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On bivariate osculatory interpolation
Journal of Computational North-Holland
and Applied
Mathematics
271
38 (1991) 271-282
On bivariate osculatory interpolation Xue-Zhang Department
Liang
of Mathematics,
Luo-Qing
Jilin University, Changchun,
130023, China
Li
Department
of Mathematics,
Hubei University,
Wuhan, China
Received 25 June 1990 Revised 31 May 1991
Abstract Liang, X.-Z. and L.-Q. Li, On bivariate Mathematics 38 (1991) 271-282.
osculatory
interpolation,
Journal
of Computational
and Applied
In this paper the straight-line-superposed interpolation which was first proposed by Liang (1965) for the bivariate Lagrange interpolation problem is generalized to set up a new bivariate interpolation method, the order-raising process. The new method expands and develops the bivariate osculatory interpolation schemes proposed by Le Mehaute (1981) and Hakopian (1984). Keywork
Bivariate
interpolation,
order-raising
process.
1. Introduction Let n be a nonnegative integer, s, = i( n + l)( n + 2) - 1. II, bivariate polynomials of the form p(x,
y) =
c
denotes the space of all real
UjjX’r’.
0 J>
O
is the two-dimensional Euclidean space.. Let Q;= (x;,
v;),
i=o,
l)...)
S,,
0.2)
be S, + 1 distinct points in lR* and consider the following bivariate problem. Given a set of real numbers {f”‘ER
(i=O,
l,...,
s,},
Lagrange interpolation 0.3)
to find a polynomial p( x, y) E II,, satisfying p(Qi)
=f(‘),
i=O,
l,...,
s,.
We call N=
{ Qi 10 < i G s, } a unisolvent set of interpolating unique solution of problem (1.3), (1.4).
0377-0427/91/$03.50
0 1991 - Elsevier
Science Publishers
(1.4) nodes for II,, if there exists a
B.V. All rights reserved
272
X.-Z. Liang L. -Q. Li / Bivariate osculatory interpolation
In [6] one of the authors interpolation.
gave the following
two theorems
on the straight-line-superposed
Theorem 1. If Mn = { Qi ]0 < i < s, } is unisolvent for IT,,, and if none of these points is on the irreducible curve of degree k: 1(x, y) = 0 (either k = 1 or k = 2; k = 1 means a straight line; k = 2 means a conic), then Nti with the (n + 3) k - 1 points being distinct and selected freely in the irreducible curve must constitute a unisolvent set of nodes for 17n+k. Straight-line-superposition process (SLSP). Step 0. In the plane Oxy freely draw a straight line L,: 10(x, y) = a,x + b, y + c,-,= 0, and freely choose a point Q, on the straight line. Set X0 = { Q, }. Step 1. In the plane Oxy draw a straight line L,: lI(x, y) = a,x + b, y + c1 = 0 such that L, n No = 8. On the straight line choose two arbitrary but mutually distinct points Q,, Q2. Set = IQ,, Q1, QzlThe rest may be deduced by analogy. We only give the last step. Step n. In the plane Oxy draw a straight line L,. * a,x + b, y + c, = 0 such that L, nJlm_* =fl. On L, choose n + 1 arbitrary but mutually distinct points Qo,2jn(n+lj,. . . , Qo,ajn(n+Jj. Set
4
JVIn={Qci> Q,,...,Q,,,,,,,,+,,>. Theorem 2. The set of points M4, given by the SLSP is a unisolvent set of interpolating IT,. The interpolating polynomial p,,( x, y ) has the following form: i-l n Pn(xy Y) = ,~~q,-i(b,-ix-a~-iy),~~l”-j(x,
nodes for
(1.5)
Y),
whereq,(t), q,-l(t), . . . , q,,(t) (being the univariate polynomials tively) can be determined successively.
by the interpolation
conditions
of degree n, n - 1,. . . ,O, respecon the straight lines L,, L,_l,. . . , L,,
In [2] Chung and Yao made a systematic study on multivariate Lagrange interpolation and defined the natural lattices and the GC condition. In [3] Gasca and Maetzu studied both Lagrange interpolation and Hermite interpolation in two variables and proposed a new scheme of bivariate osculatory interpolation. So far as the Lagrange interpolation for II, is concerned, the unisolvent sets of interpolating nodes given by the scheme are always constructed by the above SLSP, and the sets may not obey the GC condition. Recently Micchelli [9] and Cheney [l] have pointed out that the above-mentioned straightline-superposed interpolation is fully effective. The purpose of this paper is to make a generalization of the method and solve a more general osculatory interpolation problem. In the paper we will give a new bivariate osculatory interpolation method, the order-raising process. The method is quite different from the scheme in [3]. 2. Bivariate osculatory interpolation problem First of all we introduce
some notations.
Tk= {(TV,... ,T~)[T,EIW*, T=
ij k=O
Tk.
Let T, = 0 be the empty
I17iII =l,
i=1,2
,...,
k},
set,
k=l,2,3
,...,
(2-l)
213
X.-Z. Liang, L.-Q. Li / Bioariate osculatory interpolation
rk) E T, we define the directional derivative of k th order
For a==(~~,...,
D, = D,,D,.
. . QA = q
aa
a ..*G .
K
(2-2)
In particular, we define D+ = I,
the identity operator.
(2-3)
For a nonnegative integer k, we set gk=
{D,I%ET&
(2 -4)
For a point Q E IR2 and an operator D, E SSk, we define an interpolating functional D,(Q) setting D, (Q)(&
Y)) = D&(x,
Y) I cx,rj=P,
VfE
Cm(R2>-
by
(2.5)
In addition, we use the notation Da(Da(Q)>
=
D,,,(Q>,
vz,
(2.6)
BE T.
Now we describe the general bivariate osculatory interpolation problem. Suppose that n and s are two given nonnegative integers. Let Qj=(xi,
y,),
i=O,l,...,
s,
(2.7)
be s + 1 distinct points in R2, and m,, m,, . . . , m, natural numbers satisfying
c m; = +I + l)(n + 2). I=0
(2.8)
Suppose that A?‘~,Ar,...,
s?--c T
(2.9)
are finite subsets of T such that Card(dj)
=m,,
i=O, l,...,
S.
(2.10)
We consider the following osculatory interpolation problem. Given a set of real numbers {f$‘ER
l%E.&;,
i=O, l,...,
to find a polynomial p( x, y) E II, D,(Qi)p=f$‘, The polynomial p(x, 0=
‘BE&;,
s},
(2.11)
such that i=O,l,...,
S.
y) satisfying (2.12) is called the interpolating
{D,(Q,)~xEA?~,
i=O, l,...,
S}
(2.12) polynomial. The set (2.13)
is called the set of interpolating functionals. We write A--=
{Q,, Q,,---, Q,>
(2.14)
the set of interpolating nodes, and
~~={D~(Qi)l~~4}
(2.15)
X.-Z. Liang L.-Q. Li / Bivariate osculatory interpolation
274
the set of interpolating functionals at node Qi. If there exists a unique polynomial p(x, y) E IIT, satisfying (2.12), then the interpolation problem (2.11), (2.12) is called a unisolvent problem and 0 is called a unisolvent set of interpolating functionals for II,, (or say “0 is unisolvent for II,,” for short).
3. Order-raising osculatory interpolation We first introduce some concepts. Suppose that k and m are nonnegative integers, k > m. L denotes a straight, line in the plane Oxy, and r a unit tangent vector of it. Choose m + 1 mutually distinct PO, Pi,. . . , P,,, on L and m + 1 natural numbers (Y,,, (~i,. . . , a, satisfying &,=k+l. i=o For a nonnegative setting
(3-l) integer (Yand a point
P E R*, we define an interpolating
functional
D:(P)
by
(3 4 Then we call the set of interpolating 9=
{D,“(P,))O
functionals i=O,
l,...,
a set of interpolating functionals for one-dimensional straight line L. At the same time, we call JV= {PO, PI,..., the set of interpolating gi=
(3.3)
m} Hermite
interpolation
of degree
PJ
(3.4)
nodes corresponding
to 9
and
{D,*(P;)IO
(3.5)
the limitation of 9 at the point Pi. We call the natural P, on the straight line L, and write
number
(Y;the osculating
degree of 9
(Y~= Odeg( L, Pi). If P E L\JV,
k on the
at
(3.6)
then we make a supplemental
definition
Odeg( L, P) = 0;
(3.7)
Now we generalize the straight-line-superposition process and give a more general constructing the unisolvent set of interpolating functionals for the bivariate osculatory tion.
process of interpola-
Order-raising process. Step 0. In the plane Oxy freely draw a straight line Lo: l,(x, y) = a,x + boy + c,, = 0, and on the line freely choose a node Poe. Let r,, be a unit tangent vector of L,. With regard to PO0 we freely choose two linearly independent unit vectors uoo, poo E R*, which are called the first and the second order-raising directions at the node P,, respectively. We set m,=O,
2?. =.A0 =g,
4=~0=
{PO&
X.-Z. Liang, L.-Q. Li / Biuariate osculatory interpolation
275
and YO=~~o)=.@($o)(Poo)
= {1(P,)}.
1. It includes five successive stages. (i) In the plane Oxy freely draw a straight line L,. . Z,(x, y) = a,x + b,y + c1 = 0 such that L, n .A!,= 0. Let r1 E R * be a unit tangent vector of L,. (ii) If PO0E L,, then we raise the orders of all directional derivatives at the node. Namely, we define a new set of interpolating functionals at the node PO,:
Step
if PO0G L,,
kV(Poo)~ gf)
D~~.G?~)( PO,),
( p,, ) =
I Q7$%o) ( PO0>9
if PO0E L,
and if a, 117,)
if PO0E L,
and if oooH TV.
Furthermore, we set go(‘) = m; ‘Qq’( PO,). J=o
(iii) On the straight line L, freely choose a set of interpolating sional Hermite interpolation of degree 1,
functionals
for one-dimen-
3$(i) = (D;( Plj) 10 < cx < 01:” - 1, 0
O
The corresponding set of interpolating nodes is 9, = {P1,0,...7
Pi,,,},
where PI,o,. . . , P,,, are mutually distinct points on L,. With regard to each node P, j E 9, we define a pair of order-raising directions ui j, pl, E R* being linearly independent unit vectors, j=O,l ,‘..> m,. If plj E No, namely PI, = P,, then we define u1j = %J 9 Plj= POO(iv) Set ~,“‘(P~~)~(D~(P~~)~O~~~~~~~~l), Jv;=
;.9;,
y; = ;
For PI j E No k(l,
q(i).
i=O
i=O (v)
j=O,...,m,,
we define
j)=max{tIP,,EL,,O
Set
Odeg(Lk(1.j)) Plj)
pi”‘= 0 i
7
9
if
Plj E No >
if P,,GNo,
j=O, 1,.
--3
ml,
X.-Z. Liang L.-Q. Li / &variate osculatory interpolation
276
and
The rest may be deduced by analogy. We only give the last step. Step n. It includes five successive stages. (i) In the plane Oxy draw a straight line L,: In(x, y) = a,x + b,,y -t c, = 0 such that L, n A?,_,= 0. Let 7n E lR2 be a unit tangent vector of L,. (ii) If Pij E L,, then we raise the orders of all directional derivatives at the node. Namely, we define a new set of interpolating functionals at the point Pii:
.~$~)(p~~) =
I l)...)
j=o,
Furthermore,
53T;“-1)( Pii),
if Pij 4 L, ,
D,,543/“-“(P,,),
if Pij E L, and
aij 11T,,
D$$(-‘)(
if Pij E L, and
uij +t r,, ,
Pii),
i=o,
mi,
1)“‘)
n-l.
we set
F(n) = 5 JJ$“‘(Pij), I
i=o,
l)...)
n-l.
j=O
(iii) On the straight line L, freely choose sional Hermite interpolation of degree n, 3(“)
a set of interpolating
functionals
for one-dimen-
= (D;( Pnj) 10 < a < aj.“’ - 1, 0
where a;.") = Odeg( L, , P,,j),
O
2
ol(i”)=n+l.
j=O
The corresponding pn= where define j=O,l which (iv)
set of interpolating
nodes is
{P,,OY..94,,“}~
P,, o,. . . , P,,,, are mutually distinct points on L,. With regard to each node Pnj E 9,, we a ‘pair of order-raising directions anj, pnj E lR2 being linearly independent unit vectors, ,***, m,. If P, j E Nn_l, then we require that the new pair (a,, j, p, j) is equal to the old one has been defined with regard to the node already. Set ~~n)(P~j)=(D~(P,j)lOda~olj”)-l),
and n
n
Nn= Upi i=O
9” =
u 9y”). i=O
j=O,...,m,,
277
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
(v) For P,,, E Jr/-,_, k(n,
we define
j)=max{t]Odt
PnjEL,}.
Set p(n)
Odeg(L,(,,,j), P,,), if Pn,E-4’--lv j=o,
=
J
if Pni EJ+_,,
0,
l)...)
m,,
and A,
= ( P, j E 9n
1 a;“’
>
/y’
+
1>)
2?,=
;A!;. i=O
Our main result in this paper is as follows. Theorem 3. The set $!! obtained by the above process is a unisolvent set of interpolating functionals for II,,. The osculatory interpolating polynomial p,,( x, y) has the following form: Pn(x,
Y) = 2
q~-i(b~-ix-a~-iy)~~~l.-J(x,
(3.8)
Y>,
i=O where
q,,(t),
q,-l(tL..
of degree n, n - 1, ___,O, respecon the straight lines L,, L,-,, . . . , L,
. , qO(t) (being the univariate polynomials
tively) can be determined successively.
by the interpolation
conditions
Remark 4. In the above process, if the osculatory degrees at all nodes are equal to 1, and if every new straight line does not pass through any one of the nodes which have been chosen already, then the new process is simplified to be the original straight-line-superposition process. Remark 5. The above process can be generalized to the Euclidean space R”, s > 2, to be a hyperplane-superposition process. In this situation the order-raising directions at every node are s linearly independent unit vectors. Remark 6. If every straight line in the above process coincides with a side of some fixed triangle or passes through some fixed point in the triangle, then the process leads to the osculatory interpolation schemes described in [5]. Remark 7. Obviously, the osculatory interpolation schemes proposed in [4] can be obtained by using the order-raising process or the hyperplane-superposition process.
4. Proof of Theorem 3 In order to prove Theorem 3, we first set up a few lemmas. Lemma 8. Let m be a nonnegative integer and P E R2 be a given node. q( x, y) are two bivariate polynomials and q( P) z 0. Then the relation D,tP)(qp)=O,
vD,~gm,
Suppose
that p( x, y), (4.1)
278
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
implies VD,
D,(P)p=O,
lg,,,.
(4.2)
The proof is omitted. Lemma 9. Let m andj be two natural numbers, P E R2 be a given point. Suppose that 1(x, y) = 0 is a straight line passing through the point P, and that 7 is a unit tangent vector of the straight line. Let a E lR2, o # r, be another unit vector and p(x, y) be a bivariate polynomial. Then the relation D,jD,(P)(lp)=O,
VD,E~,,_~,
(4.3)
D,j-‘D,(P)p=O,
VD,E~,,_~.
(4.4)
implies
Proof. From (4.3) we have Dj+aD:(
O
P)( 1’) = 0,
Di+*(P)(I(DPp))=O,
0<(~+/3~m-l.
By means of Leibniz’ differential formula we get D’+“-‘Df(P)p=O, (I
O
Hence the relation (4.4) is true.
0
Lemma 10. Let m, i and j be nonnegative integers, P E R2 be a given node. Suppose that L, L ,, . . . , Lj are the straight lines passing through the node P, and their equations are I( x, y) = 0, 1,(x, y) =O,..., lj(x, y) = 0, respectively. Let q(x, y), p( x, y) be bivariate polynomials, q(P) f 0. Assume that u, p E R2 are two linearly independent unit vectors, u (1L, and D++Lk,
k=l,2
,...,
j.
Then the relation D;D,‘D,
(P)( l’l,l, . . . I,qp) = 0,
VD, E gW,
(4.5)
implies D%(P)p=O,
(4.6)
VD,E~~.
Proof. From (4.5) we have Di+uDi+B( P)( 1’1,12. . . I,qp) = 0, P
0 < o[ + fi 6 m.
Hence D~t”(P)(l’D,i+B(lllz
e-1 l,qp))=O,
O
By using Leibniz’ differential formula we have D;D,‘+B( P)( l,l, . . . /,qp)=O, D,/D,(P)(I,I,
.- - l/&=0,
O
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
Applying
Lemma
219
9 j times, we get
D&‘)(P)=O, Going a step further,
tfD,=%. from Lemma
D@)p=O,
8 we obtain
VD,E&&.
The proof of Lemma
•I
10 is finished.
Proof of Theorem 3. Here we only prove the first part of Theorem 3, namely that Yn is unisolvent for II,,. The validity of the second part can easily be seen from the proof of the first part. It is clear that the number of interpolating functionals in Yn is $( n + l)( n + 2) and that it is equal to the dimension of II,. Let p E II,, be a bivariate polynomial satisfying Fp = 0,
QFEY?,.
(4.7)
In order to prove the theorem, we only need to show p(x, y) = 0. The approach to prove p(x, y) = 0 on the straight lines L,, L,_l,. . . , L,, L,, successively. We first consider the interpolating functionals on L,. Notice Fp = 0,
VF E@“),
(4.8)
where %(“I is the set of interpolating degree n on L,. So we have p(P)=O,
functionals
for one-dimensional
Hermite
interpolation
Y) =hl(x~
that
Y)P,-lb,
Y),
(4.9)
where p,_ 1( x, y ) is a bivariate polynomial of degree Now we consider the interpolating functionals interpolating node on L,_,. If l,(P) # 0, then
n - 1. on L,_ 1. Let
P = P,_l, j E 9,,_ 1 be an
s?;“-‘l(P) = .Gzq!~l’( P). Since D,-,(P)(l,p,_,)
= 0,
0 G (Y < ajn-‘) - 1,
follows Fp,_,
= 0,
VF E .99;“;“(P).
If Z,(P) = 0, then write u = CT~_~,~,p = ~~_i,~, m = a$‘-l). n ,+l=k(nr, Then the straight
P),
Assume . . . . e,,
i=l,2
,...,
Lnz7 L,,,
e,, e,,
the point
P are
L,O,
that the corresponding
e,_,,...,e,,
Let n, = n - 1, n, = n,
m-l.
lines which pass through
.**> L nm’ L,,_,>...Y successively.
of
VPEL,.
It follows from Bezout’s Theorem P(%
of the proof is
order-raising
directions
at the point
P
are
280
X-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
successively,
where
Let us consider
the set of interpolating
9$!\(P)
= ij A@:‘(P) i=l
functionals
cx.
Since P $5A’,,_ 1, we have Odeg(L,,,
i=l,2
P)>m-i+l,
,...,
m.
From this we know that gJT’,( P) must involve
the following
D/&,Dl9, ’ . . De”,_,-, DTn,_,(P),
O
O
interpolating
functionals:
So from F(l,p,_J
= 0,
V’FE qy(P),
we get D,0D&‘)(4~,-,)
VD, E %,-I.
= 0,
Hence D0”D;D:(W&-*)
= 03
D{+l(I,D{p,_,)=O, By using Leibniz’ differential
O
Dg(P)(D(pn_l)=O,
Fp,_, =O,
we get
O
VP+,,_,(P).
Hence D,~_,(P)P,_~
0 < a < c$-~)-
= 0,
1.
Thus we have Fpn_* = 0,
VFE
B”“;“(P),
even if Z,(P) = 0. So we obtain Fp,_, = 0,
VF ES$(“~-?
Noticing that %L”,” is a set of interpolating functionals for one-dimensional lation of degree n - 1 on L,_ I, we get p,q(P)
= 0,
VP E L,_,.
It follows from Bezout’s Theorem n - 2 such that Pn-1 = in-IPn-2,
Hermite interpo-
that there exists a bivariate
P = IJn-,Pn-2.
polynomial
pn_*(x,
y) of degree (4.10)
281
X. -Z. Liang, L. -Q. Li / Bivariate osculatory interpolation
Next we consider the interpolating functionals on L,_,.Let P = Pn_2, jE gn_* be an u and p be a pair of order-raising directions at the node P.Then for interpolating node on Ln_2, F~k?:?~(P)wehave
F = D;D,/D;->, where O
O
P)-1.
PE~~_~, then i=j=O, 93;:gP)= .%3;"_,"(P),
and I,(
P)l,_l(P)#O. From D;_2(P)(l,l,_,p,,)
= 0,
0 G (r G Odeg(L,-,,
P)
1,
we can get D;_,(P)=O,
O
P)-1,
i.e.,
Fp,_,=O, VFESY;~;~)(P). If P42?,_,, thenweset m= Odeg( Ln-2,P) - 1.Similar to the above proof for L,_1,from DiD,jD7y_,(
P)(l,l,_,p,_,)=O, O
F(IJ,_,p,-2) =O, VFEZ, we conclude that D;D,‘D,
(P)(1,1,_,p,_,) = 0, VD, E gm.
By Lemma 10 we have D,(P)P,-,=O,
VD,=%,
D;m2(P)P,_2=0,
O
Fp,_,=O, VF’E .@:;2)(P). Hence we get Q-2
= 0,
VF E%(“;~).
Noticing that E?T2) is a set of interpolating functionals for one-dimensional lation of degree n - 2 on Ln_2, we have p,_,(P)
= 0,
VP E L,_,.
It follows from Bezout’s Theorem that there exists a bivariate polynomial n - 3 such that Pn-2
=
Hermite interpo-
in-2Pn-37
p=ll
n n-l 1n-2Pn-3.
P~_~( x,
y)
of degree (4.11)
The rest may be deduced by analogy. Finally we can prove that there exists a constant c such that p=ll n n-l . . . l,l, *c.
(4.12)
282
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
We consider the interpolating node P = PO0on La. Let u = a, and p = poo be the order-raising directions at P. Then for some two nonnegative integers a and /3, 0 G (Y+ p G n, we have D:D;( This equation
P)(l,Z,_,
. . . Z,l, . c) = 0.
can be rewritten
in the form
D;D,“( P)( l”ln,ln2 . . . lnr. q . CT)= 0, where q(P) + 0, I( x, y) = 0 is the straight line which passes through the node P and parallels f,,( x, y) = 0 is the straight line which passes through the node P and does not parallel i= 1, 2,..., Y. It follows from Lemma 10 that c = 0. So we get P(X, which completes
(4.13)
Y) = 0, the proof of Theorem
u, u,
3.
0
References [l] E.W. Cheney, Multivariate Approximation Theory: Selected Topics, CBMS-NSF Regional Conf. Ser. in Appl. Math. 51 (SIAM, Philadelphia, PA, 1986). [2] K.C. Chung and T.H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal. 14 (1977) 735-743. [3] M. Gasca and J.I. Maetzu, On Lagrange and Hermite interpolation in lRk, Numer. Math. 39 (1982) l-14. [4] H.A. Hakopian, Multivariate interpolation II of Lagrange and Hermite type, Studia Math. 80 (1984) 77-88. [5] A. Le Mehaute, Taylor interpolation of order n at the vertices of a triangle, in: Z. Ziegler, Ed., Approximation Theory and Applications (Academic Press, New York, 1981) 171-185. [6] X.-Z. Liang, On the interpolation and approximation in several variables, Postgraduate Thesis, Jilin Univ., 1965. [7] X.-Z. Liang, Properly posed nodes for bivariate interpolation and the superposed interpolation, Acta Sci. Natur. Univ. Jilin. 1 (1979) 27-32. [8] X.-Z. Liang, Lagrange representation of multivariate interpolation, Sci. Sinica Ser. A 32 (4) (1989) 385-396. [9] C.A. Micchelli, Algebraic aspects of interpolation, in: C. de Boor, Ed., Approximation Theory Short Course (Amer. Mathematical Sot., Providence, RI, 1986) 81-103.
and Applied
Mathematics
271
38 (1991) 271-282
On bivariate osculatory interpolation Xue-Zhang Department
Liang
of Mathematics,
Luo-Qing
Jilin University, Changchun,
130023, China
Li
Department
of Mathematics,
Hubei University,
Wuhan, China
Received 25 June 1990 Revised 31 May 1991
Abstract Liang, X.-Z. and L.-Q. Li, On bivariate Mathematics 38 (1991) 271-282.
osculatory
interpolation,
Journal
of Computational
and Applied
In this paper the straight-line-superposed interpolation which was first proposed by Liang (1965) for the bivariate Lagrange interpolation problem is generalized to set up a new bivariate interpolation method, the order-raising process. The new method expands and develops the bivariate osculatory interpolation schemes proposed by Le Mehaute (1981) and Hakopian (1984). Keywork
Bivariate
interpolation,
order-raising
process.
1. Introduction Let n be a nonnegative integer, s, = i( n + l)( n + 2) - 1. II, bivariate polynomials of the form p(x,
y) =
c
denotes the space of all real
UjjX’r’.
0 J>
O
is the two-dimensional Euclidean space.. Let Q;= (x;,
v;),
i=o,
l)...)
S,,
0.2)
be S, + 1 distinct points in lR* and consider the following bivariate problem. Given a set of real numbers {f”‘ER
(i=O,
l,...,
s,},
Lagrange interpolation 0.3)
to find a polynomial p( x, y) E II,, satisfying p(Qi)
=f(‘),
i=O,
l,...,
s,.
We call N=
{ Qi 10 < i G s, } a unisolvent set of interpolating unique solution of problem (1.3), (1.4).
0377-0427/91/$03.50
0 1991 - Elsevier
Science Publishers
(1.4) nodes for II,, if there exists a
B.V. All rights reserved
272
X.-Z. Liang L. -Q. Li / Bivariate osculatory interpolation
In [6] one of the authors interpolation.
gave the following
two theorems
on the straight-line-superposed
Theorem 1. If Mn = { Qi ]0 < i < s, } is unisolvent for IT,,, and if none of these points is on the irreducible curve of degree k: 1(x, y) = 0 (either k = 1 or k = 2; k = 1 means a straight line; k = 2 means a conic), then Nti with the (n + 3) k - 1 points being distinct and selected freely in the irreducible curve must constitute a unisolvent set of nodes for 17n+k. Straight-line-superposition process (SLSP). Step 0. In the plane Oxy freely draw a straight line L,: 10(x, y) = a,x + b, y + c,-,= 0, and freely choose a point Q, on the straight line. Set X0 = { Q, }. Step 1. In the plane Oxy draw a straight line L,: lI(x, y) = a,x + b, y + c1 = 0 such that L, n No = 8. On the straight line choose two arbitrary but mutually distinct points Q,, Q2. Set = IQ,, Q1, QzlThe rest may be deduced by analogy. We only give the last step. Step n. In the plane Oxy draw a straight line L,. * a,x + b, y + c, = 0 such that L, nJlm_* =fl. On L, choose n + 1 arbitrary but mutually distinct points Qo,2jn(n+lj,. . . , Qo,ajn(n+Jj. Set
4
JVIn={Qci> Q,,...,Q,,,,,,,,+,,>. Theorem 2. The set of points M4, given by the SLSP is a unisolvent set of interpolating IT,. The interpolating polynomial p,,( x, y ) has the following form: i-l n Pn(xy Y) = ,~~q,-i(b,-ix-a~-iy),~~l”-j(x,
nodes for
(1.5)
Y),
whereq,(t), q,-l(t), . . . , q,,(t) (being the univariate polynomials tively) can be determined successively.
by the interpolation
conditions
of degree n, n - 1,. . . ,O, respecon the straight lines L,, L,_l,. . . , L,,
In [2] Chung and Yao made a systematic study on multivariate Lagrange interpolation and defined the natural lattices and the GC condition. In [3] Gasca and Maetzu studied both Lagrange interpolation and Hermite interpolation in two variables and proposed a new scheme of bivariate osculatory interpolation. So far as the Lagrange interpolation for II, is concerned, the unisolvent sets of interpolating nodes given by the scheme are always constructed by the above SLSP, and the sets may not obey the GC condition. Recently Micchelli [9] and Cheney [l] have pointed out that the above-mentioned straightline-superposed interpolation is fully effective. The purpose of this paper is to make a generalization of the method and solve a more general osculatory interpolation problem. In the paper we will give a new bivariate osculatory interpolation method, the order-raising process. The method is quite different from the scheme in [3]. 2. Bivariate osculatory interpolation problem First of all we introduce
some notations.
Tk= {(TV,... ,T~)[T,EIW*, T=
ij k=O
Tk.
Let T, = 0 be the empty
I17iII =l,
i=1,2
,...,
k},
set,
k=l,2,3
,...,
(2-l)
213
X.-Z. Liang, L.-Q. Li / Bioariate osculatory interpolation
rk) E T, we define the directional derivative of k th order
For a==(~~,...,
D, = D,,D,.
. . QA = q
aa
a ..*G .
K
(2-2)
In particular, we define D+ = I,
the identity operator.
(2-3)
For a nonnegative integer k, we set gk=
{D,I%ET&
(2 -4)
For a point Q E IR2 and an operator D, E SSk, we define an interpolating functional D,(Q) setting D, (Q)(&
Y)) = D&(x,
Y) I cx,rj=P,
VfE
Cm(R2>-
by
(2.5)
In addition, we use the notation Da(Da(Q)>
=
D,,,(Q>,
vz,
(2.6)
BE T.
Now we describe the general bivariate osculatory interpolation problem. Suppose that n and s are two given nonnegative integers. Let Qj=(xi,
y,),
i=O,l,...,
s,
(2.7)
be s + 1 distinct points in R2, and m,, m,, . . . , m, natural numbers satisfying
c m; = +I + l)(n + 2). I=0
(2.8)
Suppose that A?‘~,Ar,...,
s?--c T
(2.9)
are finite subsets of T such that Card(dj)
=m,,
i=O, l,...,
S.
(2.10)
We consider the following osculatory interpolation problem. Given a set of real numbers {f$‘ER
l%E.&;,
i=O, l,...,
to find a polynomial p( x, y) E II, D,(Qi)p=f$‘, The polynomial p(x, 0=
‘BE&;,
s},
(2.11)
such that i=O,l,...,
S.
y) satisfying (2.12) is called the interpolating
{D,(Q,)~xEA?~,
i=O, l,...,
S}
(2.12) polynomial. The set (2.13)
is called the set of interpolating functionals. We write A--=
{Q,, Q,,---, Q,>
(2.14)
the set of interpolating nodes, and
~~={D~(Qi)l~~4}
(2.15)
X.-Z. Liang L.-Q. Li / Bivariate osculatory interpolation
274
the set of interpolating functionals at node Qi. If there exists a unique polynomial p(x, y) E IIT, satisfying (2.12), then the interpolation problem (2.11), (2.12) is called a unisolvent problem and 0 is called a unisolvent set of interpolating functionals for II,, (or say “0 is unisolvent for II,,” for short).
3. Order-raising osculatory interpolation We first introduce some concepts. Suppose that k and m are nonnegative integers, k > m. L denotes a straight, line in the plane Oxy, and r a unit tangent vector of it. Choose m + 1 mutually distinct PO, Pi,. . . , P,,, on L and m + 1 natural numbers (Y,,, (~i,. . . , a, satisfying &,=k+l. i=o For a nonnegative setting
(3-l) integer (Yand a point
P E R*, we define an interpolating
functional
D:(P)
by
(3 4 Then we call the set of interpolating 9=
{D,“(P,))O
functionals i=O,
l,...,
a set of interpolating functionals for one-dimensional straight line L. At the same time, we call JV= {PO, PI,..., the set of interpolating gi=
(3.3)
m} Hermite
interpolation
of degree
PJ
(3.4)
nodes corresponding
to 9
and
{D,*(P;)IO
(3.5)
the limitation of 9 at the point Pi. We call the natural P, on the straight line L, and write
number
(Y;the osculating
degree of 9
(Y~= Odeg( L, Pi). If P E L\JV,
k on the
at
(3.6)
then we make a supplemental
definition
Odeg( L, P) = 0;
(3.7)
Now we generalize the straight-line-superposition process and give a more general constructing the unisolvent set of interpolating functionals for the bivariate osculatory tion.
process of interpola-
Order-raising process. Step 0. In the plane Oxy freely draw a straight line Lo: l,(x, y) = a,x + boy + c,, = 0, and on the line freely choose a node Poe. Let r,, be a unit tangent vector of L,. With regard to PO0 we freely choose two linearly independent unit vectors uoo, poo E R*, which are called the first and the second order-raising directions at the node P,, respectively. We set m,=O,
2?. =.A0 =g,
4=~0=
{PO&
X.-Z. Liang, L.-Q. Li / Biuariate osculatory interpolation
275
and YO=~~o)=.@($o)(Poo)
= {1(P,)}.
1. It includes five successive stages. (i) In the plane Oxy freely draw a straight line L,. . Z,(x, y) = a,x + b,y + c1 = 0 such that L, n .A!,= 0. Let r1 E R * be a unit tangent vector of L,. (ii) If PO0E L,, then we raise the orders of all directional derivatives at the node. Namely, we define a new set of interpolating functionals at the node PO,:
Step
if PO0G L,,
kV(Poo)~ gf)
D~~.G?~)( PO,),
( p,, ) =
I Q7$%o) ( PO0>9
if PO0E L,
and if a, 117,)
if PO0E L,
and if oooH TV.
Furthermore, we set go(‘) = m; ‘Qq’( PO,). J=o
(iii) On the straight line L, freely choose a set of interpolating sional Hermite interpolation of degree 1,
functionals
for one-dimen-
3$(i) = (D;( Plj) 10 < cx < 01:” - 1, 0
O
The corresponding set of interpolating nodes is 9, = {P1,0,...7
Pi,,,},
where PI,o,. . . , P,,, are mutually distinct points on L,. With regard to each node P, j E 9, we define a pair of order-raising directions ui j, pl, E R* being linearly independent unit vectors, j=O,l ,‘..> m,. If plj E No, namely PI, = P,, then we define u1j = %J 9 Plj= POO(iv) Set ~,“‘(P~~)~(D~(P~~)~O~~~~~~~~l), Jv;=
;.9;,
y; = ;
For PI j E No k(l,
q(i).
i=O
i=O (v)
j=O,...,m,,
we define
j)=max{tIP,,EL,,O
Set
Odeg(Lk(1.j)) Plj)
pi”‘= 0 i
7
9
if
Plj E No >
if P,,GNo,
j=O, 1,.
--3
ml,
X.-Z. Liang L.-Q. Li / &variate osculatory interpolation
276
and
The rest may be deduced by analogy. We only give the last step. Step n. It includes five successive stages. (i) In the plane Oxy draw a straight line L,: In(x, y) = a,x + b,,y -t c, = 0 such that L, n A?,_,= 0. Let 7n E lR2 be a unit tangent vector of L,. (ii) If Pij E L,, then we raise the orders of all directional derivatives at the node. Namely, we define a new set of interpolating functionals at the point Pii:
.~$~)(p~~) =
I l)...)
j=o,
Furthermore,
53T;“-1)( Pii),
if Pij 4 L, ,
D,,543/“-“(P,,),
if Pij E L, and
aij 11T,,
D$$(-‘)(
if Pij E L, and
uij +t r,, ,
Pii),
i=o,
mi,
1)“‘)
n-l.
we set
F(n) = 5 JJ$“‘(Pij), I
i=o,
l)...)
n-l.
j=O
(iii) On the straight line L, freely choose sional Hermite interpolation of degree n, 3(“)
a set of interpolating
functionals
for one-dimen-
= (D;( Pnj) 10 < a < aj.“’ - 1, 0
where a;.") = Odeg( L, , P,,j),
O
2
ol(i”)=n+l.
j=O
The corresponding pn= where define j=O,l which (iv)
set of interpolating
nodes is
{P,,OY..94,,“}~
P,, o,. . . , P,,,, are mutually distinct points on L,. With regard to each node Pnj E 9,, we a ‘pair of order-raising directions anj, pnj E lR2 being linearly independent unit vectors, ,***, m,. If P, j E Nn_l, then we require that the new pair (a,, j, p, j) is equal to the old one has been defined with regard to the node already. Set ~~n)(P~j)=(D~(P,j)lOda~olj”)-l),
and n
n
Nn= Upi i=O
9” =
u 9y”). i=O
j=O,...,m,,
277
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
(v) For P,,, E Jr/-,_, k(n,
we define
j)=max{t]Odt
PnjEL,}.
Set p(n)
Odeg(L,(,,,j), P,,), if Pn,E-4’--lv j=o,
=
J
if Pni EJ+_,,
0,
l)...)
m,,
and A,
= ( P, j E 9n
1 a;“’
>
/y’
+
1>)
2?,=
;A!;. i=O
Our main result in this paper is as follows. Theorem 3. The set $!! obtained by the above process is a unisolvent set of interpolating functionals for II,,. The osculatory interpolating polynomial p,,( x, y) has the following form: Pn(x,
Y) = 2
q~-i(b~-ix-a~-iy)~~~l.-J(x,
(3.8)
Y>,
i=O where
q,,(t),
q,-l(tL..
of degree n, n - 1, ___,O, respecon the straight lines L,, L,-,, . . . , L,
. , qO(t) (being the univariate polynomials
tively) can be determined successively.
by the interpolation
conditions
Remark 4. In the above process, if the osculatory degrees at all nodes are equal to 1, and if every new straight line does not pass through any one of the nodes which have been chosen already, then the new process is simplified to be the original straight-line-superposition process. Remark 5. The above process can be generalized to the Euclidean space R”, s > 2, to be a hyperplane-superposition process. In this situation the order-raising directions at every node are s linearly independent unit vectors. Remark 6. If every straight line in the above process coincides with a side of some fixed triangle or passes through some fixed point in the triangle, then the process leads to the osculatory interpolation schemes described in [5]. Remark 7. Obviously, the osculatory interpolation schemes proposed in [4] can be obtained by using the order-raising process or the hyperplane-superposition process.
4. Proof of Theorem 3 In order to prove Theorem 3, we first set up a few lemmas. Lemma 8. Let m be a nonnegative integer and P E R2 be a given node. q( x, y) are two bivariate polynomials and q( P) z 0. Then the relation D,tP)(qp)=O,
vD,~gm,
Suppose
that p( x, y), (4.1)
278
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
implies VD,
D,(P)p=O,
lg,,,.
(4.2)
The proof is omitted. Lemma 9. Let m andj be two natural numbers, P E R2 be a given point. Suppose that 1(x, y) = 0 is a straight line passing through the point P, and that 7 is a unit tangent vector of the straight line. Let a E lR2, o # r, be another unit vector and p(x, y) be a bivariate polynomial. Then the relation D,jD,(P)(lp)=O,
VD,E~,,_~,
(4.3)
D,j-‘D,(P)p=O,
VD,E~,,_~.
(4.4)
implies
Proof. From (4.3) we have Dj+aD:(
O
P)( 1’) = 0,
Di+*(P)(I(DPp))=O,
0<(~+/3~m-l.
By means of Leibniz’ differential formula we get D’+“-‘Df(P)p=O, (I
O
Hence the relation (4.4) is true.
0
Lemma 10. Let m, i and j be nonnegative integers, P E R2 be a given node. Suppose that L, L ,, . . . , Lj are the straight lines passing through the node P, and their equations are I( x, y) = 0, 1,(x, y) =O,..., lj(x, y) = 0, respectively. Let q(x, y), p( x, y) be bivariate polynomials, q(P) f 0. Assume that u, p E R2 are two linearly independent unit vectors, u (1L, and D++Lk,
k=l,2
,...,
j.
Then the relation D;D,‘D,
(P)( l’l,l, . . . I,qp) = 0,
VD, E gW,
(4.5)
implies D%(P)p=O,
(4.6)
VD,E~~.
Proof. From (4.5) we have Di+uDi+B( P)( 1’1,12. . . I,qp) = 0, P
0 < o[ + fi 6 m.
Hence D~t”(P)(l’D,i+B(lllz
e-1 l,qp))=O,
O
By using Leibniz’ differential formula we have D;D,‘+B( P)( l,l, . . . /,qp)=O, D,/D,(P)(I,I,
.- - l/&=0,
O
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
Applying
Lemma
219
9 j times, we get
D&‘)(P)=O, Going a step further,
tfD,=%. from Lemma
D@)p=O,
8 we obtain
VD,E&&.
The proof of Lemma
•I
10 is finished.
Proof of Theorem 3. Here we only prove the first part of Theorem 3, namely that Yn is unisolvent for II,,. The validity of the second part can easily be seen from the proof of the first part. It is clear that the number of interpolating functionals in Yn is $( n + l)( n + 2) and that it is equal to the dimension of II,. Let p E II,, be a bivariate polynomial satisfying Fp = 0,
QFEY?,.
(4.7)
In order to prove the theorem, we only need to show p(x, y) = 0. The approach to prove p(x, y) = 0 on the straight lines L,, L,_l,. . . , L,, L,, successively. We first consider the interpolating functionals on L,. Notice Fp = 0,
VF E@“),
(4.8)
where %(“I is the set of interpolating degree n on L,. So we have p(P)=O,
functionals
for one-dimensional
Hermite
interpolation
Y) =hl(x~
that
Y)P,-lb,
Y),
(4.9)
where p,_ 1( x, y ) is a bivariate polynomial of degree Now we consider the interpolating functionals interpolating node on L,_,. If l,(P) # 0, then
n - 1. on L,_ 1. Let
P = P,_l, j E 9,,_ 1 be an
s?;“-‘l(P) = .Gzq!~l’( P). Since D,-,(P)(l,p,_,)
= 0,
0 G (Y < ajn-‘) - 1,
follows Fp,_,
= 0,
VF E .99;“;“(P).
If Z,(P) = 0, then write u = CT~_~,~,p = ~~_i,~, m = a$‘-l). n ,+l=k(nr, Then the straight
P),
Assume . . . . e,,
i=l,2
,...,
Lnz7 L,,,
e,, e,,
the point
P are
L,O,
that the corresponding
e,_,,...,e,,
Let n, = n - 1, n, = n,
m-l.
lines which pass through
.**> L nm’ L,,_,>...Y successively.
of
VPEL,.
It follows from Bezout’s Theorem P(%
of the proof is
order-raising
directions
at the point
P
are
280
X-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
successively,
where
Let us consider
the set of interpolating
9$!\(P)
= ij A@:‘(P) i=l
functionals
cx.
Since P $5A’,,_ 1, we have Odeg(L,,,
i=l,2
P)>m-i+l,
,...,
m.
From this we know that gJT’,( P) must involve
the following
D/&,Dl9, ’ . . De”,_,-, DTn,_,(P),
O
O
interpolating
functionals:
So from F(l,p,_J
= 0,
V’FE qy(P),
we get D,0D&‘)(4~,-,)
VD, E %,-I.
= 0,
Hence D0”D;D:(W&-*)
= 03
D{+l(I,D{p,_,)=O, By using Leibniz’ differential
O
Dg(P)(D(pn_l)=O,
Fp,_, =O,
we get
O
VP+,,_,(P).
Hence D,~_,(P)P,_~
0 < a < c$-~)-
= 0,
1.
Thus we have Fpn_* = 0,
VFE
B”“;“(P),
even if Z,(P) = 0. So we obtain Fp,_, = 0,
VF ES$(“~-?
Noticing that %L”,” is a set of interpolating functionals for one-dimensional lation of degree n - 1 on L,_ I, we get p,q(P)
= 0,
VP E L,_,.
It follows from Bezout’s Theorem n - 2 such that Pn-1 = in-IPn-2,
Hermite interpo-
that there exists a bivariate
P = IJn-,Pn-2.
polynomial
pn_*(x,
y) of degree (4.10)
281
X. -Z. Liang, L. -Q. Li / Bivariate osculatory interpolation
Next we consider the interpolating functionals on L,_,.Let P = Pn_2, jE gn_* be an u and p be a pair of order-raising directions at the node P.Then for interpolating node on Ln_2, F~k?:?~(P)wehave
F = D;D,/D;->, where O
O
P)-1.
PE~~_~, then i=j=O, 93;:gP)= .%3;"_,"(P),
and I,(
P)l,_l(P)#O. From D;_2(P)(l,l,_,p,,)
= 0,
0 G (r G Odeg(L,-,,
P)
1,
we can get D;_,(P)=O,
O
P)-1,
i.e.,
Fp,_,=O, VFESY;~;~)(P). If P42?,_,, thenweset m= Odeg( Ln-2,P) - 1.Similar to the above proof for L,_1,from DiD,jD7y_,(
P)(l,l,_,p,_,)=O, O
F(IJ,_,p,-2) =O, VFEZ, we conclude that D;D,‘D,
(P)(1,1,_,p,_,) = 0, VD, E gm.
By Lemma 10 we have D,(P)P,-,=O,
VD,=%,
D;m2(P)P,_2=0,
O
Fp,_,=O, VF’E .@:;2)(P). Hence we get Q-2
= 0,
VF E%(“;~).
Noticing that E?T2) is a set of interpolating functionals for one-dimensional lation of degree n - 2 on Ln_2, we have p,_,(P)
= 0,
VP E L,_,.
It follows from Bezout’s Theorem that there exists a bivariate polynomial n - 3 such that Pn-2
=
Hermite interpo-
in-2Pn-37
p=ll
n n-l 1n-2Pn-3.
P~_~( x,
y)
of degree (4.11)
The rest may be deduced by analogy. Finally we can prove that there exists a constant c such that p=ll n n-l . . . l,l, *c.
(4.12)
282
X.-Z. Liang, L.-Q. Li / Bivariate osculatory interpolation
We consider the interpolating node P = PO0on La. Let u = a, and p = poo be the order-raising directions at P. Then for some two nonnegative integers a and /3, 0 G (Y+ p G n, we have D:D;( This equation
P)(l,Z,_,
. . . Z,l, . c) = 0.
can be rewritten
in the form
D;D,“( P)( l”ln,ln2 . . . lnr. q . CT)= 0, where q(P) + 0, I( x, y) = 0 is the straight line which passes through the node P and parallels f,,( x, y) = 0 is the straight line which passes through the node P and does not parallel i= 1, 2,..., Y. It follows from Lemma 10 that c = 0. So we get P(X, which completes
(4.13)
Y) = 0, the proof of Theorem
u, u,
3.
0
References [l] E.W. Cheney, Multivariate Approximation Theory: Selected Topics, CBMS-NSF Regional Conf. Ser. in Appl. Math. 51 (SIAM, Philadelphia, PA, 1986). [2] K.C. Chung and T.H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal. 14 (1977) 735-743. [3] M. Gasca and J.I. Maetzu, On Lagrange and Hermite interpolation in lRk, Numer. Math. 39 (1982) l-14. [4] H.A. Hakopian, Multivariate interpolation II of Lagrange and Hermite type, Studia Math. 80 (1984) 77-88. [5] A. Le Mehaute, Taylor interpolation of order n at the vertices of a triangle, in: Z. Ziegler, Ed., Approximation Theory and Applications (Academic Press, New York, 1981) 171-185. [6] X.-Z. Liang, On the interpolation and approximation in several variables, Postgraduate Thesis, Jilin Univ., 1965. [7] X.-Z. Liang, Properly posed nodes for bivariate interpolation and the superposed interpolation, Acta Sci. Natur. Univ. Jilin. 1 (1979) 27-32. [8] X.-Z. Liang, Lagrange representation of multivariate interpolation, Sci. Sinica Ser. A 32 (4) (1989) 385-396. [9] C.A. Micchelli, Algebraic aspects of interpolation, in: C. de Boor, Ed., Approximation Theory Short Course (Amer. Mathematical Sot., Providence, RI, 1986) 81-103.