On bivariate fractal approximation
aa r X i v : . [ m a t h . C A ] J a n ON BIVARIATE FRACTAL APPROXIMATION
V. AGRAWAL, T. SOM, AND S. VERMA
Abstract.
In this paper, the notion of dimension preserving approximation for real -valuedbivariate continuous functions, defined on a rectangular domain ⊏⊐ , has been introduced andseveral results, similar to well-known results of bivariate constrained approximation in terms ofdimension preserving approximants, have been established. Further, some clue for the construc-tion of bivariate dimension preserving approximants, using the concept of fractal interpolationfunctions, has been added. In the last part, some multi-valued fractal operators associated withbivariate α -fractal functions are defined and studied. . 1. Introduction
Following the seminal work of Barnsley [2], Navascu´es [17, 18] studied the approximation offunctions using their fractal counterparts termed as α -fractal functions. In the same vein, Vermaand Masspoust [23] recently introduced the notion of dimension preserving approximation. Weuse dim and Gr ( f ) respectively to represent fractal dimension and graph of a function of f .Various concepts of fractal dimensions are available but we cover only those fractal dimensionsthat are suitable for this article. We only need to mention the Hausdorff dimension, the boxdimension, and the packing dimension defined for nonempty subsets of R n , n ∈ N , and denotedby dim H , dim B and dim P respectively. To know these fractal dimensions readers are suggestedto go through, for instance, [9, 15].The following relations are established between these fractal dimensions. (see [9]):dim H F ≤ dim B F ≤ dim B F and dim H F ≤ dim P F ≤ dim B F. The class of all real-valued continuous functions on ⊏⊐ := I × J is defined by C (cid:0) ⊏⊐ (cid:1) where I = [ a, b ] and J = [ c, d ] . For a bivariate function f , we denote the derivative of ( k, l )-th order by D ( k,l ) f , that is, D ( k,l ) f := ∂ k + l f∂x k ∂y l . Let C m,n ( ⊏⊐ ) = { f : ⊏⊐ → R ; D ( k,l ) f ∈ C (cid:0) ⊏⊐ (cid:1) , ∀ ≤ k ≤ m, ≤ l ≤ n } . If D ( k,l ) f ( x ) ≥ , ∀ x ∈ ⊏⊐ , then we say the function f is ( m, n )-convex. Let g ∈ C (cid:0) ⊏⊐ (cid:1) such thatdim( Gr ( g )) >
2. We may refer to [21] for the existence of such functions. The function f : ⊏⊐ → R defined by f ( x, y ) := x R a y R c g ( t, s ) dtds satisfies the following:dim( Gr ( f )) = 2 and dim Gr ( D (1 , f ) = dim( Gr ( g )) > , where dim denotes a fractal dimension.Recall that the tensor product Bernstein polynomial on ⊏⊐ is defined as: B m,n ( f )( x, y ) = m X i =0 n X j =0 f (cid:16) a + i ( b − a ) m , c + j ( d − c ) n (cid:17)(cid:18) mi (cid:19)(cid:18) nj (cid:19) ( x − a ) i ( b − x ) m − i ( y − c ) j ( d − y ) n − j . Mathematics Subject Classification.
Primary 28A80; Secondary 10K50, 41A10.
Key words and phrases. fractal dimension, fractal interpolation, fractal surfaces, Bernstein polynomials, bivari-ate constrained approximation.
Let us approximate a function f ∈ C k,l ( ⊏⊐ ) by B m,n ( f ), then (see [11] for several properties ofBernstein polynomials) we have the following: • B m,n ( f ) → f uniformly on ⊏⊐ . • (cid:16) D ( k,l ) ( B m,n ( f )) (cid:17) → D ( k,l ) f uniformly on ⊏⊐ . • Since B m,n ( f ) and D ( k,l ) ( B m,n ( f )) are polynomials, then dim (cid:16) Gr (cid:0) D ( k,l ) ( B m,n ( f )) (cid:1)(cid:17) =dim( Gr ( B m,n ( f ))) = dim( Gr ( f )) = 2 . The above items may conclude that the approximation by Bernstein polynomials maintains thesmoothness of a function but not (necessarily) the dimensions of its partial derivatives.The present paper explores the approximation perspective relative to fractal dimension of afunction and its partial derivatives.The paper is structured as follows. In Section 1, we give a brief introduction and some pre-liminaries needed for the paper. In Section 2, we start to prove some results regarding dimensionpreserving approximation. In Section 3, we define some multi-valued mappings which are definedwith the help of bivariate α -fractal functions, and establish some properties of them.2. Dimension preserving approximation of bivariate functions
Firstly, we mention the following result required for our paper:
Lemma 2.1 ( [23], Lemma 3 . . Let A ⊂ R m and f, g : A → R n be continuous functions. Then, dim H ( Gr ( f + g )) = dim H ( Gr ( g )) and dim P ( Gr ( f + g )) = dim P ( Gr ( g )) provided that f is a Lipschitz function.Remark . Note that the above lemma is also true for box dimensions.Let us denote the class of Y -valued Lipschitz functions on X by L ip ( X, Y ) , where ( X, d X ) isa compact metric space and ( Y, k . k Y ) is a normed linear space. Note that this space is a densesubset of C ( X, Y ) with respect to the supremum norm.In view of Lipschitz invariance property of dimension, one may conclude that the upcomingtheorem holds for all aforementioned dimensions.
Theorem 2.3.
Let dim( X ) ≤ β ≤ dim( X ) + dim( Y ) . Then the set S β := { f ∈ C ( X, Y ) :dim( Gr ( f )) = β } is dense in C ( X, Y ) . Proof.
Let f ∈ C ( X, Y ) and ǫ > . Using the density of L ip ( X, Y ) in C ( X, Y ), there exists g in L ip ( X, Y ) such that k f − g k ∞ ,Y < ǫ . Further, we consider a non-vanishing function h ∈ S β . Let h ∗ = g + ǫ k h k ∞ ,Y h, which immediatelygives k g − h ∗ k ∞ ,Y ≤ ǫ . This together with Lemma 2.1 implies that dim( Gr ( h ∗ )) = dim( Gr ( h )) = β. Hence, we have h ∗ ∈ S β and k f − h ∗ k ∞ ,Y ≤ k f − g k ∞ ,Y + k g − h ∗ k ∞ ,Y < ǫ. Thus, the proof of the theorem is complete. (cid:3)
To the best our knowledge, the univariate version of the next theorem is well-known, however,we could not find a proof of the theorem in bivariate setting. Hence, we write a detailed proof ofit.
Theorem 2.4.
Let (cid:0) f k (cid:1) be a sequence of differentiable functions on ⊏⊐ . Assume that for some ( x , y ) ∈ ⊏⊐ , the sequences (cid:0) f k ( x , . ) (cid:1) and (cid:0) f k ( ., y ) (cid:1) converges uniformly on [ c, d ] and [ a, b ] re-spectively. If ( D (1 , f k ) converges uniformly on ⊏⊐ , then (cid:0) f k (cid:1) converges uniformly on ⊏⊐ to afunction f , and D (1 , f ( x ) = lim k →∞ D (1 , f k ( x ) , N BIVARIATE FRACTAL APPROXIMATION 3 for every x ∈ ⊏⊐ . Proof.
Let ǫ >
0. Since ( D (1 , f k ) converges uniformly, there exists N ∈ N such that | D (1 , f k ( x ) − D (1 , f m ( x ) | < ǫ b − a )( d − c ) , ∀ x ∈ ⊏⊐ , k, m ≥ N . By the mean-value theorem, see, for instance, [20, Theorem 9 . (cid:12)(cid:12) f k ( x + h, y + k ) − f m ( x + h, y + k ) − f k ( x + h, y ) + f m ( x + h, y ) − f k ( x, y + k ) + f m ( x, y + k )+ f k ( x, y ) − f m ( x, y ) (cid:12)(cid:12) = hk (cid:12)(cid:12) D (1 , ( f k − f m )( t, s ) (cid:12)(cid:12) ≤ hk max ( t,s ) ∈ ⊏⊐ (cid:12)(cid:12) D (1 , f k ( t, s ) − D (1 , f m ( t, s ) (cid:12)(cid:12) ≤ ǫ b − a )( d − c ) hk ≤ ǫ . By the hypothesis for ( x , y ) ∈ ⊏⊐ , one can choose N ( > N ) ∈ N such that | f k ( x , y ) − f m ( x , y ) | < ǫ ∀ k, m ≥ N and | f k ( x, y ) − f m ( x, y ) | < ǫ ∀ k, m ≥ N . Now, using the above estimates and Equation 2.1 we have | f k ( x, y ) − f m ( x, y ) | ≤ ǫ | f k ( x, y ) − f m ( x, y ) | + | f k ( x , y ) − f m ( x , y ) | + | f k ( x , y ) − f m ( x , y ) | < ǫ ǫ ǫ ǫ <ǫ, for every ( x, y ) ∈ ⊏⊐ and k, m ≥ N . This immediately confirms the uniform convergence of ( f k ) . The rest part follows by routine calculations, hence omitted. (cid:3)
Lemma 2.5.
Let f : I → R be a Lipschitz map and g : J → R be a continuous function. Amapping h : ⊏⊐ → R defined by h ( x, y ) = f ( x ) + g ( y ) , then dim H ( Gr ( h )) = dim H ( Gr ( g )) + 1 . Proof.
Proof follows by defining a bi-Lipschitz mapping from Gr ( h ) to the set { ( x, y, g ( y )) : x ∈ I, y ∈ J } . (cid:3) Here, let us recall some dimensional results for univariate functions. Mauldin and Williams [16]considered the following class of functions: W b ( x ) := ∞ X n = −∞ b − αn [ φ ( b n x + θ n ) − φ ( θ n )] , where θ n is an arbitrary real number, φ is a periodic function with period one and b > , < α < . They showed that for a large enough b there exists a constant C > H ( Gr ( W b ) isbounded below by 2 − α − ( C/ ln b ) . V. AGRAWAL, T. SOM, AND S. VERMA
Further, a significant progress in dimension theory of functions is contributed by Shen [21] forthe following class of functions: f φλ,b ( x ) := ∞ X n =0 λ n φ ( b n x )where b ≥ φ is a real-valued, Z -periodic, non-constant, C -function defined on R . He provedthat there exists a constant K depending on φ and b such that if 1 < λb < K thendim H ( Gr ( f φλ,b ) = 2 + log λ log b . For f ∈ C , ( ⊏⊐ ) , we get dim( Gr ( f )) = 2 . However, no conclusion can be drawn for dimensionsof its partial derivatives. This is evident from the following example: let Weierstrass-type nowheredifferentiable continuous function W : I → R as in [21] with 1 ≤ dim( Gr ( W )) ≤
2. Now, wedefine h : ⊏⊐ → R by h ( x, y ) = W ( x ) + y. Here, by Lemma 2.5, we obtain 2 ≤ dim( Gr ( h )) = dim( Gr ( W )) + 1 ≤ . Then for the function f defined by f ( x, y ) := x Z a y Z c h ( t, s ) dtds, we have dim( Gr ( f )) = 2 and 2 ≤ dim( Gr ( D (1 , f )) = dim( Gr ( h )) ≤ . Theorem 2.6.
Let f ∈ C , ( ⊏⊐ ) such that dim( Gr ( D (1 , f )) = β for some ≤ β ≤ . Then wehave a sequence ( f k ) in C , ( ⊏⊐ ) such that dim( Gr ( D (1 , f k )) = β and f k → f uniformly on ⊏⊐ . Proof.
In view of Theorem 2.3, there exists a sequence ( g k ) in C ( ⊏⊐ ) such that dim( Gr ( g k )) = β and g k → D (1 , f uniformly on ⊏⊐ . Further, let us consider a function f k : ⊏⊐ → R defined by f k ( x, y ) := x Z a y Z c g k ( t, s ) dtds. Then D (1 , f k = g k and ( D (1 , f k ) → D (1 , f uniformly. Next, we have that (cid:0) f k ( a, y ) (cid:1) → (cid:0) f k ( x, c ) (cid:1) → I and J respectively. Now, Theorem 2.4 provides the proof. (cid:3) Theorem 2.7.
Let f ∈ C ( ⊏⊐ ) with f ( x ) ≥ ∀ x ∈ ⊏⊐ . Then, for a given ǫ > , there exists g ∈ S β satisfying the following: g ( x ) ≥ ∀ x ∈ ⊏⊐ and k f − g k ∞ < ǫ. Proof.
Let ǫ > . Theorem 2.3 yields an element h ∈ S β such that k f − h k ∞ < ǫ . We define g ( x ) := h ( x ) + ǫ , ∀ x ∈ ⊏⊐ . Then, by Lemma 2.1, g ∈ S β , and by routine calculations, we get g ( x ) = h ( x ) − f ( x ) + f ( x ) + ǫ ≥ −k f − h k ∞ + f ( x ) + ǫ > f ( x ) ≥ . Furthermore, one has k f − g k ∞ ≤ k f − h k ∞ + k h − g k ∞ < ǫ, hence the proof. (cid:3) Theorem 2.8.
Let f : ⊏⊐ → R be a ( m, n ) -convex function such that f ( a, y ) = f ( x, c ) = 0 , ∀ x ∈ I, y ∈ J. Then for ǫ > , there exists ( m, n ) -convex function g such that D ( m,n ) g ∈ S β and k f − g k ∞ < ǫ. N BIVARIATE FRACTAL APPROXIMATION 5
Proof.
Let ǫ > . Using Theorem 2.3, there exists h ∈ S β such that k D ( m,n ) f − h k < ǫ ( b − a ) m ( d − c ) n . By choosing g ( x, y ) := Z xa Z yc · · · Z x m − a Z y n − c h ( x m , y n ) dx m dy n . . . dx dy , we have k f − g k = sup ( x,y ) ∈ ⊏⊐ n(cid:12)(cid:12)(cid:12) f − Z xa Z yc · · · Z x m − a Z y n − c h ( x m , y n ) dx m dy n . . . dx dy (cid:12)(cid:12)(cid:12)o < ǫ, proving the assertion. (cid:3) Theorem 2.9.
Let f ∈ C ( ⊏⊐ ) . Then, for ǫ > there exists g ∈ S β such that g ( x ) ≤ f ( x ) ∀ x ∈ ⊏⊐ and k f − g k ∞ < ǫ. Proof.
Since f ∈ C ( ⊏⊐ ) and ǫ >
0, Theorem 2.3 generates a member h ∈ S β such that k f − h k ∞ < ǫ . Choose g ( x ) := h ( x ) − ǫ , ∀ x ∈ ⊏⊐ . Then, g ( x ) = h ( x ) − f ( x ) + f ( x ) − ǫ ≤ k f − h k ∞ + f ( x ) − ǫ < f ( x ) . Furthermore, k f − g k ∞ ≤ k f − h k ∞ + k h − g k ∞ < ǫ, establishing the proof. (cid:3) Now, we aim to show the existence of best one-sided approximation. Let β ∈ [2 , , and define C β ( ⊏⊐ ) := { f ∈ C ( ⊏⊐ ) : dim B ( Gr ( f )) ≤ β } . In view of [10, Proposition 3 . C β ( ⊏⊐ ) is a normed linear space. Let { g , g , . . . , g n } be a linearly independent subset of C β ( ⊏⊐ ) . Further, for a bounded below and Lebesgue integrablefunction f : ⊏⊐ → R , we define Y βn ( f ) := n h ∈ span { g , g , . . . , g n } : h ( x ) ≤ f ( x ) ∀ x ∈ ⊏⊐ o . Theorem 2.9 guarantees the nonemptyness of Y βn ( f ) . A function h f ∈ Y βn ( f ) is said to be a bestone-sided approximation from below to f on ⊏⊐ if Z ⊏⊐ h f ( x ) d x = sup n Z ⊏⊐ h ( x ) d x : h ∈ Y βn ( f ) o . In a similar way, we define best one-sided approximations from above. We state the next theoremfor one-sided approximation from below. Though a similar result can be proved in terms ofone-sided approximation from above, see, for instance, [7, 25].
Theorem 2.10.
For a bounded below and integrable function f : ⊏⊐ → R , there exists a memberin Y n ( f ) of best one-sided approximant from below to f on ⊏⊐ .Proof. Let ( h m ) be a sequence in Y n ( f ) such that(2.2) Z ⊏⊐ h m ( x ) d x → A as m → ∞ , where A = sup n R ⊏⊐ h ( x ) d x : h ∈ Y βn ( f ) o . With an appropriate constant M ∗ > , we have Z ⊏⊐ | h m ( x ) | d x ≤ Z ⊏⊐ (cid:12)(cid:12)(cid:12) h m ( x ) − A ( b − a )( d − c ) (cid:12)(cid:12)(cid:12) d x + Z ⊏⊐ A ( b − a )( d − c ) d x ≤ M ∗ , where I = [ a, b ] and J = [ c, d ] . Since Y βn ( f ) is a subset of finite-dimensional linear space, the closedset of radius M ∗ in Y βn ( f ) is compact. Therefore, there exist a subsequence ( h m k ) and a function h in Y βn ( f ) such that the sequence ( h m k ) converges to h in L ( ⊏⊐ ) . Recall a basic functional analysis
V. AGRAWAL, T. SOM, AND S. VERMA result that every norm is equivalent on a finite-dimensional linear space. Now, from the finite-dimensionality of Y βn ( f ), it follows that the sequence ( h m k ) also converges to h uniformly. Further,since h m ( x ) ≤ f ( x ) , ∀ x ∈ ⊏⊐ , and h m k → h uniformly, we get h ( x ) ≤ f ( x ) , ∀ x ∈ ⊏⊐ . Thus, h ∈ Y βn ( f ) . Now, by (2.2), we have Z ⊏⊐ h ( x ) d x = lim k →∞ Z ⊏⊐ h m k ( x ) d x = A, completing the task. (cid:3) Construction of dimension preserving approximants.
First, Hutchinson [14] hinted atthe generation of parameterized fractal curves. In [2], Barnsley introduced Fractal InterpolationFunctions (FIFs) via Iterated Function System (IFSs). It is important to choose IFS appropriatelythat it is fitted as an attractor for a graph of a continuous function called FIF. We refer to thereader [2] for more study regarding the construction of FIFs.Computation of dimensions of fractal functions has been an integral part of fractal geometry.In [2], Barnsley proved estimates for the Hausdorff dimension of an affine FIF. Falconer alsoestablished a similar results in [8]. Barnsley and his collaborators [3, 4, 12] computed the boxdimension of classes of affine FIFs. In [4], FIFs generated by bilinear maps have been studied.In [13], a formula for the box dimension of FIFs R n → R m was proved. A particular caseof FIFs given by Navascu´es [17], namely, (univariate) α -fractal function has been proven veryuseful in approximation theory and operator theory. Using series expansion, the box dimension of(univariate) α -fractal function is estimated in [26].Let us recall a construction of bivariate α -fractal function introduced in [24], which was influ-enced by Ruan and Xu [19], on rectangular grids.Let x = a, x N = b, y = c, y M = d, and f ∈ C ( ⊏⊐ ) . Let us denote Σ k = { , , . . . , k } , Σ k, = { , , . . . k } , ∂ Σ k, = { , k } and intΣ k, = { , , . . . , k − } . Further, a net ∆ on ⊏⊐ isdefined as follows:∆ := { ( x i , y j ) : i ∈ Σ N, , j ∈ Σ M, and x < x < · · · < x N ; y < y < · · · < y M } . For each i ∈ Σ N and j ∈ Σ M , let us define I i = [ x i − , x i ] , J j = [ y j − , y j ] and ⊏⊐ ij := I i × J j . Let i ∈ Σ N , we define contraction mappings u i : I → I i such that u i ( x ) = x i − , u i ( x N ) = x i , if i is odd , and u i ( x ) = x i , u i ( x N ) = x i − , if i is even.Similar to the above, for each j ∈ Σ M , we define v j : J → J j , and Q ij ( x ) := ( u − i ( x ) , v − j ( y )) , where x = ( x, y ) ∈ ⊏⊐ ij . Let α ∈ C ( ⊏⊐ ) be such that k α k ∞ < . Assume further that s ∈ C ( ⊏⊐ ) satisfying s ( x i , y j ) = f ( x i , y j ) , for all i ∈ ∂ Σ N, , j ∈ ∂ Σ M, . By [25, Theorem 3 . f α ∆ ,s ∈C ( ⊏⊐ ) termed as α -fractal function, such that f α ∆ ,s ( x ) = f ( x ) + α ( x ) f α ∆ ,s (cid:0) Q ij ( x ) (cid:1) − α ( x ) s (cid:0) Q ij ( x ) (cid:1) , for x ∈ ⊏⊐ ij , ( i, j ) ∈ Σ N × Σ M . Note . In this note, we recall Theorem 5 .
16 in [25]. With the metric d ⊏⊐ ( x , y ) := p ( x − y ) + ( x − y ) , where x = ( x , x ) , y = ( y , y ) , we consider f and s such that(2.3) | f ( x ) − f ( y ) | ≤ K f d ⊏⊐ ( x , y ) σ , | s ( x ) − s ( x ) | ≤ K s d ⊏⊐ ( x , y ) σ . for every x , y ∈ ⊏⊐ , and for fixed K f , K s > . Assume that for some k f > , δ > x ∈ ⊏⊐ and 0 < δ < δ there exists y such that d ⊏⊐ ( x , y ) ≤ δ and(2.4) | f ( x ) − f ( y ) | ≥ k f d ⊏⊐ ( x , y ) σ . Furthermore, we suppose N = M, x i − x i − = N , y j − y j − = M , ∀ i ∈ Σ N , j ∈ Σ M and constantscaling function α. If | α | < min n M , k f ( K fα + K s ) M σ o , then dim B (cid:0) Gr ( f α ) (cid:1) = 3 − σ. N BIVARIATE FRACTAL APPROXIMATION 7
Remark . With the assumptions in the above note, one may construct dimension preservingapproximants for a given function, see, for instance, [23, Theorem 3 . α -fractal function via so-called (univariate)fractal operator. In [24, 25], her collaborators extended some of her results in bivariate setting.On putting L = B m,n in [24, Theorem 3 . f α ∆ ,B m,n ∈ C ( ⊏⊐ ) such that(2.5) f α ∆ ,B m,n ( x ) = f ( x ) + α ( x ) f α ∆ ,B m,n (cid:0) Q ij ( x ) (cid:1) − α ( x ) B m,n ( f ) (cid:0) Q ij ( x ) (cid:1) , for x ∈ ⊏⊐ ij , ( i, j ) ∈ Σ N × Σ M . Following the work of [24], we define a single-valued fractal operator F αm,n : C ( ⊏⊐ ) → C ( ⊏⊐ ) by F αm,n ( f ) = f α ∆ ,B m,n . In [24], several operator theoretic results for fractal operator are obtained. We recall that F αm,n isa bounded linear operator, see, for instance, [24, Theorem 3 . Lemma 2.13 ( [5], Lemma 1) . Let ( X, k . k ) be a Banach space, T : X → X be a linear operator.Suppose there exist constants λ , λ ∈ [0 , such that k T x − x k ≤ λ k x k + λ k T x k , ∀ x ∈ X. Then T is a topological isomorphism, and − λ λ k x k ≤ k T − x k ≤ λ − λ k x k , ∀ x ∈ X. Note . We have the following. B m,n ( f )( x ) = 1( b − a ) m ( d − c ) n m X i =0 n X j =0 (cid:18) mi (cid:19)(cid:18) nj (cid:19) ( x − a ) i ( b − x ) m − i ( y − c ) j ( d − y ) n − j f (cid:16) a + i ( b − a ) m , c + j ( d − c ) n (cid:17) , Choosing f = 1 , we have B m,n x ) = 1( b − a ) m ( d − c ) n m X i =0 n X j =0 (cid:18) mi (cid:19)(cid:18) nj (cid:19) ( x − a ) i ( b − x ) m − i ( y − c ) j ( d − y ) n − j = 1( b − a ) m ( d − c ) n m X i =0 (cid:18) mi (cid:19) ( x − a ) i ( b − x ) m − i n X j =0 (cid:18) nj (cid:19) ( y − c ) j ( d − y ) n − j = 1( b − a ) m ( d − c ) n m X i =0 (cid:18) mi (cid:19) ( x − a ) i ( b − x ) m − i ( y − c + d − y ) n = 1( b − a ) m ( d − c ) n ( x − a + b − x ) m ( y − c + d − y ) n = 1 . This implies that k B m,n k ≥ . Now, for every f ∈ C ( ⊏⊐ ) we get | B m,n ( f )( x ) | ≤ k f k ∞ ( b − a ) m ( d − c ) n m X i =0 n X j =0 (cid:18) mi (cid:19)(cid:18) nj (cid:19) ( x − a ) i ( b − x ) m − i ( y − c ) j ( d − y ) n − j = k f k ∞ , which produces k B m,n k ≤ . Therefore, we have k B m,n k = 1 . Theorem 2.15.
The fractal operator F αm,n : C ( ⊏⊐ ) → C ( ⊏⊐ ) is a topological isomorphism.Proof. Using equation (2.5) and note 2.14, one gets (cid:13)(cid:13) f − F αm,n ( f ) (cid:13)(cid:13) ∞ ≤ k α k ∞ (cid:13)(cid:13) F αm,n ( f ) − B m,n f (cid:13)(cid:13) ∞ = k α k ∞ (cid:13)(cid:13) F αm,n ( f ) (cid:13)(cid:13) ∞ + k α k ∞ k f k ∞ . Since k α k ∞ <
1, the previous lemma yields that the fractal operator F αm,n is a topological isomor-phism. (cid:3) V. AGRAWAL, T. SOM, AND S. VERMA
Remark . The above theorem may strengthen item-4 of [24, Theorem 3 . F αm,n is a topological isomorphism if k α k ∞ < (cid:0) k I − B m,n k (cid:1) − , which is morerestricted than the standing assumption considered in the above theorem, that is, k α k ∞ < . Theorem 2.17.
Let f ∈ C ( ⊏⊐ ) be such that f ( x ) ≥ , ∀ x ∈ ⊏⊐ . Then for ǫ > , and for α ∈ C ( ⊏⊐ ) satisfying k α k ∞ < , we have an α -fractal function g α ∆ ,B m,n satisfying g α ∆ ,B m,n ( x ) ≥ , ∀ x ∈ ⊏⊐ and k f − g α ∆ ,B m,n k ∞ < ǫ. Proof.
Note that the Bernstein operator B m,n fixes the constant function 1, that is, B m,n (1) = 1 , where 1( x ) = 1 on ⊏⊐ . Consider α ∈ C ( ⊏⊐ ) such that k α k ∞ < . From Equation 2.5, we deduce k g α ∆ ,B m,n − g k ∞ ≤ k α k ∞ k g α ∆ ,B m,n − B m,n g k ∞ , ∀ g ∈ C ( ⊏⊐ ) . Choose g = 1, then the above inequality gives k f α ∆ ,B m,n − k ∞ ≤ k α k ∞ k f α ∆ ,B m,n − k ∞ , and this further yields k f α ∆ ,B m,n − k ∞ = 0 . Therefore, f α ∆ ,B m,n = 1, that is, F αm,n (1) = 1 . For ǫ > α ∈ C ( ⊏⊐ ) and f ∈ C ( ⊏⊐ ) . Using Theorem 2.3, there exists a function h α ∆ ,B m,n such that k f − h α ∆ ,B m,n k ∞ < ǫ , where F αm,n ( h ) = h α ∆ ,B m,n . Define g α ∆ ,B m,n ( x ) = h α ∆ ,B m,n ( x ) + ǫ for all x ∈ ⊏⊐ . Since F αm,n (1) = 1 ,g α ∆ ,B m,n ( x ) = h α ∆ ,B m,n ( x ) + ǫ x ) = h α ∆ ,B m,n ( x ) + ǫ α ( x ) . Further, since F αm,n is a linear operator g α ∆ ,B m,n = h α ∆ ,B m,n + ǫ α = F αm,n ( h + ǫ . Moreover, g α ∆ ,B m,n ( x ) = h α ∆ ,B m,n ( x ) + ǫ h α ∆ ,B m,n ( x ) + ǫ − f ( x ) + f ( x ) ≥ f ( x ) + ǫ − k h α ∆ ,B m,n − f k ∞ ≥ . Further, we get k f − g α ∆ ,B m,n k ∞ ≤ k f − h α ∆ ,B m,n k ∞ + k h α ∆ ,B m,n − g α ∆ ,B m,n k ∞ < ǫ ǫ ǫ, completing the proof. (cid:3) Some multi-valued mappings
First, we collect some definitions and related results which will be used in this section.
Definition 3.1. ( [1]). Let ( X, k . k X ) and ( Y, k . k Y ) be normed linear spaces. For a multi-valued(set-valued) mapping T : X ⇒ Y , the domain of T is defined by Dom( T ) := { x ∈ X : T ( x ) = ∅} . Then T : X ⇒ Y is(1) convex if λT ( x ) + (1 − λ ) T ( x ) ⊆ T (cid:0) λx + (1 − λ ) x (cid:1) , ∀ x , x ∈ Dom( T ) , λ ∈ [0 , . (2) process if λT ( x ) = T ( λx ) , ∀ x ∈ X, λ > , and 0 ∈ T (0) . (3) linear if βT ( x ) + γT ( x ) ⊆ T (cid:0) βx + γx (cid:1) , ∀ x , x ∈ Dom( T ) , β, γ ∈ R . N BIVARIATE FRACTAL APPROXIMATION 9 (4) closed if the graph of T defined by Gr ( T ) := (cid:8) ( x ) ∈ X × Y : y ∈ T ( x ) (cid:9) is closed.(5) Lipschitz if T ( x ) ⊆ T ( x ) + l k x − x k X U Y , ∀ x , x ∈ Dom( T ) , for some constant l > , where U Y = { y ∈ Y : k y k Y ≤ } .(6) lower semicontinuous at x ∈ X if there exists a δ > U ∩ T ( x ′ ) = ∅ whenever k x − x ′ k X < δ holds for a given open set U in Y satisfying U ∩ T ( x ) = ∅ . Note that the above definitions are also applicable in metric spaces with obvious modifications,see, for instance, [1].
Theorem 3.2 ( [6], Corollary 1 . . Let T : Dom ( T ) = X ⇒ Y be linear such that T (0) = { } . Then, T is single-valued. Theorem 3.3 ( [6], Corollary 2 . . Let T : Dom ( T ) = X ⇒ Y be such that T ( x ) is singleton forsome x ∈ X. Then the following are equivalent: • T is single-valued and affine. • T is convex. Our work in this part is partly motivated by [26].
Theorem 3.4.
The multi-valued mapping W α ∆ : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) defined by W α ∆ ( f ) = { f α ∆ ,B m,n : m, n ∈ N } is a Lipschitz process.Proof. Using the linearity of F αm,n , we have W α ∆ ( λf ) = { ( λf ) α ∆ ,B m,n : m, n ∈ N } = λ W α ∆ ( f ) , ∀ f ∈ C ( ⊏⊐ ) , λ > . Again by linearity of F αm,n , it is plain that W α ∆ (0) = { } . Therefore, W α ∆ is a process.Let f, g ∈ C ( ⊏⊐ ) . On applying Equation 2.5, we have (cid:12)(cid:12) f α ∆ ,B m,n ( x ) − g α ∆ ,B m,n ( x ) (cid:12)(cid:12) ≤ k f − g k ∞ + k α k ∞ k f α ∆ ,B m,n − g α ∆ ,B m,n k ∞ + k α k ∞ k B m,n ( g ) − B m,n ( f ) k ∞ , for any x ∈ ⊏⊐ . Further, we deduce k f α ∆ ,B m,n − g α ∆ ,B m,n k ∞ ≤ k α k ∞ k B m,n k − k α k ∞ k f − g k ∞ . Using k B m,n k = 1 , k f α ∆ ,B m,n − g α ∆ ,B m,n k ∞ ≤ k α k ∞ − k α k ∞ k f − g k ∞ . Consequently, we have W α ∆ ( g ) ⊆ W α ∆ ( f ) + 1 + k α k ∞ − k α k ∞ k f − g k ∞ U C ( ⊏⊐ ) , proving the Lipschitzness of W α ∆ , and hence the proof. (cid:3) Remark . For the multivalued mapping W α ∆ , let us first note the following:(1) By linearity of F α ∆ ,B m,n , we have W α ∆ (0) = { } . (2) Since if α = 0, m = k then f α ∆ ,B m,n = f α ∆ ,B k,l , hence W α ∆ : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) is not single-valued.In view of the above items, Theorems 3.2-3.3 produce that the mapping W α ∆ : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) isnot convex. Theorem 3.6.
Let a fixed net △ and m, n ∈ N , the multivalued mapping T ∆ m,n : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) by T ∆ m,n ( f ) = { f α △ ,B m,n : α ∈ C ( ⊏⊐ ) such that k α k ∞ < } is a process.Proof. Let f ∈ C ( ⊏⊐ ) and λ > ,λ T ∆ m,n ( f ) = λ { f α : α ∈ C ( ⊏⊐ ) such that k α k ∞ < } = { λf α : α ∈ C ( ⊏⊐ ) such that k α k ∞ < } = T ∆ m,n ( λf ) . Moreover, Using linearity of fractal operator, we have f α = 0 , whenever f = 0 . That is, 0 ∈T ∆ m,n (0) . Therefore, T ∆ m,n is a process. (cid:3) Remark . One may see that T ∆ m,n is not convex through the following lines. Let f, g ∈ C ( ⊏⊐ ) , T ∆ m,n ( f + g ) = { ( f + g ) α : k α k ∞ < } = { f α + g α : k α k ∞ < }⊆{ f α + g β : k α k ∞ < , k β k ∞ < } = { f α : k α k ∞ < } + { g β : k β k ∞ < }⊆T ∆ m,n ( f ) + T ∆ m,n ( g ) . Theorem 3.8.
Let a fixed net △ and m, n ∈ N , the multivalued mapping T ∆ m,n : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) defined by T ∆ m,n ( f ) = { f α △ ,B m,n : k α k ∞ ≤ q < } , satisfies the following: kT ∆ m,n k ≤ q − q k Id − B m,n k . Proof.
We have kT ∆ m,n k = sup f ∈C ( ⊏⊐ ) d (0 , T ∆ m,n ( f )) k f k ∞ = sup f ∈C ( ⊏⊐ ) inf f α ∈T ∆ m,n ( f ) k f α kk f k≤ sup f ∈C ( ⊏⊐ ) (cid:16) k α k ∞ − k α k ∞ k Id − B m,n k (cid:17) ≤ sup f ∈C ( ⊏⊐ ) (cid:16) q − q k Id − B m,n k (cid:17) =1 + q − q k Id − B m,n k , hence the proof. (cid:3) Theorem 3.9.
For a fixed net △ and operator L, the multivalued mapping T ∆ m,n : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) defined by T ∆ m,n ( f ) = { f α △ ,B m,n : k α k ∞ < } is lower semicontinuous.Proof. Let f ∈ C ( ⊏⊐ ) , let f α ∈ T ∆ m,n ( f ) and a sequence ( f k ) in C ( ⊏⊐ ) such that f k → f. Since thefractal operator is continuous, we have f αk → f α . It is clear that f αk ∈ T ∆ m,n ( f k ) . Therefore, theresult follows. (cid:3)
N BIVARIATE FRACTAL APPROXIMATION 11
Theorem 3.10.
Let △ be a net of ⊏⊐ and m, n ∈ N . The multi-valued mapping T ∆ m,n : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) defined by T ∆ m,n ( f ) = { f α △ ,B m,n : k α k ∞ ≤ q < } , is Lipschitz.Proof. Let f, g ∈ C ( ⊏⊐ ) . Equation (2.5) yields (cid:12)(cid:12) f α △ ,B m,n ( x ) − g α △ ,B m,n ( x ) (cid:12)(cid:12) = k f − g k ∞ + k α k ∞ k f α △ ,B m,n − g α △ ,B m,n k ∞ + k α k ∞ k B m,n g − B m,n f k ∞ , for every x ∈ ⊏⊐ . Further, we deduce k f α △ ,B m,n − g α △ ,B m,n k ≤ k α k ∞ k B m,n k − k α k ∞ k f − g k ∞ . Since k α k ∞ ≤ q and k B m,n k = 1 , we get k f α △ ,B m,n − g α △ ,B m,n k ≤ q − q k f − g k . Choosing l = q − q , we have T ∆ m,n ( g ) ⊂ T ∆ m,n ( f ) + l k f − g k ∞ U C ( ⊏⊐ ) , proving the assertion. (cid:3) Theorem 3.11.
For a fixed admissible scale vector α and m, n ∈ N , the multivalued mapping V αm,n : C ( ⊏⊐ ) ⇒ C ( ⊏⊐ ) defined by V αm,n ( f ) = { f α △ ,B m,n : all possible net △} is a process.Proof. Let f ∈ C ( ⊏⊐ ) and λ > , then λ V αm,n ( f ) = λ { f α △ ,B m,n : all possible net △} = { λf α △ ,B m,n : all possible net △} = { ( λf ) α △ ,B m,n : all possible net △} = V αm,n ( λf ) . The third equality follows from the fact that the fractal operator F αm,n is a linear operator. More-over, using linearity of the fractal operator, we have f α △ ,B m,n = 0 , whenever f = 0 . That is,0 ∈ V αm,n (0) . Therefore, V αm,n is a process. (cid:3) Theorem 3.12.
For a fixed admissible scale function α and m, n ∈ N , the multivalued mapping V αm,n is lower semicontinuous.Proof. Let f ∈ C ( ⊏⊐ ) , let f α △ ,B m,n ∈ V αm,n ( f ) and a sequence ( f k ) converges to f in C ( ⊏⊐ ) . Since thefractal operator is continuous, we have ( f k ) α △ ,B m,n → f α △ ,B m,n . By definition of V αm,n , ( f k ) α △ ,B m,n ∈V αm,n ( f k ) . Hence, the lower semicontinuity of V αm,n follows. (cid:3) Theorem 3.13.
The multi-valued function
Φ : [dim( X ) , dim( X ) + dim( Y )] → C ( X, Y ) defined by Φ( β ) := { f ∈ C ( X, Y ) : dim( Gr ( f )) = β } is lower semicontinuous.Proof. Let U be an open set of C ( X, Y ) . In the light of Theorem 2.3, that is, Φ( α ) = S α is a densesubset of C ( X, Y ), we obtain S ( α ) ∩ U = ∅ , ∀ α ∈ [dim( X ) , dim( X ) + dim( Y )] . Now, by the very definition of lower semicontinuous, the result follows. (cid:3)
Remark . Note that the multivalued mapping Φ is not closed. To show this, let f ∈ C ( X, Y )with dim( Gr ( f )) > dim( X ) . Consider a sequence of Lipschitz functions ( f k ) converging to f uniformly. It is obvious that dim( Gr ( f k )) = dim( X ) . Now, we have (cid:0) dim( X ) , f k (cid:1) → (cid:0) dim( X ) , f (cid:1) as n → ∞ . Using (cid:0) dim( X ) , f k (cid:1) ∈ Gr (Φ) and (cid:0) dim( X ) , f k (cid:1) → (cid:0) dim( X ) , f (cid:1) with dim( Gr ( f )) > dim( X ) , we get the result. 4. conclusion This paper has been intended to develop a newly defined notion of constrained approximationtermed as dimension preserving approximation for bivariate functions. The later work of thepaper has introduced some multi-valued operators associated with bivariate α -fractal functions.The notion of dimension preserving approximation is new, and demands further developments. Inparticular, dimension preserving approximation of set-valued mappings may be one of our futureinvestigations. References
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Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi- 221005,India
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