J. Math. Anal. Appl. 412 (2014) 416–425

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Journal of Mathematical Analysis andApplications

www.elsevier.com/locate/jmaa

Box-counting dimensions of fractal interpolation surfacesderived from fractal interpolation functions ✩

Zhigang Feng a,b,∗, Xiuqing Sun a

a Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR Chinab State Key Laboratory of Coal Resources and Safe Mining, China University of Mining & Technology, Beijing 100086, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 September 2010Available online 17 October 2013Submitted by C.E. Wayne

Keywords:Affine fractal interpolation functionVariationFractal interpolation surfaceBox-counting dimension

A construction method of Fractal Interpolation Surfaces on a rectangular domain witharbitrary interpolation nodes is introduced. The variation properties of the binary functionscorresponding to this type of fractal interpolation surfaces are discussed. Based on therelationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

In 1986 Barnsley [1,2] proposed the concept of the fractal interpolation function, then a new and more advantageousinterpolation method is provided for simulating the rough and irregular curves. In accordance with the idea of fractal inter-polation functions, Xie et al. [18,19] proposed a mathematical model of the fractal interpolation surfaces on a rectangulardomain G = [a,b]× [c,d], where their interpolation points set {(xi, y j, zi, j): i = 0,1, . . . ,m; j = 0,1, . . . ,n} are on the grids(xi, y j), and a = x0 < x1 < · · · < xm = b, c = y0 < y1 < · · · < yn = d. For m = n and xi − xi−1 = y j − y j−1 = 1/m, they de-rived the box-counting dimension formula of the fractal interpolation surfaces. Dalla [7] gave a method of constructions ofbivariate fractal interpolation functions, and investigated the case where the interpolation points on each edge are collinearand the general case. These results improve and correct a construction quoted in the paper [18]. Feng [10] proposed thecontinuity condition for the fractal interpolation surface on rectangular domain, and discussed the variation properties ofthe corresponding bivariate continuous functions. According to the relation between the box-counting dimension of thegraph of continuous function and its variation, the exact value of the box-counting dimension of the fractal interpolationsurface can be obtained. By analyzing the papers above, we found that in order to ensure the fractal interpolation surfacecontinuous, the interpolation points on each edge is assumed to be collinear, or they must meet the conditions that aredifficult to verify. Because of this, many researchers are exploring new methods of fractal interpolation. Bouboulis et al. [5]proposed recurrent bivariate fractal interpolation surfaces that generalized the notion of affine fractal interpolation surfaces.In the paper, the authors construct a recurrent iterated function system on rectangular grids associated with the given inter-polating set. The attractor of the recurrent iterated function system is the graph of a continuous function, called a recurrentbivariate fractal interpolation surface. Malysz [13] used some reflections to construct the iterated function system with same

✩ This research is supported by the National Nature Science Foundation of China (No. 51079064) and the State Key Laboratory for Coal Resources andSafe Mining, China University of Mining & Technology (No. SKLCRSM10KF).

* Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China.E-mail address: zgfeng@mail.ujs.edu.cn (Z. Feng).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmaa.2013.10.032

Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425 417

contraction factors, and then proposed a new construction method of fractal interpolation surface for arbitrary interpolationnodes on the grids {(i/n, j/n): i, j = 0,1, . . . ,n}, where n is a positive even number. Based on the construction of recurrentfractal interpolation functions, Bouboulis and Dalla [4] presented a new construction of fractal interpolation surfaces for ar-bitrary interpolation nodes on the grids {(xi, y j): i = 0,1, . . . , N, j = 0,1, . . . , M} of the rectangular domain [0,1] × [0, p].They also proved some properties of the fractal interpolation surfaces, and provided a lower bound of their box-countingdimension. Feng et al. [11] proposed a new method of construction for the fractal interpolation surfaces on a rectangulardomain with arbitrary interpolation nodes by using function vertical scaling factors. This construction method can get rid ofthe constraints that the interpolation nodes on the boundary are collinear and the vertical scaling factors are equal.

In this article we discuses a class of fractal interpolation surfaces on the rectangle domain [a,b] × [c,d] derived fromaffine fractal interpolation functions. These fractal interpolation surfaces are constructed in the same way as in [4]. Someproperties, especially about their variations, are studied. According to the relation between the box-counting dimension andthe variation of the graph of the continuous function, the box-counting dimension formulae are derived for this class offractal interpolation surfaces.

2. Affine fractal interpolation function

Let I = [a,b]. For a number m ∈ N and m � 2, let {(xi, zi): i = 0,1, . . . ,m} be a set of interpolation knots on I × R ,where a = x0 < x1 < · · · < xm = b. Define the linear mappings Li(x) = ai x + bi , such that Li(a) = xi−1 and Li(b) = xi , fori = 1,2, . . . ,m. For a given real array {si}m

i=1, where s = max{|si |: i = 1,2, . . . ,m} < 1, called vertical scaling factors, weconstruct the mappings from I × R to R , Fi(x, z) = si z + ci x + di , they satisfy the conditions

Fi(x0, z0) = zi−1, Fi(xm, zm) = zi, (1)

for i = 1,2, . . . ,m. Then we get the affine mappings from I × R to I × R:

wi

(xz

)=

(Li(x)

Fi(x, z)

)=

(ai 0ci si

)(xz

)+

(bidi

), (2)

where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ai = xi − xi−1

xm − x0,

bi = xi−1 − xi − xi−1

xm − x0x0,

ci = zi − zi−1

xm − x0− si

zm − z0

xm − x0,

di = xmzi−1 − x0zi

xm − x0− si

xmz0 − x0zm

xm − x0,

(3)

for i = 1,2, . . . ,m. Then an iterated function system associated with the interpolation knots {(xi, zi): i = 0,1, . . . ,m},

{I × R; wi: i = 1,2, . . . ,m} (4)

is constructed. According to the paper [1–3], we have the following theorems.

Theorem 2.1. The iterated function system (4) is hyperbolic, then there exits a unique invariant set K ⊂ R2 . It is a graph of a continuousfunction f on I , which interpolates the data set {(xi, zi): i = 0,1, . . . ,m}. That is, K = {(x, z): z = f (x), x ∈ I}, where f (xi) = zi , fori = 0,1, . . . ,m. If

∑mi=1 |si | > 1 and the interpolation knots do not all lie on a single straight line, then the box-counting dimension of

K is the unique real solution D of

m∑i=1

|si |aD−1i = 1. (5)

Otherwise the box-counting dimension of K is 1.

Frequently, the box-dimension of the function f is larger than one, and f is non-smooth everywhere. Then f is calledaffine fractal interpolation function, for it is generated by the iterated function system with the affine mappings wi , i =1,2, . . . ,m. With the following theorem can determine whether the function f is the fractal interpolation function (see [2]).

Theorem 2.2. f is the affine fractal interpolation fractal function generated by the iterated function system (4) if and only if fori = 1,2, . . . ,m, the f satisfied the equations

f(Li(x)

) = Fi(x, f (x)

), for x ∈ I. (6)

418 Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425

In [12,6], the authors discussed the difference between the fractal interpolation functions corresponding to the differentinterpolation nodes on interval [0,1]. In fact, a similar conclusion is also true on the general interval.

Theorem 2.3. Let the vertical scaling factors |si| < 1, i = 1,2, . . . ,m, be given. For two data sets {(xi, zi): i = 0,1, . . . ,m} and{(xi, zi): i = 0,1, . . . ,m}, let f and g be the affine fractal interpolation functions generated by the two iterated function systemsassociated with the data sets above respectively. Then

‖ f − g‖∞ � 1 + s

1 − smax

{|zi − zi |: i = 0,1, . . . ,m}, (7)

where ‖ f − g‖∞ = max{| f (x) − g(x)|: x ∈ I}.

Proof. Let x ∈ I . Suppose x ∈ [xi−1, xi]. According to Theorem 2.2, f (x) = si f (L−1i (x)) + ci L−1

i (x) + di , and g(x) =si f (L−1

i (x))+ ci L−1i (x)+ di , where ci , di , ci and di can be calculated by Eq. (3). Let Li(x) = (ci − ci)L−1

i (x)+ (di − di). BecauseLi(x) is a linear function on [xi−1, xi], and by Eq. (1), Li(xi−1) = (zi−1 − zi−1) − si(z0 − z0), Li(xi) = (zi − zi) − si(zm − zm),then we have |Li(x)| � (1+ s) ·max{|zi − zi |: i = 0,1, . . . ,m}. Therefore | f (x)− g(x)| � s‖ f − g‖∞ + (1+ s) ·max{|zi − zi |: i =0,1, . . . ,m}. Inequality (7) is true. �3. Variation of continuous function

Let G = [a1,b1] × [a2,b2] × · · · × [an,bn] be a rectangular domain in Rn , f be a continuous function on G . For a realnon-negative number γ and any x = (x1, x2, . . . , xn) ∈ G , denote G[x;γ ] = G ∩ [x1 − γ , x1 + γ ] × [x2 − γ , x2 + γ ] × · · · ×[xn − γ , xn + γ ], then the value

O Gf ;γ (x) = sup

G[x;γ ]f(x′) − inf

G[x;γ ] f(x′) = sup

x′,x′′∈G[x;γ ]∣∣ f

(x′) − f

(x′′)∣∣

is called γ -oscillation of function f at the point x about G . It is also denoted by O f ;γ (x) for short. Because f is a continuousfunction on the closed domain G , it is obvious that O f ;γ (x) is a continuous function of x on G . Then O f ;γ (x) is Riemannintegrable. The Riemann integral of O f ;γ (x) on G ,

∫G O f ;γ (x)dx, is called γ -variation of the function f on the domain G ,

denoted by V f ;γ (G).The variation of a continuous function has been studied by Dubuc, Tricot, Quiniou and others (see for example [16,8,17,

9]). Using the variation, one can calculate the box-counting dimension of the graph of a continuous function. Let f be acontinuous function on a rectangular domain G ⊂ Rn , and Γ ( f ; G) ⊂ Rn+1 be the graph of the function f on G , then wehave the box-counting dimension formulae (see [8,9]),

dimBΓ ( f ; G) = lim supγ →0+

(n + 1 − log V f ;γ (G)

logγ

), (8)

and

dimBΓ ( f ; G) = lim infγ →0+

(n + 1 − log V f ;γ (G)

logγ

). (9)

Definition 3.1. Let f be a continuous function on the closed domain G , γ be a non-negative real number. We call the value

O Gf ;γ (x) = sup

x′∈G[x;γ ]∣∣ f

(x′) − f (x)

∣∣the γ -central oscillation of f at the point x about G . And the Riemann integral of O G

f ;γ (x) on G is called γ -central variation

of f on G , denoted by V f ;γ (G).

It is obvious that O Gf ;γ (x) � O G

f ;γ (x) � 2O Gf ;γ (x). Then we have V f ;γ (G) � V f ;γ (G) � 2V f ;γ (G). According to (7) and

(8), we have the next theorem.

Theorem 3.1. Let f be a continuous function on G, V f ;γ (G) be the γ -central variation of f on G, then we have the box-countingdimension formulae

dimBΓ ( f ; G) = lim supγ →0+

(n + 1 − log V f ;γ (G)

logγ

), (10)

and

dimBΓ ( f ; G) = lim infγ →0+

(n + 1 − log V f ;γ (G)

logγ

). (11)

Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425 419

4. Construction of fractal interpolation surface

Let G = [a,b] × [c,d] be a rectangular domain in R2, {(xi, y j, zi, j) | i = 0,1, . . . ,m; j = 0,1, . . . ,n} be a data set in R3,where a = x0 < x1 < · · · < xm = b and c = y0 < y1 < · · · < yn = d. Let ui(y), i = 0,1, . . . ,m, be m + 1 continuous functionson J = [c,d], satisfying the interpolation conditions ui(y j) = zi, j , for j = 0,1, . . . ,n.

For any y ∈ [c,d], and the data set {(xi, ui(y)) | i = 0,1, . . . ,m}, according to the method given in Section 2, we can get anaffine fractal interpolation function g y(x) which interpolate the given data set {(xi, ui(y)) | i = 0,1, . . . ,m}, i.e. g y(xi) = ui(y)

for i = 0,1, . . . ,m. Let f (x, y) = g y(x), for (x, y) ∈ [a,b] × [c,d]. Then f (x, y) is a bivariate function on [a,b] × [c,d], andsatisfies the interpolation condition f (xi, y j) = zi, j , for i = 0,1, . . . ,m, j = 0,1, . . . ,n.

Theorem 4.1. f is a continuous function on G = [a,b] × [c,d].

Proof. For any (x, y) ∈ G and ε > 0, because ui(y), i = 0,1, . . . ,m, are all continuous on [c,d], then there exists δ1 > 0, suchthat for |y − y| < δ1, we have |ui(y) − ui( y)| < ε, i = 0,1, . . . ,m. According to the construction of the function f (x, y) andTheorem 2.3, ‖ f (·, y) − f (·, y)‖∞ < 1+s

1−s · ε. On the other hand, because the affine fractal interpolation function f (x, y) iscontinuous on [a,b], then there exists δ2 > 0, such that | f (x, y) − f (x, y)| < ε. Let δ = min{δ1, δ2}, then for |x − x| < δ and|y − y| < δ, we have | f (x, y)− f (x, y)| � | f (x, y)− f (x, y)|+ | f (x, y)− f (x, y)| < ( 1+s

1−s + 1)ε. So f (x, y) is continuous on G .The theorem has been proved. �

Now let ui(y), i = 0,1, . . . ,m, be also n + 1 affine fractal interpolation functions interpolating the data sets {(y j, zi, j) |j = 0,1, . . . ,n} respectively, with the same vertical scaling factors {s j}n

j=1, where s = max{|s j|: j = 0,1, . . . ,n} < 1. We canalso construct a continuous interpolation function f (x, y) on rectangular domain G with the method above. For any giveny ∈ [c,d], the section z = f (x, y) is an affine fractal interpolation curve on [a,b]. Now we discuss the sections z = f (x, y)

for any given x ∈ [a,b].

Lemma 4.1. For some x ∈ [a,b], if the section z = f (x, y), y ∈ [c,d] is an affine fractal interpolation function generated by the iteratedfunction system {[c,d] × R; wx, j: j = 1,2, . . . ,n}, where

wx, j(y, z) = (L j(y), F x, j(y, z)

) =(

y j−1 + y j − y j−1

yn − y0(y − y0), s j z + cx, j y + dx, j

),

then the section z = f (Li(x), y), y ∈ [c,d] is also an affine fractal interpolation function, which is generated by the iterated functionsystem {[c,d] × R; w Li(x), j: j = 1,2, . . . ,n}, where

w Li(x), j(y, z) = (L j(y), F Li(x), j(y, z)

) =(

y j−1 + y j − y j−1

yn − y0(y − y0), s j z + cLi(x), j y + dLi(x), j

).

Proof. For any y ∈ [y j−1, y j], because the sections z = f (·, y) and z = f (·, L−1j (y)) are both affine fractal interpolation

functions on [a,b], according to Theorem 2.2, we have the equations

f(Li(x), y

) = Fi,y(x, f (x, y)

) = si f (x, y) + ci,yx + di,y, (12)

and

f(Li(x), L−1

j (y)) = Fi,L−1

j (y)

(x, f

(x, L−1

j (y))) = si f (x, y) + ci,L−1

j (y)x + di,L−1

j (y). (13)

On the other hand, because the section z = f (x, ·) is an affine fractal interpolation function on [c,d] which iteratedfunction system is {[c,d] × R; wx, j: j = 1,2, . . . ,n}, then f (x, y) = s j f (x, L−1

j (y)) + cx, j L−1j (y) + dx, j . With Eq. (12) and

Eq. (13), it can be gotten that f (Li(x), y) = s j f (Li(x), L−1j (y))+ si cx, j L−1

j (y)+ sidx, j + (ci,y − s jci,L−1j (y)

)x+ (di,y − s jdi,L−1j (y)

).

With Eq. (3), when ξ = y and ξ = L−1j (y),

⎧⎪⎪⎨⎪⎪⎩

ci,ξ = f (xi, ξ) − f (xi−1, ξ)

xm − x0− si

f (xm, ξ) − f (x0, ξ)

xm − x0,

di,ξ = xm f (xi−1, ξ) − x0 f (xi, ξ)

xm − x0− si

xm f (x0, ξ) − x0 f (xm, ξ)

xm − x0.

(14)

Because z = f (xk, ·), i = 0,1, . . . ,m, are all affine fractal interpolation functions on [c,d], with Theorem 2.2, we havef (xi, y) − si f (xi, L−1

j (y)) = cxi , j L−1j (y) + dxi , j . Combining Eq. (14), it can be gotten that⎧⎨

⎩ci,y − s jci,L−1

j (y)= Ai, j L

−1j (y) + Bi, j,

di,y − s jdi,L−1(y)= Ci, j L

−1j (y) + Di, j,

j

420 Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425

where Ai, j , Bi, j , Ci, j and Ai, j are all independent on y. Therefore, f (Li(x), y) = s j f (Li(x), L−1j (y)) + cLi(x), j L−1

j (y) + dLi(x), j ,

where cLi(x), j and dLi(x), j are both independent on y. According to Theorem 2.2, z = f (Li(x), ·) is also an affine fractalinterpolation function. This lemma has been proved. �Theorem 4.2. If ui(y), i = 0,1, . . . ,m are all affine fractal interpolation functions on [c,d], then f (x, ·) is also an affine fractalinterpolation function on [c,d] for any x ∈ [a,b].

Proof. For any fixed x ∈ [a,b], let z = g(y) be the affine fractal interpolation function on [c,d] with the data set{(y j, f (x, y j)): j = 0,1, . . . ,n} and the vertical scaling factors {s j: j = 1,2, . . . ,n}. We need to prove f (x, y) = g(y), forany y ∈ [c,d].

It is obvious that there exists a sequence {i1, i2, . . . , ik, . . .}, where ik ∈ {1,2, . . . ,m}, such that xk = Li1 ◦ Li2 ◦· · ·◦ Lik (x0) →x, (k → ∞). Because f (x0, ·) is an affine fractal interpolation function on [c,d] with the vertical scaling factors {s j: j =1,2, . . . ,n}, according to Lemma 4.1, for k = 1,2, . . . , the sections f (xk, ·) are all affine fractal interpolation functions on[c,d] with the vertical scaling factors {s j: j = 1,2, . . . ,n}.

Because f (·, y j), j = 0,1, . . . ,n, are all continuous on [a,b], for any ε > 0, there exists N ∈ Z+ , such that if k > N ,| f (xk, y j) − g(y j)| = | f (xk, y j) − f (x, y j)| < ε for all j = 0,1, . . . ,n. Then for any y ∈ [c,d] and k > N , according to The-

orem 2.3, | f (xk, y) − g(y)| � ‖ f (xk, ·) − g‖∞ � 1+s1−s · ε. It means f (x, y) = limk→∞ f (xk, y) = g(y), with Theorem 4.1. The

proof of Theorem 4.2 is completed. �This surface construction method can get rid of the constraints that the interpolation nodes on the boundary are collinear

and the vertical scaling factors are equal. Similar to the tensor product fractal surface proposed by Massopust [14,15], thesurface constructed with the method above is the graphs of the bivariate function y = f (x, y) on rectangular domain.It’s sections parallel to the coordinate planes yoz or zox are all fractal interpolation curves. Thus, it may have any givenroughness. But the tensor product fractal surface is constructed by two fractal curves. All points on the surface are fixedonce the curves are determined. The surface constructed above is determined by the interpolation points set. It is moreflexible, so that it is more suitable for fitting the rough surfaces in the nature.

5. Box-counting dimension of fractal interpolation surface

Let ui , i = 0,1, . . . ,m be continuous functions on [c,d], and for y ∈ [c,d], the sections f (·, y) be affine fractal interpola-tion function with the data set {(xi, ui(y)): i = 0,1, . . . ,m} and vertical scaling factors {si: i = 1,2, . . . ,m}. According to themethod given in Section 4, we can obtain a fractal interpolation surface z = f (x, y), (x, y) ∈ [a,b] × [c,d].

Theorem 5.1. Let z = f (x, y), (x, y) ∈ [a,b] × [c,d], be a fractal interpolation surface above, Λ = ∑mi=1 |si| > 1, and D be the

unique solution of Eq. (5). If there exists a y ∈ [c,d] such that the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinear andD � max{dimBΓ (ui; [c,d]): i = 0,1, . . . ,m}, then

dimB Γ(

f ; [a,b] × [c,d]) = D + 1. (15)

In order to prove this theorem, we give some lemmas first.

Lemma 5.1. Let ui , i = 0,1, . . . ,m be all continuous functions on [c,d]. If there exits y ∈ [c,d] such that the data set {(xi, ui(y)): i =0,1, . . . ,m} is not collinear, then there exists interval [α,β] ⊆ [c,d], such that the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinearfor any y ∈ [α,β].

Proof. Let {(xi, ui(y0)): i = 0,1, . . . ,m} is not collinear for y0 ∈ [c,d]. Then there exists a point {(xi0 , ui0(y0))}, i0 ∈{1,2, . . . ,m − 1}, such that

ui0(y0) −[

u0(y0) + xi0 − x0

xm − x0

(um(y0) − u0(y0)

)] �= 0.

Because function F (y) = ui0 (y) − [u0(y) + xi0 −x0

xm−x0(um(y) − u0(y))] is a continuous function on [c,d], and F (y0) �= 0, there

exists an interval [α,β], such that F (y) �= 0 for y ∈ [α,β]. Therefore, the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinearfor any y ∈ [α,β]. �Lemma 5.2. (1) If g(x) is a differential function on the interval I , then 0 � V g;γ (I) � 2 maxx∈I |g′(x)| · |I| · γ .

(2) If τ (t) = λt + ς , t ∈ I , is a linear transformation, where λ and ς are constants and λ �= 0, g(x) is a continuous function onτ (I), then V g◦τ ;γ (I) = 1

|λ| V g;|λ|γ (τ (I)).(3) Let g be a continuous function on I = [a,b], a = x0 < x1 < · · · < xm = b, and Ii = [xi−1, xi], then

Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425 421

m∑i=1

V g;γ (Ii) � V g;γ (I) �m∑

i=1

V g;γ (Ii) + 2(m − 1)V g(I)γ ,

where V g(I) = supx∈I g(x) − infx∈I g(x).

Proof. (1) Because 0 � O Ig;γ (x) = supx′,x′′∈I[x γ ] |g(x′) − g(x′′)| � supx′,x′′∈I[x γ ] |g′(ξ)||x′ − x′′| � 2 supx∈I |g′(ξ)| · γ , integrating

it on I , (1) is proved.(2) Let x = τ (t), for t ∈ I . Because O I

g◦τ ;γ (t) = supt′,t′′∈I[t;γ ] |g ◦ τ (t′) − g ◦ τ (t′′)| = supx′,x′′∈L(I)[x;|λ|γ ] |g(x′) − g(x′′)| =O L(I)

g;|λ|γ (x), then V g◦τ ;γ (I) = ∫I O I

g◦τ ;γ (t)dt = 1|λ|

∫L(I) O L(I)

g;|λ|γ (x)dx = 1|λ| V g;|λ|γ (τ (I)).

(3) For any i ∈ 1,2, . . . ,m, x ∈ Ii , we have Ii[x;γ ] ⊆ I[x;γ ], then O Iig;γ (x) � O I

g;γ (x). Therefore the left inequality is true.Now let we prove the right one.

Let Ic1 = [a, x1 − γ ], Ic

m = [xm−1 + γ ,b], Ici = [xi−1 + γ , xi − γ ], i = 2,3, . . . ,m − 1, and I s

i = [xi − γ , xi + γ ] ∩ [a,b],i = 1,2, . . . ,m − 1, where assuming [α,β] is an empty set if α > β . Then

⋂mi=1 Ic

i ∩ ⋂m−1i=1 I s

i = [a,b]. It is obvious that

O Ig;γ (x) = O Ii

g;γ (x) for x ∈ Ici , and O I

g;γ (x) � V g(I) for x ∈ I si . Therefore V g;γ (I) �

∑mi=1

∫Ici

O Iig;γ (x)dx + ∑m−1

i=1

∫I si

V g(I)dx �∑mi=1 V g;γ (Ii) + (m − 1)V g(I) · 2γ . The right inequality is also true. �

Lemma 5.3. Let f (x, y), (x, y) ∈ G, be a fractal interpolation surface constructed above. There exist positive real numbers β1 and β1 ,such that for any γ � 0 any y ∈ [c,d],

m∑i=1

|si |ai V f (·,y); γai

(I) − β1γ � V f (·,y);γ (I) �m∑

i=1

|si |ai V f (·,y); γai

(I) + β2γ . (16)

Proof. Because f (·, y) is an affine fractal interpolation function with the data set {(xi, ui(y)): i = 0,1, . . . ,m} and thevertical scaling factors {si: i = 1,2, . . . ,m}. According to Theorem 2.2, if x ∈ Ii , f (x, y) = si f (L−1

i (x), y) + pi,y(L−1i (x)),

where pi,y(t) = ci,y · t + di,y , for i = 1,2, . . . ,m. Because L−1i (x) = (x − bi,y)/ai,y for x ∈ Ii , with Lemma 5.2 we

have V f (·,y);γ (I) �∑m

i=1[|si|V f (L−1i (·),y);γ (Ii) − V pi,y◦L−1

i ;γ (Ii)] = ∑mi=1[|si|ai V f (·,y);γ /ai (I) − ai V pi,y;γ /ai (I)] �

∑mi=1 |si| ·

ai V f (·,y);γ /ai (I) − 2|I|∑mi=1 |ci,y| · γ . On the other hand, with Lemma 5.2 we have V f (·,y);γ (I) �

∑mi=1[|si|V f (L−1

i (·),y);γ (Ii) +V pi,y◦L−1

i ;γ (Ii)] + 2(m − 1)V f (·,y)(I)γ �∑m

i=1 |si |ai V f (·,y);γ /ai (I) + 2|I| · ∑mi=1 |ci,y|γ + 2(m − 1)V f (·,y)(I)γ . According to

Eq. (3), ci,y = [ui(y)− ui−1(y)− si(um(y)− u0(y))]/(xm − x0), and ui are all continuous on [c,d], so there exists M > 0, suchthat |ci,y| � M for any y ∈ [c,d] and i ∈ {1,2, . . . ,m}. And because V f (·,y)(I) � sup(x,y)∈G f (x, y)− inf(x,y)∈G f (x, y) = V f (G),then we can let β1 = 2mM|I| and β2 = β1 + 2(m − 1)V f (G), which are independent on y and γ . The proof of the lemma iscomplete. �Lemma 5.4. Let Λ = ∑m

i=1 |si| > 1, a = min{ai: i = 1,2, . . . ,m}, and for any y ∈ [α,β] ⊆ [c,d], the data set {(xi, ui(y)): i =0,1, . . . ,m} be not collinear. There exists positive constant C such that for any y ∈ [α,β] and any γ ∈ [0,ak], we have

V f (·,y);γ (I) � C · Λk · γ . (17)

Proof. Let l(x, y) = u0(y) + x−x0xm−x0

(um(y) − u0(y)). For any y ∈ [α,β], l(·, y) is a linear function on I , satisfying l(a, y) =f (a, y) and l(b, y) = f (b, y). Let ρ(y) = supx∈I |l(x, y) − f (x, y)|, then ρ(y) is positive on [α,β]. Because l(x, y) and f (x, y)

are continuous on G , and for any y1, y2 ∈ [c,d], |ρ(y1) − ρ(y2)| � supx∈I |(l(x, y1) − f (x, y1)) − (l(x, y2) − f (x, y2))| �supx∈I |(l(x, y1) − (l(x, y2)| + supx∈I | f (x, y1)) − f (x, y2))|, it can be proved that ρ(y) is continuous on [α,β]. So ρ(y) hasthe minimum on [α,β] which is strictly positive. Therefore ρ = infy∈[α,β] ρ(y) > 0.

For y ∈ [α,β] and i1 ∈ {1,2, . . . ,m}, let li1 (x, y) = si1 l(L−1i1

(x), y)+ ci1,y L−1i1

(x)+di1,y . li1 (x, y) is a linear function on Ii1 =Li1 (I), and f (Li1 (x0), y) = li1 (Li1 (x0), y), f (Li1 (xm), y) = li1 (Li1 (xm), y). With Theorem 2.2, ρi1(y) = supx∈Li1 (I) |li1(x, y) −f (x, y)| = supx∈Li1 (I) |si1 ||l(L−1

i1(x), y) − f (L−1

i1(x), y)| = supξ∈I |si1 ||l(ξ, y) − f (ξ, y)| = |si1 | · ρ(y) � ρ · |si1 |. Recursively

defining lik,ik−1,...,i1 (x, y) = sik lik−1,ik−2,...,i1(L−1ik

(x), y) + cik,y L−1ik

(x) + dik,y , for ik ∈ {1,2, . . . ,m}, k = 1,2, . . . . With math-ematical induction, we can prove that lik,ik−1,...,i1 (x, y) is a linear function on Lik,ik−1,...,i1(I) = Lik ◦ Lik−1 ◦ · · · ◦ Li1 (I),f (Lik,ik−1,...,i1(x0), y) = lik,ik−1,...,i1(Lik,ik−1,...,i1(x0), y), f (Lik,ik−1,...,i1 (xm), y) = lik,ik−1,...,i1 (Lik,ik−1,...,i1 (xm), y), andρik,ik−1,...,i1(y) = supx∈Lik ,ik−1,...,i1 (I) |lik,ik−1,...,i1 (x, y) − f (x, y)| = |sik sik−1 · · · si1 |ρ(y)� ρ · |sik sik−1 · · · si1 |.

For a continuous function g(t) on [t1, t2], if 0 � γ � t2 − t1, then V g;γ ([t1, t2]) � V g([t1, t2]) · γ , where V g([t1, t2]) =supt∈[t1,t2] g(t)− inft∈[t1,t2] g(t). It is obvious that V f (·,y)(Lik,ik−1,...,i1 (I)) � ρik,ik−1,...,i1 (y)� ρ · |sik sik−1 · · · si1 |. Therefore whenγ ∈ [0,ak], with Lemma 5.2(3), V f (·,y);γ (I) �

∑mik,ik−1,...,i1=1 V f (·,y);γ (Lik,ik−1,...,i1(I)) �

∑mik,ik−1,...,i1=1 ρ · |sik sik−1 · · · si1 | · γ =

ρ · Λk · γ . The proof of Lemma 5.4 is complete. �

422 Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425

Lemma 5.5. Let f (x, y) be a fractal interpolation surface constructed above, D is the solution of Eq. (5).(1) If Λ > 1 and there exists an interval [α,β] such that for any y ∈ [α,β], the data sets {(xi, ui(y)): i = 0,1, . . . ,m} are not

collinear, then there exists positive constants C1 and γ0 , such that V f (·,y);γ (I) > C1γ2−D , for 0 < γ < γ0 and y ∈ [α,β].

(2) If Λ > 1, then there exists positive constant C2 and γ0 , such that V f (·,y);γ (I) < C2γ2−D , for 0 < γ < γ0 and y ∈ [c,d].

(3) If Λ < 1, then there exists positive constant C3 and γ0 , such that V f (·,y);γ (I) < C3γ , for 0 < γ < γ0 and y ∈ [c,d].(4) If Λ = 1, then there exists positive constant C4 and γ0 , such that V f (·,y);γ (I) < C4(− logγ )γ , for 0 < γ < γ0 and y ∈ [c,d].

Proof. (1) According to Lemma 5.4, there exists γ0 > 0, such that V f (·,y);γ (I) � 2β1γ /(Λ − 1), for 0 < γ � γ0/a and

y ∈ [α,β], where β1 is the same as in Lemma 5.3. Let k1 = β1γD−1

0 /(Λ − 1), then V f (·,y);γ (I) � β1γ /(Λ − 1) + k1γ2−D ,

for γ ∈ [γ0, γ0/a]. Denote φ(γ ) = β1γ /(Λ − 1) + k1γ2−D . Because

∑mi=1 |si|aiφ(γ /ai) − β1γ = φ(γ ), with the mathemat-

ical induction, we can prove V f (·,y);γ (I) � φ(γ ), for ak+1γ0 � γ � akγ0, where a = max{ai: i = 1,2, . . . ,m} < 1. Because1 < D < 2, there exists a positive constant C1, such that φ(γ ) > C1γ

2−D , for 0 < γ � γ0. Therefore V f (·,y);γ (I) > c1γ2−D ,

for 0 < γ � γ0.(2) Because f (x, y) is continuous on closed domain D , V f (·,y);γ is also continuous, for γ > 0 and y ∈ [c,d], then there

exists a constant k2 > 0 big enough, such that V f (·,y);γ (I) � β2γ /(1 − Λ) + k2γ2−D , for γ ∈ [γ0, γ0/a] and y ∈ [c,d], where

β2 is the same as in Lemma 5.3. Denote Φ1(γ ) = β2γ /(1 −Λ)+k2γ2−D . Because

∑mi=1 |si |aiΦ1(γ /ai)+β2γ = Φ1(γ ), with

the mathematical induction, we can prove V f (·,y);γ (I) � Φ1(γ ), for ak+1γ0 � γ � akγ0. Because 1 < D < 2, there exists apositive constant C2, such that Φ1(γ ) < C2γ

2−D , for 0 < γ � γ0. Therefore V f (·,y);γ (I) < C2γ2−D , for 0 < γ � γ0.

(3) Similar to the proof of (2), we can prove that there exists k3 such that V f (·,y);γ (I) � Φ2(γ ), for ak+1γ0 � γ � akγ0,where Φ2(γ ) = β2γ /(1 − Λ) + k3γ

2−D . It is obvious that the solution of Eq. (5) D < 1 when Λ < 1. Then there existspositive constant C3, such that Φ2(γ ) < C3 · γ , for 0 < γ � γ0. Therefore V f (·,y);γ (I) < C3 · γ , for 0 < γ � γ0.

(4) Noticing Λ = 1, the solution of Eq. (5) D = 1. Similarly, it can be prove that there exists k4 such that V f (·,y);γ (I) �Φ3(γ ), for ak+1γ0 � γ � akγ0, (0 < γ0 < 1), where Φ3(γ ) = β2 log(γ ) · γ /(

∑mi=1 |si | log ai) + k4γ . It is obvious that there

exists C4 > 0, such that Φ3(γ ) < C4(− logγ )γ . Then V f (·,y);γ (I) < C4(− logγ )γ , for 0 < γ � γ0. �The proof of Theorem 5.1. Let (x, y) ∈ G = [a,b] × [c,d], G[(x, y);γ ] = G ∩ [x − γ , x + γ ] × [y − γ , y + γ ], I[x ;γ ] = [a,b] ∩[x − γ , x + γ ] and J [y ;γ ] = [c,d] ∩ [y − γ , y + γ ] for γ � 0. Because f and f (·, y) are continuous on G and I = [a,b]respectively, for (x, y) ∈ D , there exists (x′, y′) ∈ G[(x, y);γ ] and x′′ ∈ I[x ;γ ], such that O G

f ;γ (x, y) = | f (x′, y′) − f (x, y)|and O I

f (·,y);γ (x) = | f (x′′, y) − f (x, y)|. Then O If (·,y);γ (x) � sup(x′,y′)∈G[(x,y);γ ] | f (x′, y′) − f (x, y)| = O G

f ;γ (x, y) = | f (x′, y′) −f (x, y)| � | f (x′, y′) − f (x′, y)| + | f (x′, y) − f (x, y)| � | f (x′, y′) − f (x′, y)| + O I

f (·,y);γ (x). Because f (·, y′) and f (·, y) are the

affine fractal interpolation functions with the data sets {(xi, ui(y′)): i = 0,1,2, . . . ,m} and {(xi, ui(y)): i = 0,1,2, . . . ,m}respectively, according to Theorem 2.3, we have | f (x′, y′) − f (x′, y)| � ‖ f (·, y′) − f (·, y)‖∞ � K

∑mi=0 |ui(y′) − ui(y)| �

K∑m

i=0 O Jui ;γ (y), where K = (1 + s)/(1 − s). Then we have

O If (·,y);γ (x) � O G

f ;γ (x, y) � O If (·,y);γ (x) + K

m∑i=0

O Jui;γ (y). (18)

Because there exists y ∈ [c,d] such that the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinear, according to Lemma 5.1,there exists interval [α,β] ⊆ [c,d] such that, for any y ∈ [α,β], the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinear.Integrating Eq. (16) on G , we have

β∫α

V f (·,y);γ (I)dy � V f ;γ (G) �d∫

c

V f (·,y);γ (I)dy + K (b − a)

m∑i=0

V ui;γ ( J ). (19)

Because dimBΓ (ui, J ) � D , i.e. lim supγ →0+ (2 − log V ui ,γ ( J )/logγ ) � D , for i = 0,1,2, . . . ,m, Then for any ε > 0, there

exists 0 < γ0 < 1, such that if 0 < γ < γ0, have V ui;γ < γ 2−D−ε . According to Lemma 5.5 and Eq. (19), there exist positiveconstants B1, B2 and γ0 > 0, such that

B1γ2−D � V f ;γ (G) � B2γ

2−D−ε.

With Eqs. (10) and (11) we have

D + 1 � dimBΓ ( f ; G) � dimBΓ ( f ; G) � D + 1 + ε.

Because ε > 0 can be arbitrarily small, then dimB Γ ( f ; G) = D + 1. �Theorem 5.2. Let z = f (x, y), (x, y) ∈ G = [a,b] × [c,d], be a fractal interpolation surface above. If Λ = ∑m

i=1 |si | � 1, or the datasets {(xi, ui(y)): i = 0,1, . . . ,m} are all collinear for any y ∈ [c,d], then

Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425 423

2 � dimBΓ ( f ; G) � dimBΓ ( f ; G) � max{

dimBΓ (ui; J ): i = 0,1, . . . ,m} + 1. (20)

Proof. The left inequation is obviously true. We need to prove the right inequation only. Denote � = max{dimBΓ (ui; J ): i =0,1, . . . ,m}. Because 1 � dimBΓ (ui; J )� �, for any ε > 0 there exists 0 < γ0 < 1, such that V ui ;γ ( J )� γ 2−�−ε , for 0 < γ �γ0 and i = 0,1, . . . ,m. On the other hand, if Λ � 1, with Lemma 5.5, there exist positive constant C and γ0, such thatV f (·,y);γ (I) � C(− logγ )γ , for 0 < γ < γ0 and y ∈ [c,d]; if the data sets {(xi, ui(y)): i = 0,1, . . . ,m} are all collinear forany y ∈ [c,d], it is obvious that V f (·,y);γ (I) � κγ , where κ = 2 sup{|um(y) − u0(y)|: y ∈ J }/(b − a). With the right hand

of Eq. (19), we get V f ;γ (G) � Bγ 2−�−ε , where B is a positive constant. According to Eq. (10), dimBΓ ( f ; G) � � + 1 + ε.

Because ε > 0 can be arbitrarily small, then dimBΓ ( f ; G) � � + 1. �

Now let ui(y), i = 0,1, . . . ,m, be all affine fractal interpolation functions on J = [c,d] with the vertical scaling factors{s j: j = 1,2, . . . ,n}, where |s j| < 1, associated with the interpolation knots {(y j, zi j): j = 0,1, . . . ,n} respectively. Denote

Λ = ∑nj=1 |s j|, and D is the unique solution of the equation

∑nj=1 |s j|aD−1

j = 1, where a j = (y j − y j−1)/(yn − y0).

Theorem 5.3. Let z = f (x, y) be a fractal interpolation surface derived from affine fractal interpolation function, where ui(y), i =0,1, . . . ,m, are all affine fractal interpolation functions on J = [c,d] with the vertical scaling factors {s j: j = 1,2, . . . ,n}, thenΓ ( f ; G) has box-counting dimension, and

dimB Γ ( f ; G) = max{

supy∈ J

{dimB Γ

(f (·, y); I

)}, sup

x∈I

{dimB Γ

(f (x, ·); J

)}} + 1. (21)

Proof. Suppose supy∈ J {dimB Γ ( f (·, y); I)} � supx∈I {dimB Γ ( f (x, ·); J )}. For any y ∈ J , because f (·, y) is an affine frac-tal interpolation function with the vertical scaling factors {si: i = 1,2, . . . ,m}, then the graph of the function f (·, y) onI has box-counting dimension. And according to Theorem 2.1, if Λ > 1 and the data set {(xi, ui(y)): i = 0,1, . . . ,m}is not collinear, then dimB Γ ( f (·, y); I) = D , otherwise dimB Γ ( f (·, y); I) = 1. With Theorem 5.1 we can prove that ifΛ > 1 and there exists y ∈ J such that the data set {(xi, ui(y)): i = 0,1, . . . ,m} is not collinear, then dimB Γ ( f ; G) =D + 1 = supy∈ J dimB Γ ( f (·, y); I) + 1. Eq. (21) is true. If Λ � 1 or the data set {(xi, ui(y)): i = 0,1, . . . ,m} are all collinearfor any y ∈ J , then supy∈ J dimB Γ ( f (·, y); I) = 1. So dimB Γ ( f (x, ·); J ) = 1, for any y ∈ J . According to Theorem 5.2,dimB Γ ( f ; G) = 2. Eq. (21) is also true.

Noticing that the fractal interpolation surface z = f (x, y), (x, y) ∈ G , can be constructed similarly by following steps.Let v j , j = 0,1, . . . ,n, be all affine fractal interpolation functions with the vertical scaling factors {si: i = 1,2, . . . ,m}, andassociated with the interpolation knots {(xi, zi, j): i = 0,1, . . . ,m}. Then for any x ∈ [a,b], construct f (x, ·) as an affinefractal interpolation function on J with the vertical scaling factors {s j: j = 1,2, . . . ,n}, and associated with the interpola-tion knots {(y j, v j(x)): j = 0,1, . . . ,n}. We can get the same fractal interpolation surface z = f (x, y) above. Therefore, ifsupx∈I {dimB Γ ( f (x, ·); J )} > supy∈ J {dimB Γ ( f (·, y); I)}, then there exists x0 ∈ I such that dimB Γ ( f (x0, ·); J ) > 1. It can be

deduced that Λ = ∑nj=1 |s j | > 1 and the data set {(y j, v j(x0)): j = 0,1, . . . ,n} is not collinear. According to Theorem 5.1,

we can get dimB Γ ( f ; G) = D + 1 = supx∈I {dimB Γ ( f (x, ·); J )} + 1. The proof of Theorem 5.3 is complete. �

Lemma 5.6. Let z = f (x, y) be a fractal interpolation surface derived from affine fractal interpolation function, where ui(y),i = 0,1, . . . ,m, are all affine fractal interpolation functions on J = [c,d] with the vertical scaling factors {s j: j = 1,2, . . . ,n}, and{(xi, y j, zi, j): i = 0,1, . . . ,m; j = 0,1, . . . ,n} are its interpolation knots. For any j ∈ {0,1, . . . ,n}, if the data set {(xi, zi, j): i =0,1, . . . ,m} is collinear, then for any y ∈ J , the data set {(xi, f (xi, y)): i = 0,1, . . . ,m} is collinear. Similarly, for any i ∈ {0,1, . . . ,m},if the data set {(y j, zi, j): j = 0,1, . . . ,n} is collinear, then for any x ∈ I , the data set {(y j, f (x, y j)): j = 0,1, . . . ,n} is collinear.

Proof. Because the data set {(xi, zi, j): i = 0,1, . . . ,m} is collinear, for j = 0,1, . . . ,n, there exist κ j , such that zi, j − z0, j =κ j(xi − x0). For i = 0,1, . . . ,m, because ui(y) = f (xi, y) are all affine fractal interpolation functions with affine mappingsfrom [c,d] × R to [c,d] × R ,

wi, j

(yz

)=

(L j(y)

F i, j(y, z)

)=

(a j 0ci, j s j

)(yz

)+

(b j

di, j

), (22)

where

424 Z. Feng, X. Sun / J. Math. Anal. Appl. 412 (2014) 416–425

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a j = y j − y j−1

yn − y0,

b j = y j−1 − y j − y j−1

yn − y0x0,

ci, j = zi, j − zi, j−1

yn − y0− s j

zi,n − zi,0

yn − y0,

di, j = ynzi, j−1 − y0zi, j

yn − y0− s j

ynzi,0 − y0zi,n

yn − y0,

(23)

for j = 0,1, . . . ,n. Then f (xi, L j(yk)) − f (x0, L j(yk)) = ui(L j(yk)) − u0(L j(yk)) = F i, j(yk, ui(yk)) − F0, j(yk, u0(yk)) =s j(ui(yk)−u0(yk))+(ci, j − c0, j)yk +(di, j −d0, j) = κL j(yk)

(xi −x0), where κL j(yk)is a constant independent on i ∈ {1,2, . . . ,m}.

Therefore the data set {(xi, f (xi, L j(yk))): i = 0,1, . . . ,m} is collinear for any j ∈ {1,2, . . . ,n} and k ∈ {0,1, . . . ,n}. Withmathematical induction, we can prove for any p ∈ N+ the data set {(xi, f (xi, L j1 j2··· jp (yk))): i = 0,1, . . . ,m} is collinear,

where { j1, j2, . . . , jp} ∈ {1,2, . . . ,n}p , L j1 j2··· jp = L j1 ◦ L j2 ◦ · · · ◦ L jp and k ∈ {0,1, . . . ,n}. Because the set {L j1 j2··· jp (yk)} isdense on [c,d], and ui(y) = f (xi, y) is continuous on [c,d], so for any y ∈ [c,d] the data set {(xi, f (xi, y)): i = 0,1, . . . ,m}is collinear. Similarly the rest of the lemma can be proved. �Theorem 5.4. Let z = f (x, y) be a fractal interpolation surface derived from affine fractal interpolation function, where ui(y), i =0,1, . . . ,m, are all affine fractal interpolation functions on J = [c,d] with the vertical scaling factors {s j: j = 1,2, . . . ,n}, thenΓ ( f ; G) has box-counting dimension and

(1) For j ∈ {0,1, . . . ,n}, the data set {(xi, zi, j): i = 0,1, . . . ,m} are all collinear, and there exists an i ∈ {0,1, . . . ,m}, such thatthe data set {(y j, zi, j): j = 0,1, . . . ,n} is not collinear, then dimB Γ ( f ; G) = max{D,1} + 1.

(2) For i ∈ {0,1, . . . ,m}, the data set {(y j, zi, j): j = 0,1, . . . ,n} are all collinear, and there exists a j ∈ {0,1, . . . ,n}, such that thedata set {(xi, zi, j): i = 0,1, . . . ,m} is not collinear, then dimB Γ ( f ; G) = max{D,1} + 1.

(3) For i ∈ {0,1, . . . ,m} and j ∈ {0,1, . . . ,n}, the data sets {(y j, zi, j): j = 0,1, . . . ,n} and {(xi, zi, j): i = 0,1, . . . ,m} are allcollinear, then dimB Γ ( f ; G) = 2.

(4) There exist an i ∈ {0,1, . . . ,m} and a j ∈ {0,1, . . . ,n}, such that {(y j, zi, j): j = 0,1, . . . ,n} and {(xi, zi, j): i = 0,1, . . . ,m}are not collinear, then dimB Γ ( f ; G) = max{D, D,1} + 1.

Proof. (1) According to Lemma 5.6, for any y ∈ [c,d], the data set {(xi, f (xi, y)): i = 0,1, . . . ,m} is collinear, then the fractalinterpolation function f (·, y) is a linear function on [a,b]. Therefore supy∈[c,d] dimB Γ ( f (·, y); I) = 1. If Λ = ∑n

j=1 |s j| � 1,

i.e., D � 1, then dimB Γ ( f (x, ·); J ) = 1 for any x ∈ [a,b]. If Λ = ∑nj=1 |s j| > 1, i.e., D > 1, then dimB Γ ( f (x, ·); J ) = 1 when

the data set {(y j, f (x, y j)): j = 0,1, . . . ,n} is collinear, and dimB Γ ( f (x, ·); J ) = D when the data set {(y j, f (x, y j)): j =0,1, . . . ,n} is not collinear. Noticing that there exists i ∈ {0,1, . . . ,m}, such that the data set {(y j, f (xi, y j)): j = 0,1, . . . ,n}is not collinear, we have supx∈[a,b] dimB Γ ( f (x, ·); J ) = max{D,1}. With Theorem 5.3, the result is true. (2) It can be provedsimilarly to the proof of (1). (3) Because for any y ∈ [c,d] the fractal interpolation function f (·, y) is a linear function on[a,b], and for any x ∈ [a,b] the fractal interpolation function f (·, y) is a linear function on [c,d], then

f (x, y) = f (a, c) + f (b, c) − f (a, c)

b − a(x − a) + f (a,d) − f (a, c)

d − c(y − c)

+ f (b,d) − f (b, c) − f (a,d) + f (a, c)

(b − a)(d − c)(x − a)(y − c).

It is a smooth surface on G . Then dimB Γ ( f ; G) = 2. (4) Because there exist x ∈ [a,b] and y ∈ [c,d], such that{(y j, f (x, y j)): j = 0,1, . . . ,n} and {(xi, f (xi, y)): i = 0,1, . . . ,m} are not collinear, then supx∈[a,b] dimB Γ ( f (x, ·); J ) =max{D,1} and supy∈[c,d] dimB Γ ( f (·, y); I) = max{D,1}. With Theorem 5.3, dimB Γ ( f ; G) = max{D, D,1} + 1. The proofof Theorem 5.4 is complete. �References

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