(x1 , y1 ) + b(x2 , y2 )]|
h(li, j (x, y)) = [zi−1, j−1 (1 − x)(1 − y) + zi, j−1 x(1 − y) +zi−1, j (1 − x)y + zi, j xy)],
s
R f [Ii, j ] 1 1−|s|
maxi, j=0,1,...,N |si, j (x,y)||zi, j −z0,0 |+|si, j (x,y)|(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−|s|
maxi, j=0,1,...,N |si, j (x, y)||zi, j − z0,0 |
+8 maxi, j={0,1,...,N} |zi, j | .
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
+(Lh,x aim |x1 − x2 | + Lh,y c jm |y1 − y2 |) + (Lb,x |x1 − x2 | +Lb,y |y1 − y2 |) sR f (Ii1 , j1 ;i2 , j2 ;...;im−1 , jm−1 ) +
Let I := [0, 1] × [0, 1], m ∈ N, k ∈ {1, 2, . . . , m} and
Ii1 , j1 ;i2 , j2 ;...;im , jm := lim , jm ◦ lim−1 , jm−1 ◦ · · · ◦ li1 , j1 (I) for all ik , jk ∈ {1, 2, . . . , N}.
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm ,
Lh,x aim + Lh,y c jm + Lb,x + Lb,y . Nm−1
So
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sR f (Ii1 , j1 ;i2 , j2 ;...;im−1 , jm−1 ) +
Lemma 3.3. Let f be fractal interpolation function of the data set E := {( Ni , Nj , zi, j ); i, j = 0, 1, 2, . . . , N} with function vertical scaling factor S(x, y). (1) If N1 < s, then
| f (x1 , y1 ) − f (x2 , y2 )|
Lh,x aim + Lh,y c jm + Lb,x + Lb,y . Nm−1
By induction,
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm R f (I) +(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m−1
where R is a constant such that
p=0
maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s Lh,x aim + Lh,y c jm + Lb,x + Lb,y + . s(1 − (Ns)−1 )
R := 2N2
1 N,
(2) If 0 < s
then
m Nm−1
.
f (li, j (x, y)) = h(li, j (x, y)) + S(li, j (x, y))( f (x, y) − b(x, y)) = Fi, j (x, y, f (x, y)) for (x, y) ∈ D and
| f (x1 , y1 ) − f (x2 , y2 )|,
we obtain
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) = R f (lim , jm ◦ lim−1 , jm−1 ◦ · · · ◦ li1 , j1 (I)) = = =
sup
| f (x1 , y1 ) − f (x2 , y2 )|
sup
| f (x1 , y1 ) − f (x2 , y2 )|
(x1 ,y1 ),(x2 ,y2 )∈lim , jm ◦lim−1 , jm−1 ◦···◦li1 , j1 (I)
(x1 ,y1 ),(x2 ,y2 )∈lim , jm (lim−1 , jm−1 ◦···◦li1 , j1 (I))
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
| f (lim , jm (x1 , y1 ))
− f (lim , jm (x2 , y2 ))| =
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
|Fim , jm (x1 , y1 , f (x1 , y1 ))
=
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
Lh,x aim + Lh,y c jm + Lb,x + Lb,y m−1 1 s (sN)m−1−p p=0
Lh,x aim + Lh,y c jm + Lb,x + Lb,y
s(1 − (Ns)−1 )
.
R := 2N2 maxi, j=0,1,...,N |zi, j − z0,0 |s + s(Lb,x + Lb,y ) + (Lh,x + Lh,y ) 1−s Lh,x aim + Lh,y c jm + Lb,x + Lb,y . + s(1 − (Ns)−1 ) Since
R f [Ii, j ]
maxi, j=0,1,...,N |zi, j −z0,0 |s+s(Lb,x +Lb,y )+(Lh,x +Lh,y ) 1−s
(by Lemma 3.1),
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm . (2) If s
1 N
then we obtain the estimation
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m−1
sp
p=0
|[h(lim , jm (x1 , y1 ))
+S(lim , jm (x1 , y1 ))( f (x1 , y1 ) −b(x1 , y1 ))]−[h(lim , jm (x2 , y2 ))
< 1 then
Let
−Fim , jm (x2 , y2 , f (x2 , y2 ))|
1 Ns
1 . Nm−1−p
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) sm
sm R f (I) +
Proof. Since
sup
< s, that is,
+(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
(x1 ,y1 ),(x2 ,y2 )∈I
1 N
R f (I) +
R f (I) R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Nm
R f [I] =
(1) If
sp
R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y ) Nm
m−1 p=0
1 Nm−1−p
1 1 N p Nm−1−p
R f (I) m + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y ) m−1 . Nm N
+S(lim , jm (x2 , y2 ))( f (x2 , y2 ) − b(x2 , y2 ))]| =
sup
(x1 ,y1 ),(x2 ,y2 )∈lim−1 , jm−1 ◦···◦li1 , j1 (I)
|S(lim , jm (x1 , y1 ))[ f (x1 , y1 )
− f (x2 , y2 )] + [h(lim , jm (x1 , y1 )) − h(lim , jm (x2 , y2 ))]
Theorem
3.4. (The
Minkowski
dimension)
Let
E :=
{( Ni , Nj , zi, j ) : i, j = 0, 1, . . . , N} be the data set such that the points of
358
S. Ri / Chaos, Solitons and Fractals 81 (2015) 351–358
i
j , zi , j : j = 0, 1, . . . , N N N 0 i j 0 , , zi, j0 : i = 0, 1, . . . , N or E j0 := N N Ei0 :=
0
,
are non-collinear for some i0 ∈ {0, 1, . . . , N}(or some j0 ∈ {0, 1, . . . , N}). Let f be the BFIF of the set of data E. (1) If N1 < minx,y∈[0,1] |S(x, y)| maxx,y∈[0,1] |S(x, y)| = s < 1, then
2 < 3 + logN min
x,y∈[0,1]
(2) If 0 < s
1 N,
|S(x, y)| dimM Gr( f ) 3 + logN s < 3.
then dimM Gr( f ) = 2 .
Proof. Without loss of generality, we may assume that j j i j i i p1 = ( N1 , N0 , zi1 , j0 ), p2 = ( N2 , N0 , zi2 , j0 ), p3 = ( N3 , N0 , zi3 , j0 ) are non-collinear for some i1 , i2 , i3 , j0 ∈ {0, 1, . . . , N}. Let m ∈ N, k ∈ {1, 2, . . . , m} and
wi1 , j1 ;i2 , j2 ;...;im , jm := wim , jm ◦ wim−1 , jm−1 ◦ · · · ◦ wi1 , j1
3 + logN min
x,y∈[0,1]
(2) Let 0 < s <
|S(x, y)| dimM Gr( f ) 3 + logN s.
1 N.
Since
R f (I) Nm
R f (Ii1 , j1 ;i2 , j2 ;...;im , jm )
+(Lh,x aim + Lh,y c jm + Lb,x + Lb,y )
m Nm−1
(by Lemma 3.3 (2)), we have
NN−m (Gr( f )) N2m (2 + R f (Ii1 , j1 ;i2 , j2 ;...;im , jm )Nm )
N2m 2 +
R (I) f
+Lb,x + Lb,y )
Nm
+ (Lh,x aim + Lh,y c jm
m Nm−1
Nm
= N2m [2 + R f (I) + (Lh,x aim + Lh,y c jm + Lb,x + Lb,y )mN].
for all ik , jk ∈ {1, 2, . . . , N}. Then
Thus, dimM Gr( f ) 2. Since the Minkowski dimension of any surface is at least 2, dimM Gr( f ) = 2.
wi1 , j1 ;i2 , j2 ;...;im , jm ( p1 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p2 ), wi1 , j1 ;i2 , j2 ;...;im , jm
Acknowledgment
( p3 ) ∈ wi1 , j1 ;i2 , j2 ;...;im , jm (Gr( f )). The height of the triangle with vertices
wi1 , j1 ;i2 , j2 ;...;im , jm ( p1 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p2 ), wi1 , j1 ;i2 , j2 ;...;im , jm ( p3 ) is at least h( minx,y∈[0,1] |S(x, y)|)m , where h is the vertical distance from p2 to the segment [p1 , p3 ]. (1) Let N1 < minx,y∈[0,1] |S(x, y)| maxx,y∈[0,1] |S(x, y)| = s < 1. By Lemma 3.3 (1) and the assumption N1 < minx,y∈[0,1] |S(x, y)| s < 1, we obtain
h
min
x,y∈[0,1]
m |S(x, y)| R f (Ii1 , j1 ;i2 , j2 ;...;im , jm ) Rsm .
We cover Gr(f) by boxes of side N−m . Let NN−m (Gr( f )) denote the minimal number of boxes of side N−m that cover the graph of f. Then
N2m h N
2m
min
x,y∈[0,1]
m |S(x, y)| Nm NN−m (Gr( f ))
(2 + Rsm Nm ).
Since
dimM Gr( f ) = lim
m→+∞
log NN−m (Gr( f )) m log N
(if this limit exists), we obtain that
The author would like to thank Maurice Courbage for help during preparation of this paper. References [1] Barnsley MF. Fractal functions and interpolation. Constr Approx 1986;2:303–29. [2] Barnsley MF. Fractals Everywhere. New York: Academic Press; 1988. [3] Barnsley MF, Massopust PR. Bilinear fractal interpolation and box dimension. J Approx Theory 2015;192:362–78. [4] Massopust PR. Fractal surfaces. J Math Anal Appl 1990;151:275–90. [5] Bouboulis P, Dalla L. Closed fractal interpolation surfaces. J Math Anal Appl 2007;327:116–26. [6] Bouboulis P, Dalla L. Fractal interpolation surfaces derived from fractal interpolation functions. J Math Anal Appl 2007;336:919–36. [7] Bouboulis P, Dalla L, Drakopoulos V. Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension. J Approx Theory 2006;141:99–117. [8] Xie H, Sun H. The study on bivariate fractal interpolation functions and creation of fractal interpolated surfaces. Fractals 1997;5(4):625–34. [9] Dalla L. Bivariate fractal interpolation functions on grids. Fractals 2002;10(1):53–8. [10] Feng ZG, Feng YZ, Yuan ZY. Fractal interpolation surfaces with function vertical scaling factors. Appl Math Lett 2012;25(11):1896–900. [11] Malysz R. The Minkowski dimension of the bivariate fractal interpolation surfaces. Chaos Solitons Fractals 2006;27(5):1147–56. [12] Zhao N. Construction and application of fractal interpolation surfaces. Vis Comput 1996;12:132–46.