A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This pape is concened with a genealization of the classical Benstein polynomials whee the function is evaluated at intevals which ae in geometic pogession. It is shown that, when the function is convex, the genealized Benstein polynomials ae monotonic, as in the classical case. 1991 Mathematics subject classification: 41A10. 1. Intoduction Recently the second autho poposed (see [7]) the following genealization of the Benstein polynomials, based on the q-integes. Fo each positive intege n, we define n [ n 1 n B n (f; x) = f ]x (1 q s x), (1.1) =0 whee an empty poduct denotes 1 and f = f([]/[n]). The notation equies some explanation. The function f is evaluated at atios of the q-integes [] and [n], whee q is a positive eal numbe and (1 q )/(1 q), q 1, [] =, q = 1. (1.2) Suppoted by Dokuz Eylül Univesity, Izmi, Tukey
2 Halil Ouç and Geoge M Phillips / Benstein polynomials Then, in a natual way, we define the q-factoial []! by [].[ 1]...[1], = 1, 2,..., []! = 1, = 0 and the q-binomial coefficient by [n]! = []![n ]! fo integes n 0. The Pascal identities [ ] [ ] [ ] n n 1 n 1 = q n + 1 (1.3) (1.4) (1.5) and = [ ] [ ] n 1 n 1 + q 1 ae eadily veified fom (1.4). The q-binomial coefficients ae also called Gaussian polynomials (see Andews [1]) and an induction agument using eithe (1.5) o (1.6) eadily shows that is a polynomial of degee (n ) in q with positive integal coefficients. When q = 1, the q-binomial coefficient educes to the odinay binomial coefficient and (1.1) gives the classical Benstein polynomial. (See, fo example, Cheney [2], Davis [3], Rivlin [10].) It is clea fom (1.1) that, as in the case when q = 1, the genealized Benstein polynomial intepolates the function f at x = 0 and 1 and that, fo 0 < q 1, B n is a monotone linea opeato. The genealized Benstein polynomial defined by (1.1) can be expessed in tems of q-diffeences. Fo any function f we define fo i = 0, 1,...n and, ecusively, 0 f i = f i (1.6) k+1 f i = k f i+1 q k k f i (1.7) fo k = 0, 1,..., n i 1, whee f i denotes f([i]/[n]). See Schoenbeg [11], Lee and Phillips [6]. When q = 1, these q-diffeences educe to odinay fowad diffeences and it is easily established by induction that k f i = k [ ] k ( 1) q ( 1)/2 f i+k. (1.8) =0
Poceedings of Edinbugh Mathematical Society 42 (1999) 403-413 3 Then we may wite, as shown in Phillips [7], B n (f; x) = n =0 f 0 x (1.9) which genealizes the well known esult (see, fo example, Davis [3]) fo the classical Benstein polynomial. We may deduce fom (1.9), as in Phillips [7], that if f P k, the linea space of polynomials of degee at most k, then B n (f; x) P k. In paticula, if f P 1 its second and highe q-diffeences ae zeo and we may deduce fom (1.9) that, fo any eal numbes a and b, Fo what follows, we also equie the Eule identity B n (ax + b; x) = ax + b. (1.10) (1 + x)(1 + qx)...(1 + q k 1 x) = k [ ] k q ( 1)/2 x. (1.11) We obseve that this identity, which may be veified by induction, genealizes the binomial expansion. In [7] thee is a discussion on convegence and a Voonovskaya type theoem on the ate of convegence. =0 Results concening the convegence of deivatives of the genealized Benstein polynomials ae given in [8]. The following de Casteljau type algoithm (see [9]) may be used fo evaluating genealized Benstein polynomials iteatively. ALGORITHM fo = 0 to n f [0] next := f([]/[n]) fo m := 1 to n fo := 0 to n m f [m] next next m := (q q m 1 x)f [m 1] + xf [m 1] +1 It is shown in [9] that f [n] 0 = B n (f; x). This genealizes the well known de Casteljau algoithm (see [5]) fo evaluating the classical Benstein polynomials.
4 Halil Ouç and Geoge M Phillips / Benstein polynomials 2. Non-negative diffeences In Davis [3] it is shown that, fo any convex function f, the classical Benstein polynomial (that is, (1.1) with q = 1) is also convex and the sequence of Benstein polynomials is monotonic deceasing. It is also shown in [3] that if the kth odinay diffeences of f ae non-negative then the kth deivative of the classical Benstein polynomial B n (f; x) is non-negative on [0,1]. We will discuss extensions of these esults to the genealized Benstein polynomials in this and the following section. We begin by ecalling the following definition. Definition 2.1 A function f is said to be convex on [0,1] if, fo any t 0, t 1 such that 0 t 0 < t 1 1 and any λ, 0 < λ < 1, f(λt 0 + (1 λ)t 1 ) λf(t 0 ) + (1 λ)f(t 1 ). (2.1) Geometically, this definition states that no chod of f lies below the gaph of f. With λ = q/(1 + q), t 0 = [m]/[n] and t 1 = [m + 2]/[n] in (2.1), whee 0 < q 1, we see that, if f is convex, fom which we deduce that f m+1 q 1 + q f m + 1 1 + q f m+2 f m+2 (1 + q)f m+1 + qf m = 2 f m 0. Thus the second q-diffeences of a convex function ae non-negative, genealizing the well known esult fo odinay diffeences (whee q = 1). Fo any fixed natual numbe k we now constuct a set of n k +1 piecewise polynomials whose kth q-diffeences take the value 1 at a given knot, say ([m]/[n]), and the value 0 at all the othe knots. Fo 0 m n k define g k,m 0 0 x [m + k 1]/[n], (x) = γ k,m (x), [m + k 1]/[n] < x 1, whee γ k,m (x) = m+k 1 =m+1 (2.2) ( ) [n]x []. (2.3) [2 m] [] When k = 1 in (2.3) the empty poduct denotes 1 and then (2.2) is the piecewise constant function g 1,m 0, 0 x [m]/[n], (x) = (2.4) 1, [m]/[n] < x 1,
Poceedings of Edinbugh Mathematical Society 42 (1999) 403-413 5 fo 0 m n 1. Fo a geneal value of k, the values of these piecewise polynomials at the knots ae given by g k,m j = g k,m ([j]/[n]) = 0, 0 j m + k 1, ] (2.5), m + k j n. [ j m 1 k 1 Since the polynomial pat of g k,m (x) is of degee k 1, the kth q-diffeences involving knots fom that pat of the domain ae zeo. Fom this and (1.8), and noting whee g k,m (x) is zeo, we see that k g k,m j = 1, j = m, 0, j m, 0 j n k. We can use the functions g k,m (x), 0 m n k, and the monomials 1, x,..., x k 1 as a basis fo the space of functions whose kth q-diffeences ae non-negative on the knots ([j]/[n]), 0 j n. Let p k 1 P k 1 denote the polynomial which intepolates f on the fist k of these knots, ([j]/[n]), 0 j k 1, and let us wite f(x) = p k 1 (x) + n k m=0 (2.6) k f m g k,m (x). (2.7) This is a piecewise polynomial of degee k 1 with espect to the knots. [0, [k 1]/[n]], all of the n k + 1 functions g k,m (x) ae zeo and thus On the inteval f([j]/[n]) = p k 1 ([j]/[n]) = f([j]/[n]), 0 j k 1, (2.8) so that Also, we deduce fom (2.7) and (2.6) that f0 = f 0, 0 k 1. (2.9) k fm = k f m, 0 m n k, and so Combining (2.9) and (2.10), we deduce that f0 = f 0, k n. (2.10) f([j]/[n]) = f([j]/[n]), 0 j n. (2.11)
6 Halil Ouç and Geoge M Phillips / Benstein polynomials Thus the function f, a piecewise polynomial of degee k 1, takes the same values as f on all n + 1 knots. When k = 1, f is a step function which intepolates f on all n + 1 knots and, when k = 2, the function f is the linea spline which intepolates f. Fo a geneal value of k, we deduce that B n ( f; x) = B n (f; x) (2.12) and thus, fom (2.7) and the lineaity of the Benstein opeato B n, say, whee B n (f; x) = B n (p k 1 ; x) + n k m=0 k f m C k,m (x) (2.13) C k,m (x) = B n (g k,m ; x). (2.14) We now state: Theoem 2.1 The kth deivatives of the genealized Benstein polynomials of ode n ae nonnegative on [0, 1] fo all functions f whose kth q-diffeences ae non-negative if and only if the kth deivatives of the genealized Benstein polynomials of the n k+1 functions g k,m (x), 0 m n k, ae all non-negative. Poof This follows fom (2.13) and (2.14). We will find it useful to deive an altenative expession fo the kth deivative of B n (g k,m ; x). We begin by expessing highe ode q-diffeences (of ode not less than k) in tems of the kth q-diffeences. Fo 0 s n k, we may wite s [ ] s s+k f i = ( 1) t q t(t+2k 1)/2 k f s+i t. (2.15) t t=0 This is easily veified by induction on s, using the ecuence elation fo q-diffeences (1.7) and the Pascal identities. We now wite the q-diffeence fom of the genealized Benstein polynomial (1.9) as k 1 B n (f; x) = f 0 x + =0 n k s+k f 0 x s+k. s + k Using (2.15) to eplace the highe ode diffeences in the second summation and eaanging the esulting double summation, we obtain k 1 [ n k n B n (f; x) = ] f 0 x k f m D k,m (x) (2.16) =0 m=0
Poceedings of Edinbugh Mathematical Society 42 (1999) 403-413 7 say, whee D k,m (x) = n m k t=0 [ ][ ] ( 1) t q t(t+2k 1)/2 n m + t x m+t+k. (2.17) m + t + k t On compaing (2.13) and (2.16), which hold fo all functions f, we deduce that d k dx k C k,m(x) = dk dx k D k,m(x). (2.18) Thus, given that we ae inteested only in thei kth deivatives, the sets of polynomials C k,m and D k,m ae equivalent. It is well known (see Davis [3]) that, with q = 1, the kth deivatives of D k,m ae non-negative. This is easily veified fom (2.17) since with q = 1 we have d k dx k D k,m(x) = so that, mindful of (2.18), n m k n! m!(n m k)! xm t=0 ( 1) t ( n m k t ) x t, d k dx k D k,m(x) = dk dx k C n! k,m(x) m!(n m k)! xm (1 x) n m k 0 fo 0 x 1. Fom (2.17) we can also see that, as q tends to zeo fom above, each q-intege tends to 1 and we have the limiting fom D k,m (x) = x m+k and so its kth deivative is non-negative. We conjectue that the kth deivative of each D k,m is non-negative fo 0 < q < 1, but have not found a poof, except fo cetain values of m which we will mention below. We will now wok with C k,m athe that D k,m. Fom (2.14), (1.1) and (2.5) we have C k,m (x) = n =m+k fo 0 m n k. With m = n k, we have [ ][ n m 1 k 1 C k,n k (x) = x n, ]x n 1 (1 q s x), (2.19) whose kth deivative is clealy non-negative on [0, 1]. With m = n k 1, we obtain fom (2.19) that C k,n k 1 (x) = [n]x (1 x) + [k]x n
8 Halil Ouç and Geoge M Phillips / Benstein polynomials and, with a little wok, we find that the kth deivative of the latte polynomial is also non-negative on [0, 1]. We can expess C k,m (x) in anothe way, as follows. Since B n is a linea opeato, we may wite C k,m (x) = B n (g k,m ; x) = B n (γ k,m ; x) + B n (g k,m γ k,m ; x), (2.20) whee γ k,m is defined in (2.3). Let B n (γ k,m ; x) = p k,m (x) say, whee p k,m (x) P k 1. Then we obtain fom (2.20) that C k,m (x) = p k,m (x) + q (2m+k)(k 1)/2 ( 1) k S k,m (2.21) say, whee S k,m = In paticula, (2.21) gives m =0 q (k 1) [ m + k 1 k 1 ][ n ]x C k,0 (x) = p k,0 (x) + q k(k 1)/2 ( 1) k (1 q s x). (1 q s x). (2.22) Since, fo 0 < q < 1, the zeos of the function ( 1) k (1 qs x) ae all geate than unity, the epeated application of Rolle s theoem shows that this is tue of each of its fist n deivatives. Also, Eule s identity (1.11) shows that its kth deivative is positive at x = 0 and so is positive on [0, 1]. Since p k,0 (x) P k 1 it follows that kth deivative of C k,0 is also positive on [0, 1]. 3. Monotonicity fo convex functions It is well known (see Davis [3]) that, when the function f is convex on [0, 1], its Benstein polynomials ae monotonic deceasing, in the sense that B (f; x) B n (f; x), n = 2, 3,..., 0 x 1. We now show that this esult extends to the genealized Benstein polynomials, fo 0 < q 1. In Figue 1, which illustates this monotonicity, the function is concave athe than convex and thus the Benstein polynomials ae monotonic inceasing. Figue 1 hee is modelled on Fig. 6.3.1
Poceedings of Edinbugh Mathematical Society 42 (1999) 403-413 9 in Davis [3], which elates to the classical Benstein polynomials. The function is the linea spline which joins up the points (0,0), (0.2,0.6), (0.6,0.8), (0.9,0.7) and (1,0) and the Benstein polynomials ae those of degees 2, 4 and 10, with q = 0.8 in place of q = 1 in [3]. 1 f y B 4 B 10 B 2 x 0 1 Figue 1: Monotonicity of genealized Benstein polynomials fo a concave function. The polynomials ae B 2, B 4 and B 10, with q = 0.8 Theoem 3.1 Let f be convex on [0, 1]. Then, fo 0 < q 1, B (f; x) B n (f; x) fo 0 x 1 and all n 2. If f C[0, 1] the inequality holds stictly fo 0 < x < 1 unless f is linea in each of the intevals between consecutive knots []/[n 1], 0 n 1, in which case we have the equality B (f; x) = B n (f; x). Poof The key to the poof in Davis [3] fo the case q = 1 is to expess the diffeence between the consecutive Benstein polynomials in tems of powes of x/(1 x). Since the genealized Benstein polynomials involve the poduct n 1 (1 q s x) athe than (1 x) n we need to modify the poof somewhat. Fo 0 < q < 1 we begin by witing (1 q s x) 1 (B (f; x) B n (f; x)) = f =0 ( [] [n 1] ) [ n 1 ]x s=n 1 (1 q s x) 1
10 Halil Ouç and Geoge M Phillips / Benstein polynomials n f =0 ( [] [n] ) [ n ]x (1 q s x) 1. s=n We now split the fist of the above summations into two, witing x s=n 1 (1 q s x) 1 = ψ (x) + q n 1 ψ +1 (x), whee ψ (x) = x s=n The esulting thee summations may be combined to give (1 q s x) 1. (3.1) (1 q s x) 1 (B (f; x) B n (f; x)) = a ψ (x), (3.2) =1 say, whee a = ( ) [n ] [] n [] f + q [n] [n 1] [n] f ( ) [ 1] f [n 1] ( ) []. (3.3) [n] Fom (3.1) it is clea that each ψ (x) is non-negative on [0, 1] fo 0 q 1 and thus, in view of (3.2), it suffices to show that each a is non-negative. We etun to (2.1) and put t 0 = [ 1]/[], t 1 = []/[n 1] and λ = q n []/[n]. Then 0 t 0 < t 1 1 and 0 < λ < 1 fo 1 n 1 and, compaing (2.1) and (3.3), we deduce that, fo 1 n 1, a = λf(t 0 ) + (1 λ)f(t 1 ) f(λt 0 + (1 λ)t 1 ) 0. Thus B (f; x) B n (f; x). Of couse we have equality fo x = 0 and x = 1 since all Benstein polynomials intepolate f on these end-points. The inequality will be stict fo 0 < x < 1 unless each a = 0 which can only occu when f is linea in each of the intevals between consecutive knots []/[], 0, when we have B (f; x) = B n (f; x) fo 0 x 1. This completes the poof. Fo a convex function, Goodman, Ouç and Phillips [4] show that the genealized Benstein polynomials ae also monotonic in the paamete q, fo 0 < q 1.
Poceedings of Edinbugh Mathematical Society 42 (1999) 403-413 11 REFERENCES [1] G.E. Andews, The Theoy of Patitions, Addison-Wesley, Reading, Mass., 1976. [2] E.W. Cheney, Intoduction to Appoximation Theoy, McGaw-Hill, New Yok, 1966. [3] P.J. Davis, Intepolation and Appoximation, Dove, New Yok, 1976. [4] T.N.T. Goodman, H. Ouç and G.M. Phillips, Convexity and genealized Benstein polynomials, Poc. Edin. Math. Soc. (submitted). [5] J. Hoschek and D. Lasse, Fundamentals of Compute-Aided Geometic Design, A.K. Petes, 1993. [6] S.L. Lee and G.M. Phillips, Polynomial intepolation at points of a geometic mesh on a tiangle, Poc. Roy. Soc. Edin., 108A (1988) 75-87. [7] G.M. Phillips, Benstein polynomials based on the q-integes, The heitage of P. L. Chebyshev: a Festschift in hono of the 70th bithday of T.J. Rivlin. Ann. Nume. Math. 4 (1997) 511-518. [8] G.M. Phillips, On genealized Benstein polynomials, Numeical Analysis: A.R. Mitchell 75th Bithday Volume, D.F. Giffiths and G.A. Watson (eds.), Wold Scientific, Singapoe, 1996, 263-269. [9] G.M. Phillips, A de Casteljau algoithm fo genealized Benstein polynomials, BIT 36:1 (1996) 232-236. [10] T.J. Rivlin, An Intoduction to the Appoximation of Functions, Dove, New Yok, 1981. [11] I.J. Schoenbeg, On polynomial intepolation at the points of a geometic pogession, Poc. Roy. Soc. Edin., 90A (1981) 195-207.