Some Results for a Class of Generalized

Some Results for a Class of Generalized Polynomials by

Serena Morigi and Marian Neamtu 1)

2)

March 31, 1999

Abstract. A class of generalized polynomials is considered consisting of the null spaces of certain dierential operators with constant coecients. This class strictly contains ordinary polynomials and appropriately scaled trigonometric polynomials. An analog of the classical Bernstein operator is introduced and it is shown that generalized Bernstein polynomials of a continuous function converge to this function. A convergence result is also proved for degree elevation of the generalized polynomials. Moreover, the geometric nature of these functions is discussed and a connection with certain rational parametric curves is established.

1. Introduction A number of recent papers in the area of constructive approximation theory deal with properties of trigonometric polynomials and their application in computeraided geometric design and data tting, see [1,3,11,12,16,20,21,23]. In this paper we consider a larger class of functions which strictly contains ordinary polynomials and certain appropriately scaled trigonometric polynomials, de ned as span f1; sin(2x=n); cos(2x=n); sin(4x=n); cos(4x=n); : : : ; sin(x); cos(x)g ; ; Tn := > span fsin(x=n); cos(x=n); sin(3x=n); cos(3x=n); : : : ; sin(x);ncos(even x)g ; > : n odd; 8 > >
0 and a subsequence fFk` g of fFk g such that kFk` k  ", for all `. By the Ascoli-Arzela Theorem [24], this sequence contains a subsequence fFk`m g converging to a P continuous function nk F , say. Clearly, kF k  " and hence F 6= 0. By Theorem 3.3, nk i=0 Bi converges uniformly to unity as k ! 1, and hence it is uniformly bounded with respect to k. Moreover, the convergence Fk`m ! F is uniform, and hence it is not dicult to show that nk `m X i=0

dki `m Bink`m =

nk `m X i=0

Fk`m (ih=nk`m )Bink`m

!

nk `m X i=0

F (ih=nk`m )Bink`m ;

as m ! 1, where the convergence is uniform. However, by Theorem 3.3 the sum on the right converges uniformly to F , whereas the sum on the left converges pointwisely to zero by assumption (a). Thus we see that necessarily F  0 which is a contradiction. We conclude that Fk ! 0 uniformly and hence maxi=0;:::;nk jdki j ! 0, as k ! 1. nk nk Proposition 4.3. Let P and cnk i be de ned as in (4.1). Then jci ? ci? j  C=k , 1

where C is a constant independent of k. Proof: Using the Taylor expansion of the polar form p = pnk of P , we have

p(x1 + h1; : : : ; xnk + hnk ) ? p(x1 ; : : : ; xnk ) =

nk

@p(x1 ; : : : ; xnk ) h j @xj j =1

X

+ 12

nk

@ 2p(1 ; : : : ; nk ) h h ; j ` @xj @x` j;`=1 X

where j = xj + thj ; j = 1; : : : ; nk, for some t 2 (0; 1). Specializing this identity to the case xj = 0; j = 1; : : : ; nk ? i + 1, xj = h; j = nk ? i + 2; : : : ; nk, and hj = nk?i+1;j h; j = 1; : : : ; nk, and employing (2.5), we obtain nk?i+1

z

}|

{ z

i?1 }|

{

2 nk = @p(0; : : : ; 0; h; : : : ; h) h + 1 @ p(1 ; : : : ; nk ) h2 : cnk ? c i i?1 @xnk?i+1 2 @x2nk?i+1

nk The estimate jcnk i ? ci?1 j  C=k now follows from Lemma 2.1.

9

Proof of Theorem 4.1: Observe rst that it follows from Theorem 3.3 and (4.1) that

nk X nk k nk di Bi := (P (ih=nk) ? cnk i )Bi ! 0; i=0 i=0 nk

X

uniformly as k ! 1. By Proposition 4.3 and the fact that since P is dierentiable it is also Lipschitz continuous, we thus have maxi=0;:::;nk jdki ? dki?1j  C=k, for some C independent of k. Now the assertion follows from Proposition 4.2.

5. Control Curves

Functions in the spaces Pn have a number of surprising geometric properties. To Pn describe these properties we rst introduce some terminology. Let P = i=0 ciBin be a P -polynomial and let fr; sg be a xed basis of P . The function P gives rise to the curve P := fP (x)(r(x); s(x)); x 2 I g, hereafter called the P -curve of P . The points Ci := ci (r(i ); s(i )); i = 0; : : : ; n, will be termed the control points of P. Here the values i are de ned as in (3.1). The points Ci are analogs of the classical control points [8] since by Proposition 3.1, if all Ci lie on a P -curve P associated with a polynomial P 2 P , then the P -curve corresponding to these control points is the curve P itself. To de ne an analog of the familiar notion of a control polygon for P -curves, let qi 2 P ; i = 1; : : : ; n, be the unique functions interpolating the values ci?1; ci at i?1 ; i, respectively, that is (5:1) qi (x) := d(d(x ??i) ) ci?1 + dd((x ?? i?1)) ci; x 2 [i?1; i ]: i?1 i i i?1 The curve Q consisting of the pieces fqi (x)(r(x); s(x)); x 2 [i?1; i ]g, i = 1; : : : ; n, will be called the control curve of P (see Fig. 1). Next, let C := f(r(x); s(x)); x 2 I g. Since under the restriction (3.2) P is a Chebyshev space, the curve C does not pass through the origin. Moreover, every line passing through the origin intersects C at most once. Hence every point v 2 V := f(r(a); s(a))+ (r(b); s(b)); ;  0gnf(0; 0)g can be expressed uniquely as v = c(v)(r(x(v)); s(x(v ))), for some x(v) 2 I and c(v) > 0. Let T : V ! V be the mapping de ned by Tv := (r(x(v))c(;vs)(x(v))) = c2v(v) : Clearly, the points on C are invariant under T . In the special case where r(x) = cos(x), s(x) = sin(x), the mapping T is just the inversion with respect to the unit circle. It is well known that this inversion maps circles passing through the origin onto straight lines. The next lemma asserts that the mapping T has a similar property, and hence it can be viewed as a generalized inversion. 10

Lemma 5.1. Let fr; sg be a basis for P and suppose that t 2 P does not vanish on I = [a; b]. Then the set of points fP(x) := (r(x); s(x))=t(x); x 2 I g is a line segment connecting the points P(a) and P(b). Proof: It will be sucient to prove that P(x) lies on the line segment connecting P(a) and P(b). This is the case if and only if the area of the triangle with vertices P(a); P(b), and P(x) is zero, or if 2

3

r(x)=t(x) r(a)=t(a) r(b)=t(b) 4 det s(x)=t(x) s(a)=t(a) s(b)=t(b) 5 = 0: 1 1 1 This determinant equals 2

3

r(x) r(a) r(b) 1 4 5 t(x)t(a)t(b) det st((xx)) st((aa)) st((bb)) ; which is zero because P is two-dimensional. It will be useful to consider the image of a P -curve P under the mapping T , called an inverse P -curve. That is, assuming that P 2 Pn is positive on I , the inverse P -curve of P is the set T P = f(r(x); s(x))=P (x); x 2 I g. In general, the control curve Q of P is not a curve consisting of segments of straight lines. However, it turns out that the image of Q under T is polygonal. In order to make sure that T Q is well de ned, the functions qi must be positive in the corresponding intervals [i?1; i ]. Hence from now on we shall assume that the coecients c0; : : : ; cn are positive. Note that this also guarantees the positivity of P on I . Proposition 5.2. Let P 2 Pn and let Q be the control curve of the associated P -curve P. Then the curve T Q consists of segments of straight lines connecting the pairs of points TCi?1 ; TCi, i = 1; : : : ; n. Proof: Since Q is determined by the functions qi 2 P , by Lemma 5.1 the curve T Q is assembled from the line segments f(r(x); s(x))=qi (x); x 2 [i?1; i ]g, connecting the points TCi?1 = (r(i?1 ); s(i?1 ))=ci?1 and TCi = (r(i ); s(i ))=ci , i = 1; : : : ; n.

Example. Fig. 1 shows an example of a P -curve together with the associated control and inverse curves, corresponding to n = 3, d(x) = e x ? ex, I = [0; 1], and the coecients c = 0:9; c = 0:6; c = 0:5; c = 0:7. The reference curve C 2

0

1

2

3

is determined by r(x) = b0;1(x) and s(x) = b1;1 (x), see (2.1). Fig. 2 illustrates convergence of degree elevation, addressed in Section 4. The gure displays a 11

TC3 TQ TC2

TP C3 C P

Q C2

TC1

C1 C0

TC0

Fig. 1. A P -curve of degree n = 3 along with its inverse P -curve. 3.5

3.4

3.3

3.2

3.1

3

2.9

2.8

2.7

2.6

2.5 2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Fig. 2. Control polygons of an inverse P -curve of a degree-raised polynomial. sequence of three inverse control polygons (corresponding to the values k = 2; 4; 8), converging to the inverse P -curve of a curve of degree n = 2, de ned on I = [0; 1], with coecients c0 = 0:4; c1 = 0:4; c2 = 1. Here the space P is the same as in Fig. 1, that is P = spanfex; e2x g. To compute the points of the inverse P -curve T P, associated with a function P 2 Pn, we can proceed as follows. Given the coecients c0; : : : ; cn of P and a value x 2 I , we can employ the de Casteljau scheme (2.6) (with x1 = : : : = xn = x) to obtain P (x) = cnn. The corresponding point of the inverse P -curve is therefore given by T P(x) = TCnn := T (cnn(r(x); s(x))) = (r(x); s(x))=cnn . It will be instructive to discuss the geometric nature of this algorithm. Consider the functions qik 2 P 12

de ned as

k?1 ) d(x ? ik??11 ) k?1 d ( x ?  k ? 1 k i qi (x) := k?1 k?1 ci?1 + k?1 k?1 ci ; d(i?1 ? i ) d(i ? i?1 )

where

x 2 [ik??11; ik?1 ];

ik := kx + (n ? in)a + (i ? k)b ; i = k; : : : ; n; k = 0; : : : ; n;

(5:2)

and where the cki are de ned by (2.6). It is not dicult to see from (2.6) that qik (ik ) = cki . Hence by Lemma 5.1, the point TCik lies on the line segment between the points TCik??11 and TCik?1, where Cik := cki (r(ik ); s(ik )). Thus we proved the so-called convex hull property of the inverse P -curves, a well-known property of the classical Bezier curves [8]. Proposition 5.3. The inverse of a P -curve with control points C0; : : : ; Cn satis es the convex hull property i.e., it is contained in the convex hull of the points TC0; : : : ; TCn. The above property may be useful in the design of P -curves since it suggests constructing the inverse P -curves rst, with a geometrically more intuitive control structure (see [2,19] for some examples).

6. Connection with Rational Curves

The results of the previous section are not surprising in view of the following remarkable property of inverse P -curves. By Proposition 3.1, the inverse of a P -curve corresponding to a polynomial P 2 Pn can be expressed in the form

T P(x) = (r(xP)(; xs()x)) =

n (r( ); s( ))B n (x) i i i=0Pn i = n i=0 ci Bi (x)

P

n c TC B n (x) i=0 i i i Pn ; n i=0 ci Bi (x)

P

x 2 I: (6:1) As explained below, this means that inverse P -curves are rational parametric curves. First observe that for every basis polynomial Bin 2 Pn there exists a unique bivariate homogeneous polynomial Hin of degree n such that Bin (x) = Hin (r(x=n); s(x=n)); x 2 I:

(6:2)

This is a consequence of the fact that since r and s are linearly independent, every function in Pn can be written in a unique way as a linear combination of the products rn?i (x=n)si (x=n), i = 0; : : : ; n. The functions Hin are commonly referred to as homogeneous Bernstein polynomials, see e.g., [20]. De ning the curve Cn := f(r(x=n); s(x=n)); x 2 I g, (6.2) implies that every function P 2 Pn can be viewed as the restriction of a bivariate homogeneous polynomial to Cn . This is an extension of the well-known fact that trigonometric functions are restrictions of bivariate 13

homogeneous polynomials to the circle. We also note that the curve Cn allows the following interpretation of the functions b0;n and b1;n (see (2.1)). Consider the points v0 = (r(a=n); s(a=n)), v1 = (r(b=n); s(b=n)), and v = (r(x=n); s(x=n)); x 2 I , which lie on Cn . It follows from (2.4) that

v = b0;n(x)v0 + b1;n (x)v1 : This means that b0;n and b1;n can be viewed as generalized barycentric coordinates of v with respect to v0 and v1 . On account of (6.2) and the homogeneity of Hin, the curve (6.1) can also be written in the form Pn n i=0n ci TCi Hi (;  ) ; P (6:3) c H n(; ) i=0 i i

where (; ) varies over all points from the wedge U := fv0 + v1; ;  0gnf(0; 0)g. Using homogeneity again, the same curve can be expressed in the form (6.3) but where now (; ) 2 U such that  +  = 1. However, it is well known that the restriction of Hin to the line  +  = 1 is an ordinary Bernstein polynomial of degree n, which means that the curve (6.1) is indeed rational. Another consequence of (6.3) is that the curve T Q coincides with the usual control polygon of the rational curve T P [8], a fact that is not obvious from the de nitions of P and Q. We summarize our observations in Proposition 6.1. Let P 2 Pn be a P -polynomial with coecients ci; i = 0; : : : ; n. Then the inverse P -curve T P, associated with P and with a basis fr; sg of P , is a rational parametric curve with control points TCi and weights ci , i = 0; : : : ; n. The properties of P -curves described above generalize similar results obtained in [20] for the trigonometric case. In this special case the resulting rational curves, called focal Bezier curves, have been studied in [6,20,23]. An explicit representation of focal curves in terms of ordinary Bernstein basis polynomials has been given in [2]. Moreover, it has been shown in [20] that the rational curves that are dual to focal Bezier curves have the striking property that their osets are also rational. At present it is not clear whether a similar property carries over to the more general situation discussed in this paper. Proposition 6.1 suggests an alternative to the method described in Section 5 for computing inverse P -curves. Namely, since T P is rational, we can apply the standard de Casteljau algorithm to evaluate the points of T P. A natural question is then whether the two methods lead to the same auxiliary control points TCik = T (cki (r(ik ); s(ik ))) = (r(ik ); s(ik ))=cki , i = k; : : : ; n; k = 1; : : : ; n. This question is of importance if one wants to utilize de Casteljau algorithm to subdivide T P (or the curve P) since the auxiliary points are known to give rise to the control points for the subdivided curves [8]. 14

To settle the above question, consider the rational curve of the form Pn n i =0 wi Di Ai (t) P R(t) := n w An (t) ; i=0 i i

(6:4)

with control points D0 ; : : : ; Dn 2 IR2 and weights w0; : : : ; wn 2 IR, where Ani denote the ordinary Bernstein basis polynomials on interval [0; 1]. For any t 2 [0; 1], the corresponding point R(t) on the curve can be obtained by de Casteljau algorithm [8]:

wik??11 k?1 wik?1 k?1 k Di := (1 ? t) wk Di?1 + t wk Di ; i i

i = k; : : : ; n; k = 1; : : : ; n;

where the weights are given as

wik := (1 ? t)wik??11 + twik?1 ;

(6:5)

with wi0 := wi ; i = 0; : : : ; n. To formulate our next result, we set Di := TCi, wi := ci; i = 0; : : : ; n, and t := b1;n(x)=(b0;n (x) + b1;n(x)) in (6.4). Evidently, R(t) = T P(x), hence (6.1) and (6.4) are dierent representations of the same parametric curve. We next show that the de Casteljau algorithm commutes with the generalized inversion, a fact that can also be derived, at least in the case of trigonometric P -curves, from some observations in [23]. Proposition 6.2. The de Casteljau algorithm commutes with the inversion T in the sense that Dik = TCik; i = k; : : : ; n; k = 1; : : : ; n: Proof: Clearly, (2.6) combined with (6.5) implies wik = (b0;n (x) + b1;n(x))k cki , i = k; : : : ; n; k = 1; : : : ; n. The proof of the proposition will be done by induction on k. The assertion is trivial for k = 0. As for the induction step, we obtain

wik??11 k?1 wik?1 k?1 k Di = (1 ? t) wk Di?1 + t wk Di i i k?1 cki??11 k?1 c = b0;n (x) k TCi?1 + b1;n i k TCik?1 ci ci k ? 1 k ? 1 b0;n (x)(r(i?1 (x)); s(i?1 (x))) + b1;n(x)(r(ik?1 (x)); s(ik?1 (x)) = cki k (x)); s( k (x))) ( r (  i i = TCik : = k ci

The second to last equality follows from

b0;n(x)p(ik??11 (x)) + b1;n(x)p(ik?1 (x)) = p(ik (x)); 15

(6:6)

which holds for every p 2 P . To see this, assume rst that  6=  (cf. (1.2)). Since P is two-dimensional, it will be sucient to prove (6.6) for p(x) = ex , where  2 f;  g. It follows from (5.2) and the de nition of b0;n(x), b1;n(x) that both sides of (6.6) belong to the same two-dimensional space spanned by e(+(k?1))x=n and e(+(k?1))x=n. Thus identity (6.6) will be established once we have shown that it holds for at least two values of x, namely the values x 2 fa; bg. However, for these values, the validity of (6.6) is a consequence of

b0;n(a) = 1; b0;n(b) = 0; b1;n(a) = 0; b1;n(b) = 1; and

ik??11 (a) = (n ? i + k)na + (i ? k)b = ik (a); ik?1 (b) = (n ? in)a + ib = ik (b): The proof of (6.6) for the case  =  is analogous. Given the result of the last proposition, it comes as a surprise that inversion need not commute with degree elevation, as the next example shows. Example. Let [a; b] = [0; =2] and P = spanfsin; cosg. Consider the function P (x) = c0B01 (x) + c1B11 (x)p= c0 cos x + c1 sin x; c0; c1 2 IR. Raising the degree of P gives P (x) = c0B02(x) + 22 (c0 + c1)B12 (x) + c1B22(x), where B02(x) = 2 sin2 ( 4 ? x ); B 2 (x) = 4 sin(  ? x ) sin( x ); B 2 (x) = 2 sin2 ( x ). Thus the corresponding inverse 1 2 2 4 2 2 2 p P -curve T P is a rational curve with weightspw0 p= c0; w1 = 22 (c0 + c1); w2 = c1 and control points TC0 = (1; 0)=w0; TC1 = ( 22 ; 22 )=w1 ; TC2 = (0; 1)=w2 . On the other hand, degree elevating the rational curve (see [8])

c0 (1c;00) A10(t) + c1 (0c;11) A11(t) c0A10(t) + c1A11 (t) gives a rational curve with weights w00 = c0; w10 = (c0 + c1)=2; w20 = c1 and control ;1) =w0 ; D = (0; 1)=w0 . Thus, degree elevation points D0 = (1; 0)=w00 ; D1 = (1;0)+(0 2 1 2 2 does not commute with inversion. Of course, the two quadratic rational curves, while having a dierent analytic form, represent the same curve in the plane. For a more detailed discussion of degree elevation in the trigonometric case, we refer the reader to [19].

Acknowledgments. We thank both anonymous referees of this paper for a very

careful reading of the manuscript and for suggesting many improvements in the presentation. 16

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