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Rev. Anal. Num´er. Th´eor. Approx., vol. 34 (2005) no. 1, pp. 37–45 ictp.acad.ro/jnaat

BIERMANN INTERPOLATION OF BIRKHOFF TYPE

MARIUS BIROU

Abstract. IfP0, P1, ..., Pr andQ0, Q1, ..., Qr are Birkhoff univariate projectors which form the chains

P0P1≤ · · · ≤Pr, Q0Q1≤ · · · ≤Qr, we can define the Biermann operator of Birkhoff type

BBr =P00Q00r P10Q00r−1⊕ · · · ⊕Pr0Q000, whereP10, . . . , Pr0,Q001, . . . , Q00r are the parametric extension.

We give the representations of Biermann interpolant of Birkhoff type for two particular cases (Abel-Goncharov and Lidstone projectors) and we calculate the approximation order of Biermann interpolant in these cases.

MSC 2000. 41A63, 41A05, 41A80.

Keywords. Biermann interpolation, Birkhoff interpolation, chains of projec- tors, approximation order.

1. PRELIMINARIES

Let X, Y be the linear space onR orC. The linear operator P defined on space X is called projector if P2 =P. The operator Pc=IP, where I is identity operator, is called the remainder projector of P. IfP is projector on space X then the range space of projector P is denoted by

(1) R(P) ={P f|fX}.

The set of interpolation points of projectorP is denoted byP(P).

Proposition 1. If P, Q are comutative projectors then (2) R(P⊕Q) =R(P) +R(Q),

P(P⊕Q) =P(P)∪ P(Q).

IfP1 and P2 are projectors on space X, we define relation “≤” by P1P2P1P2 =P1.

LetfC(X×Y) and xX. We definefxC(Y) by fx(t) =f(x, t), tY.

ForyY we define yfC(X) by

yf(s) =f(s, y), sX.

(2)

LetP be a linear and bounded operator onC(X). The parametric extension P0 ofP is defined by

(3) (P0f)(x, y) = (Pyf)(x).

IfP is a linear and bounded operator onC(Y) then the parametric extension Q00 of Qis defined by

(4) (Q00f)(x, y) = (Qfx)(y).

Proposition 2. Let r ∈N, P0, P1, . . . , Pr be univariate interpolation pro- jectors on C(X) and Q0, Q1, . . . , Qr univariate interpolation projectors on C(Y). Let P00, . . . , Pr0, Q000, . . . , Q00r be the corresponding parametric extension.

We assume that

(5) P0P1 ≤ · · · ≤Pr, Q0Q1 ≤ · · · ≤Qr. Then

(6) Br=P00Q00rP10Q00r−1⊕ · · · ⊕Pr0Q000 is projector and it has representation

(7) Br =

r

X

m=0

Pm0 Q00r−m

r−1

X

m=0

Pm0 Q00r−m−1. Moreover, we have

(8) Brc=Pr0c+Pr−10c Q000c+· · ·+P00cQ00r−1c +Q00rc−(Pr0cQ000c+· · ·+P00cQ00rc), where Pc=IP, I the identity operator.

For the proof of Proposition 2 one can see [6].

Remark 3. If Pi, i = 0, r, and Qj, j = 0, r, are Lagrange interpolation projectors which form the chains with respect to relation “≤”, the projector Br is called Biermann interpolation projector (see [6]). In [10] we instead the Lagrange projectors by Hermite projectors. In this article, our objective is to construct chains of Birkhoff interpolation projectors and, with their aid, the Biermann interpolant of Birkhoff type.

2. MAIN RESULT

Let be the univariate projectors of Birkhoff interpolation P0, . . . , Pr, Q0, . . . , Qr

given by relations

(Pmf)(x) =

km

X

i=0

X

p∈Ii,m

f(p)(xi)bmip(x), 0≤mr, (9)

(Qng)(y) =

ln

X

j=0

X

q∈Jj,n

g(q)(yj)ebnjq(y), 0≤nr.

(3)

Assume that

{x0, . . . , xkr} ⊆[a, b], {y0, . . . , ylr} ⊆[c, d]

with

(10) 0≤k0k1 ≤ · · · ≤kr, 0≤l0l1 ≤ · · · ≤lr

and

Ii,mIi,m+1, i= 0, km, m= 0, r−1, (11)

Jj,nJj,n+1, j = 0, ln, n= 0, r−1.

The cardinal functionsbmip,m= 0, r, and ebnjq,n= 0, r, satisfy ( bmip (j)(xν) = 0, ν 6=i, jIν,m

bmip (j)(xi) =δjp, jIi,m

forpIi,m,ν, i= 0, km and, respectively, (

ebnjq(i)(yν) = 0, ν 6=j, iJν,n ebnjq(i)(yj) =δiq, iJj,n

forqJj,n,ν, j = 0, ln.

Theorem 4. The parametric extensions

P00, . . . , Pr0, Q000, . . . , Q00r

are bivariate interpolation projectors which form the chains P00 ≤ · · · ≤Pr0, Q000 ≤ · · · ≤Q00r. Proof. Let be 0≤m1m2r. Then

km1km2, (12)

Ii,m1Ii,m2, ikm1, We have that

(Pm0 1Pm0 2f)(x, y) = (13)

=

km1

X

i1=0

X

p1∈Ii1,m1

km2

X

i2=0

X

p2∈Ii2m2

f(p2,0)(xi2, y)bmi 2 (p1)

2p2 (xi1)

bmi 1

1p1(x).

But

(14) bmi 2 (p1)

2p2 (xi1) =δi1i2δp1p2. From (12), (13) and (14) we have

(Pm0 1Pm0 2f)(x, y) =

km1

X

i1=0

X

p1∈Ii1,m1

f(p1,0)(xi1, y)bmi1p11(x) = (Pm0 1f)(x, y),

(4)

i.e.,

Pm0 1Pm0 2.

ThusP00, P10, ..., Pr0form a chain. AnalogousQ000, Q001, ..., Q00rare projectors which

form a chain.

Moreover, we have

Pm0 Q00n=Q00nPm0 , 0≤m, nr.

The tensor product projectorPm0 Q00nof bivariate interpolation has represen- tation

(Pm0 Q00nf)(x, y) =

km

X

i=0

X

p∈Ii,m

ln

X

j=0

X

q∈Jj,n

f(p,q)(xi, yj)hmip(x)ehnjq(y) and it has the interpolation properties

(Pm0 Q00nf)(p,q)(xi, yj) =f(p,q)(xi, yj), 0≤ikm, 0≤jln, pIi,m, qJj,n.

The projectors P00, . . . , Pr0, Q000, . . . , Q00r generate a distributive lattice ξ of projectors onC([a, b]×[c, d]). A special element in this lattice is

BBr =P00Q00rP10Q00r−1⊕ · · · ⊕Pr0Q000, r∈N,

called Biermann projector of Birkhoff type and which has the interpolation properties

(BrBf)(p,q)(xi, yj) =f(p,q)(xi, yj), 0≤ikm, 0≤jlr−m, 0≤mr, (15)

pIi,m\Ii,m−1, qJj,r−m

withIi,s=∅,ks< iks+1,s=−1, r−1 with k−1=−1.

Let αi =|I0,i|+· · ·+|Iki,i|, βi =|J0,i|+· · ·+|Jli,i|, 0≤ir. The range space of projector BrB is

(16) R(BrH) = Πα0−1⊗Πβr−1+· · ·+ Παr−1⊗Πβ0−1. The properties (15) and (16) result using Proposition 1.

Let be h =ba=dc and q = min{αr−m−1+βm, −1 ≤mr} with α−1 = 0, β−1= 0. From (8) we have

(17) f(x, y)−(BrBf)(x, y) =O(hq), h→0.

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3. PARTICULAR CASES

3.1. Biermann interpolation of Abel-Goncharov type. Let be the uni- variate Abel-Goncharov interpolation projectors

(Pmf)(x) =

m

X

i=0

gi,m(x)f(i)(xi), m= 0, r, (Qnf)(y) =

n

X

j=0

egj,n(y)f(j)(yj), n= 0, r, i.e.,

km=m, m= 0, r, ln=n, n= 0, r,

Ij,i={j}, j= 0, r, i=j, r, Ji,j ={i}, i= 0, r, j=i, r.

According to [5], the cardinal functions gi,m,i = 0, m, are given by recur- rence relations

g0,m(x) = 1, g1,m(x) = 1−x0, gi,m(x) = i!1xii−1P

j=0

gi,m(x) jixi−jj , i= 2, m, form= 0, r.

One remark that the cardinal functions gi,m depend only the nodes x0, x1, . . . , xi−1.It follows that the functionsgi,m,i= 0, r, are the same form=i, r.

One denotes

gi(x) :=gi,m(x), i= 0, r, m=i, r.

We have analogous relations for the cardinal functions egj,n and we denote egj(x) :=egj,m(x), j= 0, r, m=j, r.

The indices sets of derivatives satisfy the chains of equalities Ij,j =Ij,j+1=. . .=Ij,r, j= 0, r,

Ji,i=Ji,i+1=. . .=Ji,r, i= 0, r.

It follows that relations (11) hold.

We can define the Biermann operator of Abel-Goncharov type (18) BAGr =P00Q00rP10Q00r−1. . .Pr0Q000,

wherePi0, i= 0, r, andQ00j, j = 0, r, are the parametric extensions.

(6)

Taking into account (7) we obtain the following representation for Biermann interpolantBAGr

(BrAGf)(x, y) =

r

X

m=0

(Pm0 (Q00r−mQ00r−m−1)f)(x, y) (19)

=

r

X

m=0 m

X

i=0

gi(x)ger−m(y)f(i,r−m)(xi, yr−m), whereQ00−1 = 0.

Using Proposition 1, the Biermann interpolant BrAG has the interpolation properties

(20) (BrAGf)(i,r−j)(xi, yr−j) =f(i,r−j)(xi, yr−j), i= 0, r, j = 0, i.

Remark 5. The set of nodes form a triungular grid (x0, yr)

(x0, yr−1), (x1, yr−1) . . .

(x0, y1), (x1, y1), . . . (xr−1, y1)

(x0, y0), (x1, y0), . . . (xr−1, y0), (xr, y0) Leth=ba=dc. According to

kf(x)−Pmf(x)k=O(hm+1), iffCm+1[a, b], kf(y)−Qnf(x)k=O(hn+1), if fCn+1[c, d],

and relation (8), the approximation order of Biermann interpolant of Abel- Goncharov type is r+ 1, i.e.,

(21) f(x, y)BrAGf(x, y)=O(hr+1), if fCr+1([a, b]×[c, d]).

Remark 6. The approximation order of Biermann operator BrAG is the same with the approximation order of tensor product operator Pr0Q00r. IfP is an interpolation projector we denote byIP(f) the set of data aboutf (values of function f and/or certain of its partial derivatives at nodes). We have

|IBAG

r (f)|= (r+1)(r+2)2 ,

|IP0

rQ00r(f)|= (r+ 1)2, and

IBAG

r (f)⊂IPr0Q00r(f).

It follows that the Biermann operatorBAGr is more efficient than tensor prod-

uct operatorPr0Q00r.

(7)

3.2. Biermann interpolation of Lidstone type. Let be the univariate Lid- stone interpolation projectors

(Pmf)(x) =

m

X

i=0

[l0,im(x)f(2i)(x0) +l1,im(x)f(2i)(x1)], m= 0, r, (Qng)(y) =

n

X

j=0

[eln0,j(y)f(2j)(y0) +eln1,j(y)f(2j)(y1)], n= 0, r, i.e.,

k0 =k1 =. . .=kr= 1, l0 =l1=. . .=lr = 1,

I0,i =I1,i ={0,2, . . . ,2i}, i= 0, r, J0,j =J1,j ={0,2, . . . ,2j}, j= 0, r.

If we denoteh=x1x0 =y1y0,from [1] we can obtain cardinal functions by relations

lm0,i(x) = ∆i(x1h−x)h2i, lm1,i(x) = ∆i(x−xh 0)h2i,

m = 0, r, i = 0, m, where the functions ∆i, i = 0, r, are given by recurrence relations

0(x) =x,

n(x) = Z 1

0

gn(x, s)sds, n≥1, with

g1(x, s) =

(x−1)s, s≤x, (s−1)x, x≤s, gn(x, s) =

Z 1 0

g1(x, t)gn−1(t, s)dt, n≥2.

One remark that the cardinal function lm0,i and l1,im don’t depend of m. We can make notation

l0,i(x) :=l0,im(x), i= 0, r, m=i, r, l1,i(x) :=l1,im(x), i= 0, r, m=i, r.

We have analogous relations for cardinal functionsel0,jn ,eln1,j and we denote el0,j(y) :=eln0,j(y), j = 0, r, n=j, r,

el1,j(y) :=eln1,j(y), j = 0, r, n=j, r.

(8)

The indices sets of derivatives verify relations (11) because

Ij,i={0,2, . . . ,2i} ⊂ {0,2, . . . ,2i,2i+ 2}=Ij,i+1, j= 0,1, i= 0, r−1, Ji,j ={0,2, . . . ,2j} ⊂ {0,2, . . . ,2j,2j+ 2}=Ii,j+1, i= 0,1, j= 0, r−1.

We can define the Biermann operator of Lidstone type (22) BrL=P00Q00rP10Q00r−1. . .Pr0Q000,

wherePi0, i= 0, r, andQ00j, j = 0, r, are the parametric extensions.

From (7), we obtain the following representation for Biermann operatorBLr

(BrLf)(x, y) = (23)

=

r

X

m=0

(Pm0 (Q00r−mQ00r−m−1)f)(x, y)

=

r

X

m=0 m

X

i=0

l0,i(x)[el0,2r−2m(y)f(2i,2r−2m)(x0, y0) +el1,2r−2m(y)f(2i,2r−2m)(x0, y1)]

+

r

X

m=0 m

X

i=0

l1,i(x)[el0,2r−2m(y)f(2i,2r−2m)(x1, y0) +el1,2r−2m(y)f(2i,2r−2m)(x1, y1)], whereQ00−1 = 0.

Using Proposition 1, it folows that the Biermann projector BrL has the interpolations properties

(24)

(BrLf)(2i,2(r−j))(xk, yl) =f(2i,2(r−j))(xk, yl), i= 0, r, j= 0, i, k, l∈ {0,1}. From [1] we have

kf(x)−Pmf(x)k=O(h2m), iffC2m[a, b], kg(y)−Qng(y)k=O(h2n), ifgC2n[c, d].

Taking into account (8), the approximation order of Biermann operatorBLr is 2r, i.e.

(25) f(x, y)−BrLf(x, y)=O(h2r), iffC2r([a, b]×[c, d]).

Remark7. The approximation order of Biermann operatorBrLis the same with the approximation order of tensor product operator Pr0Q00r. We have

|IBL

r(f)|= 2(r+ 1)(r+ 2),

|IP0

rQ00r(f)|= 4(r+ 1)2 and

IBL

r(f)⊂IPr0Q00r(f),

(9)

where IP(f) is the set of data about f used by interpolation projector P. It follows that the Biermann operator BBr is more efficient than tensor product

operatorPr0Q00r.

REFERENCES

[1] Agarwal, R., Wong, P.J.Y., Error inegalities in polynomial interpolation and their applications, Kluwer Academic Publishers, Dordrecht, 1993.

[2] Biermann, O., Uber naherungsweise Cubaturen, Monatshefte fur Mathematik und Physic,14, pp. 211–225, 1903.

[3] Coman,Gh.,Multivariate approximation schemes and the approximation of linear func- tionals, Mathematica,16, pp. 229–249, 1974.

[4] Coman,Gh. and all,Interpolation operators, Casa C˘art¸ii de S¸tiint¸˘a, 2004.

[5] Davis, Ph. J., Interpolation and approximation, Blaisdell Publishing Company, New york, 1963.

[6] Delvos, F.-J. andSchempp, W.,Boolean methods in interpolation and approximation, Pitman Research Notes in Mathematics, Series 230, New York 1989.

[7] Delvos, F.-J. andPosdorf, H.,Generalized Biermann interpolation, Resultate Math., 5, no. 1, pp. 6–18, 1982.

[8] Gordon,WilliamJ.,Distributive lattices and the approximation of multivariate func- tions, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ.

of Wisconsin, Madison, Wis., 1969), pp. 223–277, Academic Press, New York.

[9] Gordon, William J.; Hall, Charles A. Transfinite element methods: blending- function interpolation over arbitrary curved element domains, Numer. Math., 21, pp. 109–129, 1973/74.

[10] Birou, M.,Biermann interpolation with Hermite information, Studia Univ. “Babe¸s- Bolyai”, to appear.

Received by the editors: February 15, 2005.

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