-AI67 519 ININAI. SUPPORT FOR IIRTE SPLINES(U) urSCmNINz VI iI UNXY-NADIS2N NATHENATICS RESEARCH CENTERI C ~DE BOR ETA. E 6 NC-TSR-2 18 DRR029-S0-C-0041IUNCLASSIFIED F/O 12/1 ML

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MICROCOPY RESOLUTION rESIC.HARTIMIUNAL Mlt'A l A F 0 ANr.i,

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AD-A167 519MRC Technical Summary Report *2918

MINIMAL SUPPORT FORBIVARIATE SPLINES

Carl do Boor and Klaus H8llig

Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut Street D I

DTICt3Madison, Wisconsin 53705 EET

February 1986 M Y2S

LA.. r:~C(Received December 30, 1985)

Approved for public releaseDistribution unlimited

e Sponsored by '

U. S. Army Research Office National Science FoundationP. 0. Box 12211 Washington, DC 20550Research Triangle Park*'..".North Carolina 27709

UNIVERSITY OF WISCONSIN - MADISONMATHEMATICS RESEARCH CENTER

MINIMAL SUPPORT FOR BIVARIATE SPLINES

Carl de Boor1 and Klaus Htllig

1,2

Technical Sumaary Report #2918 .

February 1986

ABSTRACT

Let S denote the space of piecewise polynomials of degree < k and

smoothness p on the regular partition of R2 which is generated either by

the three directions (1,0), (1,1), (0,1) or by the four directions (1,0),

(1,1), (0,1), (-1,1). For the choice '

p - p(k) :. max{p : dim S1 _,,1 2 ' o(N 2 )} ,

(which is the maximal smoothness for which the space S is nondegenerate), we

determine the functions which have minimal support in S. Moreover, we show"- -

that these functions form a basis for

S(n) :- {f C S : supp f C n)

'. , . °

AMS (MOS) Subject Classifications: 41A15, 41A63

Key Words: bivariate, splines, minimal support

Work Unit Number 3 - Numerical Analysis and Scientific Computing

2Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. .2Supported by International Business machines Corporation and National ScienceFoundation Grant No. DMS-8351187.

V-.r . .

iWWWWMW: JUUL VTW

SIGWIVICA=C AND n~LiATImmI

- I [PIC 2A1] - iplie~*results on box-splines [EMC #23201 t

analyze the approximation properties of bivariate smooth piecewise polynomials

on the three direction mesh. In this report we obtain similar results for the

other natural triangulation of which is generated by four directions. InI' /;If .,, " '

particular we extend e results on minimality of support which are useful for

constructing bases with good computational properties. ,r,, IP

Accesion For ,

NTIS CRA&IDTIC TAB "Unannounced -.

Justification

By ......Distribution.

Availability Codes

Dist IAvail and/orSpecial

SEcry

The responsibility for the wording and views expressed in this descriptive

summary lies with MC, and not with the authors of this report.

%

owvi:-::,:;,: ,: :,,::-.:: -:-.> ..::....:.::..:-:.. ...-: -..., .; .:.:...-:1, .--,, .. : v ':.-- -. - : :.-- .:.:.:::.<,- :--,:::: .: :. .--.-. c..

KINIMAL SUPPORT FOR SIVARIATE SPLINRS

Carl do Door I and Klaus 5llig

1'2

1. Introduction and statement of results.

Let S i- denote the space of bivariate spline functions of smoothness p and

(total) degree (_ k on a partition A of 32. In this note we determine the spline

functions of mims'al port for the two regular partitions A1, 2 which are generated by

the unit vectors al, 02 and their sum and difference eI + 02, 02 -

hi A2

(figure 1>

These minimal support elements provide a canonical basis for the subspace of functions in

S with compact support. From a practical point of view, small support of basis functions

is desirable for finite element approximations and quasi-interpolant schemes.

If the degree k of the spline space S - W is large compared to the smoothness

p, elements of minimal support can be easily constructed using Hermite interpolation.

However, in applications one often wants to achieve a certain smoothness with as few . .?

parameters as possible. When k is small compared to P, the smoothness requirements

lead to nonlocal constraints which complicate the analysis. We consider in this note the

extreme case of minimal agree k(P), i.e. the smallest degree k for which the family of

1Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.2Supported by International Business Machines Corporation and National Science PoundationGrant No. DNS-8351187.

7 .;"

Spaces 8h l- {((-/h) f ( 8), h > 0, is donee in C (R2 ). Obviously, the degree kip)

is the most "economical" choice for a given smoothness p (if one wants to minimize the

local dimension of 8). For the two partitions in Figure 1 we have (c.f. [41 for A, and

section 2 for 62)

(1) kv(O) - r(2 + v)(p + 1)/(v + )l , v - 1,2,

where fxl s- sup(n c 2: n < x}. Rughly speaking, the (minimal) degree increases by

2 + V if the smoothness increases by 1 + v. The first values of k are listed in the

table below.

P -1 0 1 2 3

k (0) 0 1 3 4 6

k2 (P) 0 1 2 4 5

2K

To state our results, we need addition notation. For a set n C ,2 we denote by

S) the subspace of functions in S which have support in n. (Note that this differs Je

from 81n, the restrictions of f e S to 0.) By span F we denote the linear span of

the set F. We say that a function K has (uniqua minimal support in S iff S.

span (f) () C S(supp f)

(2) and

n supp f - dm S ) 0

we write SIV as abbreviation for I Py -1,2, we denote the functionsv-2 (0),Ah."

V Vwith unique minimal support in Sv -2 normalized by the condition U . (W is

piecewise constant, 92,, is piecewise linear).

p -2-

%.,d

5'-

4?/

?- S * . . .5.5

//

% %

(00)

upp Uapp 11 1,2 supp U2 , 1 sUpp W2 , 2

<Pigure 2>

The simplest nontrivial examples of minimal support elements are the mhat-function

IN 0 and the Uwart element 1151 K12 C(2 both functions are normalized to satisfy

NMI. - 1. p

/% /

supp~~ >< IIPK" /4%.

her exmls cn b find i n 1141. e elmnt MI,d , defined blow, apeared in ,l

-::-

(L) e functions(.0

d-tilmes

-3- "N,

%%

• -: .': ' -: .: S , : : : : : : : : -_ ---: -, -s- -- -' : ; .-.-.: -.s- v : . : : :p p.. .: . ..: : . .: .: : : .; .... ..: :: : : "

have unique minimal support in

- d(V+1)-2%y,d t lvll2

(ii) The functions

%, i-N %, i-1,2,

have unique minimal support in

- _d(v+l)+v-2

iD,d :-SV

14ere, f * g - fR2 f(. - y)g(y)dy denotes the convolution of two functions f and g.

Figure 4 below shows the supports of the minimal support elements.

Theorem 2. For any convex set 0 C a2 , the integer translates of the functions 14 ,d and

V-Vd with support entirely in Q form a basis for the spaces SVd(O) and S ),d(n)

respectively.

we have not completed our investigations for the spaces A, P - 2 wad 3. One would

expect that convolution of N 2 with the characteristic functions with minimal support in

2 1

yields the sequence of minimal support elements. owever. this Is already false for

F22. F. Sablonniere (14] constructed a C2

quartic element with the same support as H2 .

For the three-direction mesh (V - 1) Theorems I and 2 have been proved in (4]. This

case is included here for completeness. The analysis for the four-diroction mesh A2 is

more complicated because of the two different types of vertices, Z2 and T + Z-, with

SI- (-1/2 , 1/2 ). However, some of the techniques developed in (4] are still applicable.

If the necessary modifications are straightforward we shall only outline the arguments and

refer to 14]. In particular the proof of Theorem 2 for v - 2 is completely analogous to,b.,

the case of the three-direction mesh 14, Prop. 4.2] and will not be repeated here.

In section 2 we obtain a few general results about the spaces wk A . Sections 3 andkA2.

4 are devoted to the proof of Theorem I (for the four-direction mesh).

A version of this report was issued in May, 1984, as C.A.T. report *97, Mathematics

Department, Texas A N N University, College Station, TX.

-4-4.'.

.- * %*..- 4

(0,0) I;

su 1,4 supp

8+1

supp N Supp N 2 22,91,d ,,

(Figure 0>

-5-

2. Auxiliary results.

In this section we obtain a representation of functions in 8 s- vp

in terms of

translates of truncated powers. The four-direction mesh A2 has two vertex types, the one

exemplified by 0 and the one exemplified by

The two differ in that the latter is "singular", i.e. formed as the intersection of two

meshlines, hence is less likely to be on the boundary of the support of elements of S.

For a set of vectors = - the truncated power Ta can be inductively

defined by

(3) CWe~ denot by T C with

We denote by Ti, p + Z+, the truncated power corresponding to the directions -

(CIC2,C3,C4) z- (e1,o1+e21e2,e2 -el) occurring with multiplicities P1 1 P2 1 P3 1 P4

respectively. For example we have

T1 , 0 , 1 , 0 f = 2 " - X, 2 ))d)URh--

i.e. for Ep. - 2, pU < 2, Tp is the characteristic function of the cone spanned by the

appropriate two directions. The second relation in (3) becomes

(4) T p+p, - Tp * Tp. -

It is easy to see that the truncated power Tp is a homogeneous piecewise polynomial of

degree Ep. - 2, with smoothness I pO - 2 across the ray generated by the J-th

lirection and with support in the cone generated by the vectors Pr~r, r - 1,...,4.

Denote by C the cone generated by Ell E4 and let W :- ((u,v)t *(u,v)1 :-

max~jujjj) <_ 1/21. Then S(C)iw can be decomposed into its homogeneous components (cf.

(4, Lemm 2]), i.e.

(5) S(C)fw . •U.k

where :- {f C S(C)IW: f(X-) X 'f). The restriction of functions in S(C) to thesegment r :- I,41/2 is an isomorphism from QO onto the univariate spline space Q'

segment is iomrhinfrm,

5% -6-

5%

; ' . -; .- : : - -? .- ; .- .-? -? ..i - .:i '" - : - :: :: - :; -- -. - 7/ ..- . "i " . '. . .. - : 4 . ? --: .-: ..: - ' - .. . i " : -- .-: .i ' - .q .: " ": : ';, '7 ": --: : . -' 'a -5 , .; , , ' ' ' , ., : ., - -: . ' ' _' ., -.: .--..: .--: -, , , -.' : ' -, ' , ., ? ' --.., ,

of degree I vith the knot sequence (0, 1/3, 1/2, 1), each knot occurrinq with

multiplicity I - p (i.e., Q' has smoothness p). From the smoothness and support of

the truncated powere it is clear that their restrictions to r are B-spline and in Q'.

we identify each 3-splino with a vector q e 4 where q, is the multiplicity of the v-th

knot. Lot AP denote the collection of all such vectors q for the standard B-spline

basis for QI e.g. A' -((2, 2, 1, 0), (1, 2, 2, 0), (0, 2, 2, 1), (0, 1, 2, 2)). it

follows that

(6) Q 0 span Tq~qeA

Denote by the cone spanned by 92' 94 , but with vertex T - (-1/2, 1/2), and let

- r + W. In a similar manner one concludes that

(7) 8

(6 0 span T( 0-r

where Ae ((0. v, 0, is): v, < * - p, v + v - A + 2).

The subepace 8(C) of elements of S - vp having support entirely in C ish,6 2

. infinite-dimensional, but we can specify a truncated power basis for it in the spirit

'pfamiliar from univriate spline theory. xplicitly, we can specify a sequence of truncated

powers with the property that every f c 8(C) has a unique expansion in terms of this

sequence, with the expansion converging uniformly (in fact finitely) on any bounded set.

The formal statement below, in Lemma 1, is to be interpreted in this sense.

Lmu 1.

S(C) - 0 span ((T C, - j): q C AP, i 4 k, j e 32 n C)

(9)

U(T(. - - ), , it, it, j e 2 C)q

Proof. in outline, the proof is as follows. Associate with each vertex v in C the

cone

-7-

4, *,,..*

C, v C 2

2This induces a partial order

w t V e We C e p

end we give a linear ordering of the vertices in C which refines this one. The promised

truncated power basis consists of the relevant truncated powers for each vertex, ordered

according to this vertex ordering.

Obviously, the truncated powers, Tp(. - i), (pi) - (q,j) or (q,J+r), appearing on

the riqht hand side of (9) are elements of S(C). Their linear independence follows from

(5)-(8) and the fact that

(i + W) supp Tp(" i supp Tp,"-i)- ,

if i2 < ij or if (i2 - ij and i < i).

Let f e S(C). We claim that there exist functions f. e 0 span (T . V,), q e A

k) k, v e Z+, such that the support of g : f - ZfV is contained in the union a of

the cones vCI + Z, V C Z+. To show this, we assume that have been definedV-1

and that gv :- f " fU has support in 0 U (vC1 + C). It is clear that

gV(- + vCN)IW e S(C)IW and we define fV as the extension of the truncated power 0

representation for• 4.

From the definition of 0 we see that q(. + vCl)I, e S(C)I-. Therefore by (7) and %

.* (8) there exist functions

TP spa (7. T -CrI4 ),VC

qsuch that g - Ehv has support in C4 + C.

By repeating the above Procedure we can find inductively linear combinations of

truncated powers which agree with f on the cones UC4 + C, V - 1,2,.... This completes

the proof of the Lemma.

It is clear from the above proof that translates of any functions which agree with the

truncated powers near zero and have smaller support also provide a basis for S(C).

Moreover, an analogous version of Lemma I is valid for any cone which is the image of C

%-.,-

V',.',,,,, , " , -,'. '- '- ,', .'. .'. '..,, "- ',,,," ',.'-.% -,,.-.:.''.''.'-.--.".. -.".."-." ." v - v- " ,'- "- "- "-""-''-',', "- """ -" -" v -" -" -" ." -',.' .'' ,.

under an affine mapping which leaves the partition A2 invariant.

From Lemma I we obtain what may be called the "local dimension" of S by counting the

number of elements in the sets A. We have

* A, 14(L - t)-I - 1)+, v .

(10) local dimension at v :-

# A" (2( - ) - L - 1), v t + Z2

It follows in particular that dim S(C) = 0 iff 4(k -p) - k - 1 ( 0. This yields

formula (1) for k2 since a nonzero local dimension is necessary and sufficient for the , .

denseness of Sh in C-(22) [2).

We now specialize the above results for the spaces S wp of minimal.kV(P 2

degree. We have

A3 d-2 - f(d,d,d,d))k2 (3d-2)

(11)3d

A; {(d+1,d+1,d+1,d), (d,d+l,d+1,d+1)}

and denote the corresponding truncated powers by td and ,d' P = 1,2, respectively.

in both cases,

# 1P - 0 for 1 4 k2 (P)

# AP• 0 for I < k2(0)

2

In particular, the "secondary" vertices, I.e., v Cr + Z2, are not active. Therefore

identity (9) reduces to

S2,d(C) - 0 span {td( - J): j c 32 C)

(9,) . '

2S (C) - 0 span {td(* - J): 11 1,2, J C 2 c C)

2, d ~

rrom (4) and the definitions of M, N and t one sees that for x c ,

-9- .

. . . . . . , . - . . - - . - . . . - . . - . . . - . - . - . .

%- -%-.

td(x) - 2 ,d(x)

(12)tId(N) - W2

64 (x),

Therefore we can replace the truncated powers in (9') by the corresponding elements .,.,.

and N respectively.

For later reference we note that for (u,v) e C, v + 0,

td(uv) - udv3d-1 + O(vI

(13) tld(u,v) = Budv3d+ + aud-lv3d+2 + O(v3d+3) ,

t2 ,d(ulv) - Yud-v 3d+2 + O(v3d+3)

where a, B and y are positive constants.ft

.-.

1

-10-

,.--,,

' :f'ff..'. ~ * . -*' ~ ,f.,

% %

,-y2

•U

3. Proof of Theorem 1 (1)

Denote by cony A the convex hull of the met A. We first prove

LOm 2.- for A C 5 1.We set 0il I- conv(0, L 1 , £411+ 121 4)1 02 "-

conv(O, AC21 £12 + 131 -1w) and define Zt (f1la f e 82,d' supp fC(uNv)I

V ) 1) U 10 #Z2 I- (f 9 2 t C 82,dt Supp f C ((ulv)i U - v 1 1) U 0). Then we have

dim Zi - (I + 1 - d)+, £ - 1,2 •

The cases I - 1,2 are not geometrically equivalent since the pattern of the mesh for *-

Sand 2 is slightly different.

Wroof. Consider, e.g., the case i - 1. Let

I- f + conv(O, 1/2, C112 1

since *upp td(- - 1) - + C, it follows from (9) that

21 "f -" %td(. - V1) , % C 2, fie 1 01

Since f vanishes on 0 we obtain from (13) that

1 d-1SAva(u- v) - O, 1 < + IV0

These are min{dW,+) linearly independent constraints on the coefficients a., which

implies dim 1 4 (9 + 1 - d)+. The reverse inequality follows since 1 2 1 1 " VC 1) 10.1

V - 0,....,-d, are linearly independent and in Z1.

7b prove that N2 , d has unique minimal support in s2,d , assune that

supp f Csupp M2,d for some f e 82 ,d. Tma 2, with t - d, implies that f - c 2, dX

on the set ho '- conv(O, d C , +1+ '2t C41. We define inductively a sequence of sets

Aa,A20... am follows. For I - 1,2,... we choose a shortest segment ri with respect toi-1

I I of the piecewise linear boundary of 9, 1- supp N2 , d \ U . Then we define

VMOAi s- {x c si dist.,(x,r±) 4 1/2). This procedure Is illustrated in Figure 5 below for

d -2.

The sets AIN I 0f are contained in sets of the type described in Lant, 2 with

A d. Therefore, we inductively conclude that f - c 2,d vanishes on A1 ,A2 .

-11-

*- ..*..... . .. ..* .**. o.,**. .. **,..... .,.,. . .... ,.

~4d

21

(0,0)0

(Figure S>

ll.4% -

-fw --r~j- FU 7A MA .A .WR i K M% _MM ZM nXm

4. Proof of Theorm I (ii)

We need two ioas.

Lem 3. Let 01 and i b dfine a in / 2 bt with 82, d replaced by -2,"'

Then we have

din Zi a (I - d)+ + (9 + I - d)+ _,

Proof. Similarly as in the proof of Leia 2 we conclude that

- - . .% %,s, - VI,) M a. - 0)

ui- 1,2

where 6 is defined an before. Coparing the coefficients of v3d+ 1 and v 3 d + 2 in the

expression for f on the triangle 6 we obtain, using (13), for I < u < I + 1

£ dav,,Olu - v) - 0,

(%,,a + d- I 0

X;O (a 16+ v2y)(u-v -0 ..

These are mLn{I+1,d+l) + min{1+1.d) linearly independent constraints on the coefficients

4jv, which implies dim Z'I (I - d)+ + (9 + I - d)+,. The reverse inequality follows

Since |2,ld(" " i)Io1' V - 6,...,(A - I - 8)+, and V2,2,d(* - "VO 1)jo V

s ... #(L - d)+,. are linearly independent and in Zj.

La 4. Let A 1- ConvO, d+2, d4+C3 , d[4) and 0' :- ((uv)t v A 1,

u + v ) 2). Zt f U '), then fIt\gj, - 0.

Proof. On the not 0\0, the function f can be written as linear combination of

truncated powers,

f a0,1 t ,d +a0,2 t2,d8

IA 1. 2 9

The truncated powers t I'(O - A 4 ) as well as the function f vanimh on the triangle

e. By using (13) the coefficient of v36+ 1, for (uv) C 6, is

-13- .1

6'..

% %q

0(v~d

This implies a01 -, ad,, - 0. Applying the analogous argument for the triangle

8' :-dt + con(.T we conclude that a0,2 a ai,2 - -- a, 2 - 0. Using in

particular that a0,2 = 0 and again relation (13) it follows that for (u,v) e 0,

f(u,v)- I a v2Y(u -v) d-vd2 +O(V 3d+3v,2-C

This implies that a,,2 a d,2 ' 0 and finally, by using the analogous argument

for 8', that a,',, .. aa.1 - 0. .

Tfo prove that N N2u has minimal support in I 2,d , assume that supp f( C suppz V

f or some f e S 2, Let r be a segment of the piecewise linear boundary of supp N with 0*1

diam, r - 4/2. The set A0 e x supp N: diut.(x,r) -c 1/2) is of the type considered in

Lema 3 with I - d and we conclude that f - c N on A.. We define inductively ai-i

sequence of sets A 1 1A2 ... as follows. If Di :- supp W\ U A Vhas a corner y withVi-0

angle < w/2 we set Ai t- {x: Ix - yl. < 1/2). otherwise we choose two adjacent segments

r, r, of the boundary of Bi with diameter -C d/2 and set Ai :- N a

dist.(x,r u r,) < 1/2). This procedure is illustrated in Figure 6 below forN2,1

4

51

6*

3

2 ~ 0L

<Figure 6>

-14-

%

The sot. 11 aie ootained in sets of the type considered inla Zme 3 with A - 4 - I

(3,6 in Figure 6) or Laem. 4. In either came we i~nductcively conclude that f - cli

,.* ~vanishes cnm ,2,..

r.,.

- s- i!

I ' ,A9

References

1. C. de Door and R. DeVore, Approximtion by smooth multivariate splines, Trans. Amer.

Math. Soc., 176 (1983), 775-795.

2. C. de Door and R. DeVore, Approximation and partitions of unity for certain

translation invariant spaces, Proc., NASA workshop on multivariate splines, Texas

A a N Univ., 1984, 6-13.

3. C. do Boor, and K. H51lig, 9-splnas from parallelepipeds, 3. d'Anal. Math. 42 (1983),

99-115.

4. C. de Boor and K. R81lig, Bivariate box splines and smooth pp functions on a three

direction mesh, J. Coop. Appl. Math. 9 (1983), 13-28.

5. C. de Boor, K. H51lig and S. D. Uiemenuchneider, Divariate cardinal spline

interpolation by splines on a three direction mesh, Illinois 3. Math. 29 (1985),

533-566.

6. C. K. Chui and R. H. Wang, Multivariate spline spaces, 3. Math. Anal. Appl., to

appear.

7. C. K. Chui and R. H. Wang, Spaces of bivariate cubic and quartic splines on type-1

triangulations, 3. Math. Anal. Appl., to appear.

S. W. Dabmen and C. A. Nicchelli, Translates of ultivariate splines, Linear Algebra and

its Applications, 53 (1983), 217-234.

9. W. Dahmen and C. A. Nticchalli, Recent progress in sultivariate splines, Approximation

Theory IV, ed. by C. X. Chui, L. L. Schumaker and 3. ward, Academic Press, New York

1984, 27-121.

10. G. Farn, Subsplines Uber Dreiecken, Ph.D. Thesis, Braunschweig (1979).

11. P. 0. Frederickson, Generalized triangular splines. Mathematics Report 7-71,

Lakehead University (1971).

12. R. Q. Jia, Approximation by smooth bivariate splines on a three direction mesh, to K>

appear.*

13. M. A. Sabin, The use of piecewise forms for the numerical representation of shape,

Ph.D. Dissertation, Mungar. Acad. of Science, Budapest (1977).

-16-

'-,".

: ". " '..'. .' .' .' • . . . . .. . . . .. " . . ... " . . . . .. -... - """ . ''. .' / /..r. . . . . , . • '. ." * -,. . P,.,T -. ,.,.',''. ,, . % - .

14. P. ISblonniers, Do 1'existence do splines A support born& our une triangulation

&pilat~ral 4du plan, Vublication AIO-30, U.R.R. d'I.E.R.A.-Informatique, Univ. do

Lille I (Wob. 1981).

15. P. Zvart, multivariate splines with nondogenerato partitions, SIAN J. Num. Anal., 10

(1973), 665-673.

'

-'17-

.. ,.

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REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

1. REPORT NUMBER ' GOVT ACCESSION No 3. RECIPIENTS CATALOG MUMVER

#2918 1 6

4. TITLE (And 5abilife S. TYPE OF REPORT & PERIOD COVEREDSummary Report - no specific

MINIMAL SUPPORT FOR BIVARIATE SPLINES reporting periodS. PERFORMING ORG. REPORT NUMBER

7. AUTHOR() 6. CONTRACT OR GRA%'T NUMBER(a)

DMS-8351187

Carl de Boor and Klaus H8llig DAAGZ9-80-C-O041S. PERFORMING ORGANIZATON NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

Mathematics Research Center, University of AREA WO uNIT NUMrERSI ~Work Unit Number 3 - .

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February 1986See Item 18 below IS. NUMBER OF PAGES

17I&4 MONITORING AGENCY NAMES AOORESS(if dtteortet boo Contr lliE Office) IS. SECURITY CLASS. (of Lhis repast)

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Approved for public release; distribution unlimited.

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I. SUPPLEMENTARY NOTES

U. S. Army Research Office National Science FoundationP. 0. Box 12211 Washington, DC 20550Research Triangle ParkNorth Carolina 27709

iS. KEY WORDS (Continu an reveride old. of pecseary and Identi y by black bmer)

* bivariate, splines, minimal support

0 ABSTRACT (Continue on reverse side It necoosr and Identi t by block onOb.,)

Let 8 denote the space of piecewise polynomials of degree < k andsmoothness p on the regular partition of It which is generated either bythe three directions (1,0), (1,1), (0,1) or by the four directions (1,0),(1,1), (0,1), (-1,1). For the choice

P - P(k) :a max(p : dim SI [ ] 2 # o(N 2 ) ,

DO, , 1473 tDITION Oil I NOV IS OBSOLETE UNCLASSIFIED (continued)

SECURITY CLASSIFICATION OF THIS PAGE (len Dal* EntereQ

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ABSTRACT (continued)

(which is the maximal smoothness for which the space S is nondegenerate),we determine the functions which have minimal support in S. Moreover, weshow that these functions form a basis for

S(O) : {f C S supp f C 1)

IJ

t.F.

.- %

'ft

1

b

~Ip

F. -

I C

j~I

'ft

U

'I

'ft

'V

'V.-'V

I.,..,

*1.

-. ft **ft

aft. - ~ -. - ft ftft'*ft~** ~ - ft ft ftft~ ft. .~%'V% ~ftqW~.%*~ ~I.ft ~ W* *~ ~ V,, -A 'N *~. ~ V.A. ft .*%~'%.*~. *:.-A*A%~-ft'. ft J.1 ~ \-. .,.I'Vftft.~I9~