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Improving certain Bernstein-type approximation processes
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Mathematics and Computers in Simulation 77 (2008) 170–178
Improving certain Bernstein-type approximation processes夽 D. C´ardenas-Morales ∗ , F.J. Mu˜noz-Delgado Departamento de Matem´aticas, Universidad de Ja´en, 23071 Ja´en, Spain Available online 31 August 2007
Abstract This paper deals with a modification of the classical Bernstein polynomials defined on the unit simplex. It introduces a new sequence of non-polynomial linear operators which hold fixed the polynomials x2 + αx and y2 + βy with α, β ∈ [0, +∞). We study the convergence properties of the new approximation process and certain shape properties that are preserved. Finally, we compare it with Bernstein polynomials and show an improvement of the error of convergence in certain subsets of the simplex. © 2007 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Bivariate Bernstein polynomials; Shape preserving properties; Error of approximation
1. Introduction Given n ∈ N = {1, 2, . . . }, the Bernstein polynomial of order n on the simplex S ≡ {(x, y) ∈ R2 ; x, y ≥ 0, x + y ≤ 1} is given for f ∈ C(S) by n n−k n n−k k l Bn f (x, y) = xk yl (1 − x − y)n−k−l f , . (1) n n k l k=0 l=0
No word is necessary to say about the important role that these two-dimensional polynomials play in many fields inside applied mathematics. On the other hand several extensions of this type of approximation processes have appeared in literature (see for instance Refs. [1,5] and references therein). Many properties of these operators are very well-known. Amongst them, if we denote p0 (x, y) = 1, p1 (x, y) = x, p2 (x, y) = y: Bn pi = pi ,
i = 0, 1, 2.
(2)
In this paper, following a pioneer idea of King [6], further developed in Ref. [2], we modify the operators above so that they hold fixed some polynomials different from pi (x, y). The resulting approximation processes turn out to have an order of approximation at least as good as the one of Bn f (x, y) in certain subsets of S. In addition to this, the new operators present good shape preserving properties which together with the fact that their computation do not require too much additional cost, endow them with certain interest. 夽 ∗
This work is partially supported by Junta de Andaluc´ıa (FQM-0178) and by Ministerio de Ciencia y Tecnolog´ıa (MTM 2006-14590). Corresponding author. E-mail address: [email protected] (D. C´ardenas-Morales).
0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2007.08.009
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
171
For each integer n > 1 let rn : [0, +∞) × [0, 1] → R be the function defined by (nγ + 1)2 nγ + 1 n(γz + z2 ) rn (γ, z) := − + , + 2 2(n − 1) n−1 4(n − 1) α,β
and for α, β ∈ [0, +∞), let Bn be the sequence of linear operators defined on C(S) by n n−k k l α,β Bnα,β f (x, y) := f , pn,k,l (x, y), n n
(3)
k=0 l=0
where α,β pn,k,l (x, y)
=
n n−k k
l
rn (α, x)k rn (β, y)l (1 − rn (α, x) − rn (β, y))n−k−l .
For each n > 1 and x, y ∈ [0, 1], if we let α and β go to infinity, then rn (α, x) and rn (β, y) tend to x and y, respectively, α,β and Bn f becomes the classical Bernstein polynomials Bn f (x, y) presented in Eq. (1). α,β In the following sections we study the operators Bn . We show the functions they fix and the shape preserving and convergence properties they posses. In the last section we compare them with Bn f (x, y). 2. Shape preserving properties Let α, β ≥ 0 and n > 1. Proceeding as it is usually done for the classical Bernstein polynomials it is easily obtained that Bnα,β p0 = p0 ,
Bnα,β p1 (x, y) = rn (α, x),
Bnα,β p2 (x, y) = rn (β, y),
(4)
rn (α, x) n − 1 + rn (α, x)2 , n n rn (β, y) n − 1 + rn (β, y)2 , Bnα,β p22 (x, y) = n n From the definition of rn one can check the validity of the following. Bnα,β p21 (x, y) =
(5) (6)
α,β
Proposition 1. The operators Bn hold fixed the polynomials p21 + αp1 and p22 + βp2 , i.e. Bnα,β (p21 + αp1 )(x, y) = (p21 + αp1 )(x, y) = x2 + αx,
Bnα,β (p22 + βp2 )(x, y) = (p22 + βp2 )(x, y) = y2 + βy.
On the other hand, for each γ ∈ [0, +∞), z → rn (γ, z) is an increasing and convex real function satisfying rn (γ, 0) = α,β 0, rn (γ, 1) = 1 and 0 < rn (γ, z) < z < 1 for 0 < z < 1. As a direct consequence, for α, β ∈ [0, +∞), Bn is a positive operator which interpolates f at the vertices of S. As regards other visual shape preserving properties, we first recall a usual definition of convexity for bivariate functions. For f ∈ C(S), (x, y) ∈ S and h ∈ R+ we define (whenever it has sense): (1,0)
f (x, y) = f (x + h, y) − f (x, y),
(1,1)
f (x, y) = f (x + h, y + h) + f (x, y) − f (x + h, y) − f (x, y + h),
(2,0)
f (x, y) = f (x + 2h, y) − 2f (x + h, y) + f (x, y),
(0,2)
f (x, y) = f (x, y + 2h) − 2f (x, y + h) + f (x, y).
h h h h
(0,1)
h
f (x, y) = f (x, y + h) − f (x, y),
Definition 2. f (x, y) is convex of order (i, j), i, j ∈ N, 0 < i + j ≤ 2, if for h ∈ R+ , h f ≥ 0. (i,j)
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Remark 3. If f ∈ Ci+j (S) and for all (x, y) ∈ S ∂i+j f (x, y) ≥ 0, ∂xi ∂yj then f (x, y) is convex of order (i, j). α,β
With the aim of obtaining the shape preserving properties that the operators Bn posses, now we show expressions α,β of the derivatives of Bn f we are interested in. To derive them requires tedious but elementary computation (for details see Ref. [4]). n−1 n−k−1 α,β ∂Bn f ∂rn (α, x) α,β k+1 l k l (x, y) = n pn−1,k,l (x, y) f , −f , , ∂x ∂x n n n n k=0 l=0
α,β ∂ 2 Bn f ∂x2
(x, y) = n
n−1 n−k−1 ∂2 rn (α, x) α,β k+1 l k l p (x, y) f , − f , n−1,k,l ∂x2 n n n n k=0 l=0
n−2 n−k−2 ∂rn (α, x) 2 α,β + n(n − 1) pn−2,k,l (x, y) ∂x k=0 l=0 k+2 l k+1 l k l × f , − 2f , +f , , n n n n n n n−1 n−k−1 α,β ∂Bn f ∂rn (β, y) α,β k l+1 k l (x, y) = n pn−1,k,l (x, y) f , −f , , ∂y ∂y n n n n k=0 l=0
α,β ∂ 2 Bn f ∂y2
(x, y) = n
n−1 n−k−1 ∂2 rn (β, y) α,β k l+1 k l p (x, y) f , − f , n−1,k,l ∂y2 n n n n k=0 l=0
n−2 n−k−2 ∂rn (β, y) 2 α,β pn−2,k,l (x, y) + n(n − 1) ∂y k=0 l=0 k l+2 k l+1 k l × f , − 2f , +f , , n n n n n n n−2 n−k−2 α,β α,β ∂ 2 Bn f ∂rn ∂rn pn−2,k,l (x, y) (x, y) = n (α, x) (β, y) ∂x∂y ∂x ∂y k=0 l=0 k+1 l+1 k l k l+1 k+1 l × f , +f , −f , −f , . n n n n n n n n
From these expressions, taking into account the aforementioned properties of the function z → rn (γ, z) and Remark 3, the following result follows directly. Proposition 4. Let α, β ∈ [0, +∞) and f ∈ C(S). α,β
(i) If f (x, y) is convex of order (1, 0) (resp. (0, 1)), then so is Bn f . α,β (ii) The convexity of order (2, 0) (resp. (0, 2)) of the function f (x, y) does not imply the one of Bn f . α,β (iii) If f (x, y) is simultaneously convex of order (1, 0) and (2, 0) (resp. (0, 1) and (0, 2)) then Bn f is convex of order (2, 0) (resp. (0, 2)). α,β (iv) If f (x, y) is convex of order (1, 1), then so is Bn f . Remark 5. For a proof of (ii) choose for instance f (x, y) = −x (resp. f (x, y) = −y).
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3. Convergence properties α,β
Let α, β ∈ [0, +∞) and f ∈ C(S). As regards the convergence of Bn f (x, y) we may state that lim Bα,β f (x, y) n→∞ n
= f (x, y),
(x, y) ∈ S.
Indeed, if suffices to recall that p0 , p1 , p2 and p21 + p22 form a system of test functions for C(S), to use Eqs. (4)–(6) and to check that for γ ∈ [0, +∞), rn (γ, z) converges to z as n tends to infinity. Moreover, the convergence is uniform since √ n(1 + γ) − n(1 + nγ(2 + γ)) rn (γ) := sup {z − rn (γ, z)} = (7) 2n(n − 1) z ∈ [0,1] converges to 0 as n tends to infinity. α,β Searching for a quantitative result on the convergence of Bn f towards f ∈ C(S), we can use a well-known theorem of Censor [3]. It informs us that for (x, y) ∈ S and δ > 0 |Bnα,β f (x, y) − f (x, y)| ≤ |f (x, y)| · |Bnα,β p0 (x, y) − p0 (x, y)| α,β 2 2 ((p − xp ) + (p − yp ) )(x, y) B n 1 0 2 0 ω(f, δ). + Bnα,β p0 (x, y) + δ2 Here ω(f, δ) is the usual bivariate Euclidean modulus of continuity which is defined by ω(f, δ) = sup{|f (x1 , y1 ) − f (x2 , y2 )| : (xi , yi ) ∈ S, (x2 − x1 )2 + (y2 − y1 )2 ≤ δ}. By using Eqs. (4)–(6) we can rewrite the estimate above as follows: 2x2 + αx − rn (α, x)(α + 2x) 2y2 + βy − rn (β, y)(β + 2y) α,β |Bn f (x, y) − f (x, y)| ≤ 1 + ω(f, δ). + δ2 δ2
(8)
4. Comparison with Bernstein polynomials: graphical examples The classical Bernstein polynomials on S presented in Eq. (1) have nice well-known shape properties. They preserve the convexity of all the aforementioned orders and in addition to Eq. (2) they satisfy Bn p2i =
1 n−1 2 pi + pi , n n
i = 1, 2.
On the other hand, they are known not to preserve the classical convexity in two variables, but the so-called axial convexity, that is to say, convexity in the directions which are parallel to the edges of S. Moreover, this weaker form of convexity suffices to guarantee the monotonicity of successive Bernstein polynomials (for details see Ref. [7]). As regards the quantitative aspect, Censor’s result gives the estimate x(1 − x) + y(1 − y) |Bn f (x, y) − f (x, y)| ≤ 1 + ω(f, δ). nδ2 α,β
Thus, the order of approximation towards a function f ∈ C(S) given by the sequence Bn f (see Eq. (8)) will be at α,β least as good as that of Bn f whenever the following function Qn (x, y) is non-negative: 2 2 Qα,β n (x, y) = rn (α, x)(α + 2x) − 2x − αx + rn (β, y)(β + 2y) − 2y − βy + α,β
x(1 − x) y(1 − y) + . n n
Figs. 1–3 show some graphs of Qn (x, y) for n = 5, α = 0 and different illustrative values of β. We have plotted them together with the plane z = 0 to emphasize the regions of positivity. We have also included ‘topographic maps’ of the functions, which are shaded in such a way that regions with higher values are lighter.
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Fig. 1. Graph of Q0,0 5 (x, y).
Fig. 2. Graph of Q0,0.4 (x, y). 5 α,β
The non-negativity of Qn (x, y) is obviously fulfilled at these points (x, y) where simultaneously rn (α, x)(α + 2x) − 2x2 − αx +
x(1 − x) ≥0 n
rn (β, y)(β + 2y) − 2y2 − βy +
y(1 − y) ≥ 0. n
and
Fig. 3. Graph of Q0,10 5 (x, y).
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
175
Some calculations with the aid of the software Mathematica© state the validity of these inequalities when and only when (x, y) lies in the subset of S given by the rectangle
√ 1 + n − 2nα + 1 + 2n + n2 + 8n2 α + 4n2 α2 1 + n − 2nβ + 1 + 2n + n2 + 8n2 β + 4n2 β2 0, × 0, . 2(1 + 3n) 2(1 + 3n) As n goes to infinity, the end points of these intervals decrease, respectively, to √ 1 − 2α + 1 + 8α + 4α2 6 and 1 − 2β +
1 + 8β + 4β2 . 6 α,β
As a consequence the order of approximation of Bn f (x, y) towards f (x, y) is at least as good as the order of approximation to f (x, y) given by Bn f (x, y) whenever (x, y) lies in
√ 1 − 2α + 1 + 8α + 4α2 1 − 2β + 1 + 8β + 4β2 0, × 0, . 6 6 Notice that the rectangle above becomes [0, 1/3] × [0, 1/3] when α = 0 and β = 0 and enlarges to [0, 1/2] × [0, 1/2] as α and β go to infinity. α,β The fact that the error bound for Bn f (x, y) − f (x, y) improves the one for Bn f (x, y) − f (x, y) in certain region of S does not provide us with valuable information about the actual errors. It would be desirable to be able to state a result analogous to ([2], Theorem 1), where in the univariate setting sufficient conditions on a function were provided under which the proposed modified Bernstein operators approximated the function better than the classical polynomials. Unfortunately we cannot go so further. However, in the sequel we show how the new approximation process may actually approximate a function f ∈ C(S) better than Bn f (x, y) preserving at the time certain shape properties. Let us assume that a function f ∈ C(S) is axially convex (in particular it is (2, 0)-convex and (0, 2)-convex), then α,β we know that for (x, y) ∈ S, Bn f (x, y) ≥ f (x, y). Besides, Bn f (x, y) converges uniformly in S to Bn f (x, y) as α and β go to infinity (for a proof it suffices to recover (7) and check that rn (γ) converges to 0 as γ tends to infinity), and if in α,β α,β addition we assume that f is (1, 0)-convex and (0, 1)-convex, then the functions α → Bn f (x, y) and β → Bn f (x, y) are increasing as it is derived from the following expressions one may compute easily ∂ n 2 + 2nγ + 4(n − 1)z −2 , rn (γ, z) = ∂γ 4(n − 1) (nα + 1)2 + 4(n − 1)n(γx + x2 ) n−1 n−k−1 α,β ∂Bn f k+1 l ∂rn (α, x) α,β k l pn−1,k,l (x, y) f (x, y) = n , −f , , ∂α ∂α n n n n k=0 l=0
n−1 n−k−1 α,β ∂rn (β, y) α,β k l k l+1 ∂Bn f (x, y) = n , −f , . pn−1,k,l (x, y) f ∂β ∂β n n n n k=0 l=0
α,β
On the other hand, under the assumptions on f (x, y) above, Bn f (x, y) is (2, 0)-convex, (0, 2)-convex, (1, 0)-convex and (0, 1)-convex. α,β Summing up, Bn f (x, y) is convex of orders (1, 0), (0, 1), (2, 0) and (0, 2) (as it is f (x, y)), it converges uniformly in S towards Bn f (x, y) as α, β → ∞, this convergence is monotone from below and finally Bn f ≥ f in S. A question α,β α,β naturally arises: do there exist constants α and β such that Bn f (x, y) ≥ f (x, y)? If that was the case, then Bn f (x, y) would represent a good shape preserving approximation. Finally, as particular cases, we are going to consider two exponential type functions which naturally may appear when modelling a variety of processes. From top to bottom Fig. 4 2 2 represents the functions f (x, y) = ex +y , B5 f (x, y) and B50,0 f (x, y). This function meets all the shape properties above.
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D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
Fig. 4. From top to bottom graphs of f (x, y) = ex
2 +y2
, B5 f (x, y) and B50,0 f (x, y).
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
Fig. 5. Graph of B50,0 f (x, y) − f (x, y) for f (x, y) = ex
2 +y2
Fig. 6. Graph of B50,0 f (x, y) − f (x, y) for f (x, y) = ex
2 +y
177
.
. α,β
Fig. 5 shows the graph of B50,0 f (x, y) − f (x, y), which is positive. Hence for all α, β ∈ [0, ∞), ex +y ≤ B5 f (x, y) ≤ B5 f (x, y). 2 The situation is different for f (x, y) = ex +y , though this function also fulfills the aforementioned properties. Fig. 6 0,0 shows the graph of B5 f (x, y) − f (x, y), which is not positive, and its level sets. In Fig. 7 we have increased the value of β till 100. Now B50,100 f (x, y) − f (x, y) is positive. 2
Fig. 7. Graph of B50,100 f (x, y) − f (x, y) for f (x, y) = ex
2 +y
.
2
178
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
Acknowledgements The authors wish to thank the referees for their helpful suggestions. References [1] J.A. Adell, J. de la Cal, M. San Miguel, On the property of monotonic convergence for multivariate Bernstein-type operators, J. Approx. Theory 80 (1995) 132–137. [2] D. C´ardenas-Morales, F.J. Mu˜noz-Delgado, P. Garrancho, Shape preserving approximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput. 182 (2006) 1615–1622. [3] E. Censor, Quantitative results for positive linear approximation operators, J. Approx. Theory 4 (1971) 442–450. [4] Z. Ditzian, Inverse theorems for multidimensional Bernstein operators, Pac. J. Math. 121 (2) (1986) 293–319. [5] H.H. Gonska, J. Meier, A bibliography on approximation of functions by Bernstein type operators, in: C.K. Chui, L.L. Xchumakder, J.D. Ward (Eds.), Approximation Theory IV, Academic Press, New York, 1983, pp. 739–785. [6] J.P. King, Positive linear operators which preserve x2 , Acta Math. Hungar. 99 (3) (2003) 203–208. [7] T. Sauer, Multivariate Bernstein polynomials and convexity, Comput. Aided Geom. Des. 8 (1991) 465–478.
Mathematics and Computers in Simulation 77 (2008) 170–178
Improving certain Bernstein-type approximation processes夽 D. C´ardenas-Morales ∗ , F.J. Mu˜noz-Delgado Departamento de Matem´aticas, Universidad de Ja´en, 23071 Ja´en, Spain Available online 31 August 2007
Abstract This paper deals with a modification of the classical Bernstein polynomials defined on the unit simplex. It introduces a new sequence of non-polynomial linear operators which hold fixed the polynomials x2 + αx and y2 + βy with α, β ∈ [0, +∞). We study the convergence properties of the new approximation process and certain shape properties that are preserved. Finally, we compare it with Bernstein polynomials and show an improvement of the error of convergence in certain subsets of the simplex. © 2007 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Bivariate Bernstein polynomials; Shape preserving properties; Error of approximation
1. Introduction Given n ∈ N = {1, 2, . . . }, the Bernstein polynomial of order n on the simplex S ≡ {(x, y) ∈ R2 ; x, y ≥ 0, x + y ≤ 1} is given for f ∈ C(S) by n n−k n n−k k l Bn f (x, y) = xk yl (1 − x − y)n−k−l f , . (1) n n k l k=0 l=0
No word is necessary to say about the important role that these two-dimensional polynomials play in many fields inside applied mathematics. On the other hand several extensions of this type of approximation processes have appeared in literature (see for instance Refs. [1,5] and references therein). Many properties of these operators are very well-known. Amongst them, if we denote p0 (x, y) = 1, p1 (x, y) = x, p2 (x, y) = y: Bn pi = pi ,
i = 0, 1, 2.
(2)
In this paper, following a pioneer idea of King [6], further developed in Ref. [2], we modify the operators above so that they hold fixed some polynomials different from pi (x, y). The resulting approximation processes turn out to have an order of approximation at least as good as the one of Bn f (x, y) in certain subsets of S. In addition to this, the new operators present good shape preserving properties which together with the fact that their computation do not require too much additional cost, endow them with certain interest. 夽 ∗
This work is partially supported by Junta de Andaluc´ıa (FQM-0178) and by Ministerio de Ciencia y Tecnolog´ıa (MTM 2006-14590). Corresponding author. E-mail address: [email protected] (D. C´ardenas-Morales).
0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2007.08.009
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
171
For each integer n > 1 let rn : [0, +∞) × [0, 1] → R be the function defined by (nγ + 1)2 nγ + 1 n(γz + z2 ) rn (γ, z) := − + , + 2 2(n − 1) n−1 4(n − 1) α,β
and for α, β ∈ [0, +∞), let Bn be the sequence of linear operators defined on C(S) by n n−k k l α,β Bnα,β f (x, y) := f , pn,k,l (x, y), n n
(3)
k=0 l=0
where α,β pn,k,l (x, y)
=
n n−k k
l
rn (α, x)k rn (β, y)l (1 − rn (α, x) − rn (β, y))n−k−l .
For each n > 1 and x, y ∈ [0, 1], if we let α and β go to infinity, then rn (α, x) and rn (β, y) tend to x and y, respectively, α,β and Bn f becomes the classical Bernstein polynomials Bn f (x, y) presented in Eq. (1). α,β In the following sections we study the operators Bn . We show the functions they fix and the shape preserving and convergence properties they posses. In the last section we compare them with Bn f (x, y). 2. Shape preserving properties Let α, β ≥ 0 and n > 1. Proceeding as it is usually done for the classical Bernstein polynomials it is easily obtained that Bnα,β p0 = p0 ,
Bnα,β p1 (x, y) = rn (α, x),
Bnα,β p2 (x, y) = rn (β, y),
(4)
rn (α, x) n − 1 + rn (α, x)2 , n n rn (β, y) n − 1 + rn (β, y)2 , Bnα,β p22 (x, y) = n n From the definition of rn one can check the validity of the following. Bnα,β p21 (x, y) =
(5) (6)
α,β
Proposition 1. The operators Bn hold fixed the polynomials p21 + αp1 and p22 + βp2 , i.e. Bnα,β (p21 + αp1 )(x, y) = (p21 + αp1 )(x, y) = x2 + αx,
Bnα,β (p22 + βp2 )(x, y) = (p22 + βp2 )(x, y) = y2 + βy.
On the other hand, for each γ ∈ [0, +∞), z → rn (γ, z) is an increasing and convex real function satisfying rn (γ, 0) = α,β 0, rn (γ, 1) = 1 and 0 < rn (γ, z) < z < 1 for 0 < z < 1. As a direct consequence, for α, β ∈ [0, +∞), Bn is a positive operator which interpolates f at the vertices of S. As regards other visual shape preserving properties, we first recall a usual definition of convexity for bivariate functions. For f ∈ C(S), (x, y) ∈ S and h ∈ R+ we define (whenever it has sense): (1,0)
f (x, y) = f (x + h, y) − f (x, y),
(1,1)
f (x, y) = f (x + h, y + h) + f (x, y) − f (x + h, y) − f (x, y + h),
(2,0)
f (x, y) = f (x + 2h, y) − 2f (x + h, y) + f (x, y),
(0,2)
f (x, y) = f (x, y + 2h) − 2f (x, y + h) + f (x, y).
h h h h
(0,1)
h
f (x, y) = f (x, y + h) − f (x, y),
Definition 2. f (x, y) is convex of order (i, j), i, j ∈ N, 0 < i + j ≤ 2, if for h ∈ R+ , h f ≥ 0. (i,j)
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Remark 3. If f ∈ Ci+j (S) and for all (x, y) ∈ S ∂i+j f (x, y) ≥ 0, ∂xi ∂yj then f (x, y) is convex of order (i, j). α,β
With the aim of obtaining the shape preserving properties that the operators Bn posses, now we show expressions α,β of the derivatives of Bn f we are interested in. To derive them requires tedious but elementary computation (for details see Ref. [4]). n−1 n−k−1 α,β ∂Bn f ∂rn (α, x) α,β k+1 l k l (x, y) = n pn−1,k,l (x, y) f , −f , , ∂x ∂x n n n n k=0 l=0
α,β ∂ 2 Bn f ∂x2
(x, y) = n
n−1 n−k−1 ∂2 rn (α, x) α,β k+1 l k l p (x, y) f , − f , n−1,k,l ∂x2 n n n n k=0 l=0
n−2 n−k−2 ∂rn (α, x) 2 α,β + n(n − 1) pn−2,k,l (x, y) ∂x k=0 l=0 k+2 l k+1 l k l × f , − 2f , +f , , n n n n n n n−1 n−k−1 α,β ∂Bn f ∂rn (β, y) α,β k l+1 k l (x, y) = n pn−1,k,l (x, y) f , −f , , ∂y ∂y n n n n k=0 l=0
α,β ∂ 2 Bn f ∂y2
(x, y) = n
n−1 n−k−1 ∂2 rn (β, y) α,β k l+1 k l p (x, y) f , − f , n−1,k,l ∂y2 n n n n k=0 l=0
n−2 n−k−2 ∂rn (β, y) 2 α,β pn−2,k,l (x, y) + n(n − 1) ∂y k=0 l=0 k l+2 k l+1 k l × f , − 2f , +f , , n n n n n n n−2 n−k−2 α,β α,β ∂ 2 Bn f ∂rn ∂rn pn−2,k,l (x, y) (x, y) = n (α, x) (β, y) ∂x∂y ∂x ∂y k=0 l=0 k+1 l+1 k l k l+1 k+1 l × f , +f , −f , −f , . n n n n n n n n
From these expressions, taking into account the aforementioned properties of the function z → rn (γ, z) and Remark 3, the following result follows directly. Proposition 4. Let α, β ∈ [0, +∞) and f ∈ C(S). α,β
(i) If f (x, y) is convex of order (1, 0) (resp. (0, 1)), then so is Bn f . α,β (ii) The convexity of order (2, 0) (resp. (0, 2)) of the function f (x, y) does not imply the one of Bn f . α,β (iii) If f (x, y) is simultaneously convex of order (1, 0) and (2, 0) (resp. (0, 1) and (0, 2)) then Bn f is convex of order (2, 0) (resp. (0, 2)). α,β (iv) If f (x, y) is convex of order (1, 1), then so is Bn f . Remark 5. For a proof of (ii) choose for instance f (x, y) = −x (resp. f (x, y) = −y).
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3. Convergence properties α,β
Let α, β ∈ [0, +∞) and f ∈ C(S). As regards the convergence of Bn f (x, y) we may state that lim Bα,β f (x, y) n→∞ n
= f (x, y),
(x, y) ∈ S.
Indeed, if suffices to recall that p0 , p1 , p2 and p21 + p22 form a system of test functions for C(S), to use Eqs. (4)–(6) and to check that for γ ∈ [0, +∞), rn (γ, z) converges to z as n tends to infinity. Moreover, the convergence is uniform since √ n(1 + γ) − n(1 + nγ(2 + γ)) rn (γ) := sup {z − rn (γ, z)} = (7) 2n(n − 1) z ∈ [0,1] converges to 0 as n tends to infinity. α,β Searching for a quantitative result on the convergence of Bn f towards f ∈ C(S), we can use a well-known theorem of Censor [3]. It informs us that for (x, y) ∈ S and δ > 0 |Bnα,β f (x, y) − f (x, y)| ≤ |f (x, y)| · |Bnα,β p0 (x, y) − p0 (x, y)| α,β 2 2 ((p − xp ) + (p − yp ) )(x, y) B n 1 0 2 0 ω(f, δ). + Bnα,β p0 (x, y) + δ2 Here ω(f, δ) is the usual bivariate Euclidean modulus of continuity which is defined by ω(f, δ) = sup{|f (x1 , y1 ) − f (x2 , y2 )| : (xi , yi ) ∈ S, (x2 − x1 )2 + (y2 − y1 )2 ≤ δ}. By using Eqs. (4)–(6) we can rewrite the estimate above as follows: 2x2 + αx − rn (α, x)(α + 2x) 2y2 + βy − rn (β, y)(β + 2y) α,β |Bn f (x, y) − f (x, y)| ≤ 1 + ω(f, δ). + δ2 δ2
(8)
4. Comparison with Bernstein polynomials: graphical examples The classical Bernstein polynomials on S presented in Eq. (1) have nice well-known shape properties. They preserve the convexity of all the aforementioned orders and in addition to Eq. (2) they satisfy Bn p2i =
1 n−1 2 pi + pi , n n
i = 1, 2.
On the other hand, they are known not to preserve the classical convexity in two variables, but the so-called axial convexity, that is to say, convexity in the directions which are parallel to the edges of S. Moreover, this weaker form of convexity suffices to guarantee the monotonicity of successive Bernstein polynomials (for details see Ref. [7]). As regards the quantitative aspect, Censor’s result gives the estimate x(1 − x) + y(1 − y) |Bn f (x, y) − f (x, y)| ≤ 1 + ω(f, δ). nδ2 α,β
Thus, the order of approximation towards a function f ∈ C(S) given by the sequence Bn f (see Eq. (8)) will be at α,β least as good as that of Bn f whenever the following function Qn (x, y) is non-negative: 2 2 Qα,β n (x, y) = rn (α, x)(α + 2x) − 2x − αx + rn (β, y)(β + 2y) − 2y − βy + α,β
x(1 − x) y(1 − y) + . n n
Figs. 1–3 show some graphs of Qn (x, y) for n = 5, α = 0 and different illustrative values of β. We have plotted them together with the plane z = 0 to emphasize the regions of positivity. We have also included ‘topographic maps’ of the functions, which are shaded in such a way that regions with higher values are lighter.
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Fig. 1. Graph of Q0,0 5 (x, y).
Fig. 2. Graph of Q0,0.4 (x, y). 5 α,β
The non-negativity of Qn (x, y) is obviously fulfilled at these points (x, y) where simultaneously rn (α, x)(α + 2x) − 2x2 − αx +
x(1 − x) ≥0 n
rn (β, y)(β + 2y) − 2y2 − βy +
y(1 − y) ≥ 0. n
and
Fig. 3. Graph of Q0,10 5 (x, y).
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
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Some calculations with the aid of the software Mathematica© state the validity of these inequalities when and only when (x, y) lies in the subset of S given by the rectangle
√ 1 + n − 2nα + 1 + 2n + n2 + 8n2 α + 4n2 α2 1 + n − 2nβ + 1 + 2n + n2 + 8n2 β + 4n2 β2 0, × 0, . 2(1 + 3n) 2(1 + 3n) As n goes to infinity, the end points of these intervals decrease, respectively, to √ 1 − 2α + 1 + 8α + 4α2 6 and 1 − 2β +
1 + 8β + 4β2 . 6 α,β
As a consequence the order of approximation of Bn f (x, y) towards f (x, y) is at least as good as the order of approximation to f (x, y) given by Bn f (x, y) whenever (x, y) lies in
√ 1 − 2α + 1 + 8α + 4α2 1 − 2β + 1 + 8β + 4β2 0, × 0, . 6 6 Notice that the rectangle above becomes [0, 1/3] × [0, 1/3] when α = 0 and β = 0 and enlarges to [0, 1/2] × [0, 1/2] as α and β go to infinity. α,β The fact that the error bound for Bn f (x, y) − f (x, y) improves the one for Bn f (x, y) − f (x, y) in certain region of S does not provide us with valuable information about the actual errors. It would be desirable to be able to state a result analogous to ([2], Theorem 1), where in the univariate setting sufficient conditions on a function were provided under which the proposed modified Bernstein operators approximated the function better than the classical polynomials. Unfortunately we cannot go so further. However, in the sequel we show how the new approximation process may actually approximate a function f ∈ C(S) better than Bn f (x, y) preserving at the time certain shape properties. Let us assume that a function f ∈ C(S) is axially convex (in particular it is (2, 0)-convex and (0, 2)-convex), then α,β we know that for (x, y) ∈ S, Bn f (x, y) ≥ f (x, y). Besides, Bn f (x, y) converges uniformly in S to Bn f (x, y) as α and β go to infinity (for a proof it suffices to recover (7) and check that rn (γ) converges to 0 as γ tends to infinity), and if in α,β α,β addition we assume that f is (1, 0)-convex and (0, 1)-convex, then the functions α → Bn f (x, y) and β → Bn f (x, y) are increasing as it is derived from the following expressions one may compute easily ∂ n 2 + 2nγ + 4(n − 1)z −2 , rn (γ, z) = ∂γ 4(n − 1) (nα + 1)2 + 4(n − 1)n(γx + x2 ) n−1 n−k−1 α,β ∂Bn f k+1 l ∂rn (α, x) α,β k l pn−1,k,l (x, y) f (x, y) = n , −f , , ∂α ∂α n n n n k=0 l=0
n−1 n−k−1 α,β ∂rn (β, y) α,β k l k l+1 ∂Bn f (x, y) = n , −f , . pn−1,k,l (x, y) f ∂β ∂β n n n n k=0 l=0
α,β
On the other hand, under the assumptions on f (x, y) above, Bn f (x, y) is (2, 0)-convex, (0, 2)-convex, (1, 0)-convex and (0, 1)-convex. α,β Summing up, Bn f (x, y) is convex of orders (1, 0), (0, 1), (2, 0) and (0, 2) (as it is f (x, y)), it converges uniformly in S towards Bn f (x, y) as α, β → ∞, this convergence is monotone from below and finally Bn f ≥ f in S. A question α,β α,β naturally arises: do there exist constants α and β such that Bn f (x, y) ≥ f (x, y)? If that was the case, then Bn f (x, y) would represent a good shape preserving approximation. Finally, as particular cases, we are going to consider two exponential type functions which naturally may appear when modelling a variety of processes. From top to bottom Fig. 4 2 2 represents the functions f (x, y) = ex +y , B5 f (x, y) and B50,0 f (x, y). This function meets all the shape properties above.
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Fig. 4. From top to bottom graphs of f (x, y) = ex
2 +y2
, B5 f (x, y) and B50,0 f (x, y).
D. C´ardenas-Morales, F.J. Mu˜noz-Delgado / Mathematics and Computers in Simulation 77 (2008) 170–178
Fig. 5. Graph of B50,0 f (x, y) − f (x, y) for f (x, y) = ex
2 +y2
Fig. 6. Graph of B50,0 f (x, y) − f (x, y) for f (x, y) = ex
2 +y
177
.
. α,β
Fig. 5 shows the graph of B50,0 f (x, y) − f (x, y), which is positive. Hence for all α, β ∈ [0, ∞), ex +y ≤ B5 f (x, y) ≤ B5 f (x, y). 2 The situation is different for f (x, y) = ex +y , though this function also fulfills the aforementioned properties. Fig. 6 0,0 shows the graph of B5 f (x, y) − f (x, y), which is not positive, and its level sets. In Fig. 7 we have increased the value of β till 100. Now B50,100 f (x, y) − f (x, y) is positive. 2
Fig. 7. Graph of B50,100 f (x, y) − f (x, y) for f (x, y) = ex
2 +y
.
2
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Acknowledgements The authors wish to thank the referees for their helpful suggestions. References [1] J.A. Adell, J. de la Cal, M. San Miguel, On the property of monotonic convergence for multivariate Bernstein-type operators, J. Approx. Theory 80 (1995) 132–137. [2] D. C´ardenas-Morales, F.J. Mu˜noz-Delgado, P. Garrancho, Shape preserving approximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput. 182 (2006) 1615–1622. [3] E. Censor, Quantitative results for positive linear approximation operators, J. Approx. Theory 4 (1971) 442–450. [4] Z. Ditzian, Inverse theorems for multidimensional Bernstein operators, Pac. J. Math. 121 (2) (1986) 293–319. [5] H.H. Gonska, J. Meier, A bibliography on approximation of functions by Bernstein type operators, in: C.K. Chui, L.L. Xchumakder, J.D. Ward (Eds.), Approximation Theory IV, Academic Press, New York, 1983, pp. 739–785. [6] J.P. King, Positive linear operators which preserve x2 , Acta Math. Hungar. 99 (3) (2003) 203–208. [7] T. Sauer, Multivariate Bernstein polynomials and convexity, Comput. Aided Geom. Des. 8 (1991) 465–478.