Acta Math. Acad. Sci. Hungar.

40 (1--2), (1982), 179--189.

SOME THEOREMS ON UNICITY

OF MULTIVARIATE L1-APPROXIMATION By A. K R O 0 (Budapes0

Introduction

Let X be a normed linear space and consider a finite dimensional subspace

UcX. We say that pE U is a best approximation of xEX if [[x-p[[ =inf {llx-ql] : qEU}. Since U is finite dimensional each xEX possesses a best approximation; moreover, ff the norm in X is strictly convex, this best approximation is unique. But a number of important norms (Chebyshev norm, Lx-norm) are not strictly convex, therefore the study of uniqueness of best approximation in these cases needs deeper considerations. In the present note we shall study the unicity of Ll-approximation. Let K c R m be a compact convex set in the real Euclidian space R m (m_~l) having nonempty interior. Consider the space X=CI(K) of real valued continuous functions on K with norm [[f[[ = (~,, denotes the Lebesgue measure on K.) Let U, be an

f[fld~m.

K

n-dimensionalsubspace of CI(K). U, is called a unicity subspace of C~(K) ff each fECI(K) possesses a unique best approximation out of On. Furthermore, we say that U, is a Haar subspace ff zero is the only function in U, vanishing more than n - 1 times on K. The classical theorem of Jackson and Krein (see [11], p. 236) states that ff m = 1 and K = I = [ 0 , 1] then any Haar subspace of CI(K) is a unicity subspace. Some generalizations of this result for complex and vector valued functions can be found in [5] and [6]. We shall study the unicity of/-a-approximation of functions of more than one variable. By a wellknown result of M~Jgmm~g [8] there are no Haar subspaces of dimension greater than 1 in CI(K), when m > 1. This fact is an essential difficulty in the extension of the theory of Chebyshev approximation to functions of several variables, because the Haar property is a necessary and sufficient condition for the uniqueness of Chebyshev approximation. On the other hand it is known that different families of spline functions are unicity subspaces of C~(I) where I=[0,1], m = 1 (see [1], [3], [12]). Thus it turnes out that in contrast to Chebyshev approximation, the Haar property does not characterize the unicity subspaces of C~(K). This fact gives a hope that in spite of the absence ofHaar subspaces in CI(K), when m > l , there may exist unicity subspaces in CI(K). In the present note we shall give several results on uniqueness of best L~-approximation of continuous functions of more than one variable. In the first section we consider continuous functions of m variables and prove the unicity of Ll,approximation by linear functions. We also give a general uniqueness theorem for separating functions. 12"

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A. K R O 6

In the second section we study functions of two variables. The main result of this section is the uniqueness theorem for Ll-approximation by algebraic polynomials which are linear in one variable and of arbitrary degree with respect to the other variable.

w Let GcCI(K) be a linear subspace of CI(K) and let U c G be a finite dimensional subspace of G. Then U is called a unicity subspace of G if each fEG possesses a unique best approximation out of U. In what follows we shall denote by I~m(A) the Lebesgue measure of A c R m (m=>l). The subspace UcCI(K) is called a Zm subspace if for any pEUN,{0},/Zm(Z(p))=0, where Z(p)={xEg:p(x)=O}. The following lemma gives a sufficient condition for a Zm subspace to be a unicity subspace.

L E ~ 1. Let mEN and let G be a linear subspace of Cl(K). Moreover, assume that U is a finite dimensional Zm subspace of G, which is not a unicity subspace of G. Then there exist fEG and pEU\,{0} such that Z ( f ) c Z ( p ) and

f q sgnfd#m = 0

(1)

K

for any qE U. PROOf. Since U is not a unicity subspace of G, somef*EG possesses two distinct best approximations px,paE U. Then (p~+p2)/2E U is also a best approximation. Therefore 2 If*-(pl+p~)/2l dl~m= I f * - p l l diem+ f If*-P2[ dl~m.

f

f

K

This yields (2)

K

K

2[f*-(px+pz)]21 = I f * - p l l + [f*-pzl

pm-a-e, on K. By continuity of the functions involved, (2) holds for any interior point of K. Gut Kis convex and compact, hence K is equal to the closure of its interior. Thus, finally, we obtain that (2) holds for any xEK. Set f=f*-(p~+p~)/2, p=pl-p~. Then fEG, pEU\{O} and (2) implies that Z ( f ) c Z ( p ) . Moreover by definition of f, 0 is a best approximation off. Then by a wellknown characterization theorem for best L~-approximation (see [11], p. 46) (3)

I\zf(K$) q sgnf dl~m <=z ~ lql dl~m

for any qEU. But using that Z ( f ) c Z ( p ) and U is a Zm subspace we have /~m(Z.( f ) ) = 0 . Thus (1) follows immediately from (3). The lemma is proved. Let Lm+l be the subspace of linear functions on K, i.e. Lm+l={fECl(K): m

f(x)--f(xx, x~, ...,xm)=~aix~+am+x, a~ER, 1-<_i<--m+l}. Evidently, Lm+l is an m + l dimensional Zm subspace of C~(K), THEOREM 1. Let K be a compact convex subset of R r" with nonempty interior (m=>l). Then Lm+1 is a unicity subspace of CI(K). A c t a Ma~hema~ica A c a c l e m i a e S c i e n t i a r u m H u n g a r i c a e 40, 1982

181

UNICITY OF MULTIVARIATE Lx-APPROXIMA.TION

PROOF. Assume the contrary. Then by Lemma 1 there exist fECI(K) and p ( L ~ + l \ { 0 } such that Z ( f ) c Z ( p ) and (1)holds for any q(Lm+~. Set & = ={xER":(-1)ip(x)>O}, K~=KNS~(i=I,2). It can be easily shown that St is convex, thus Ki is also convex (i=1, 2). Since Z ( f ) c Z ( p ) , f d o e s not vanish on the convex set K~. Therefore f has constant sign on K~, i.e. sgnf=~i on Ki (1~,1 = l, i=1,2). If Y1=72, then sgnf=7~ /zm-a.e. on K, a contradiction to (1). On the other hand; if 7x=-y~, then setting ~=7~p we obtain that sgn ~ = s g n f p~-a.e. on K. This again contradicts (1). The theorem is proved. Set now m > l ; 1=[0, 1], K=Im={x=(x~,x~ .... ,x~)~R~: xi~I, l<=i<=rn}. For a given f~Cl(l") we put (4)

f sgnf(x~, ..., xm)d&...dxi_~ dx~+v..dxm

f~*(x~)=

(1 <= i ~_ m).

1,7-~-

LEMMA 2. I f fECI(I m) ( m ~ l )

and for given x'CI

(5) #m-1 {(xl, ..., xi-1, xi+z, ..., Xm)CI'~-l:f(xl ..... Xi-z, x', xi+l, ..., X,,) = O} = 0

then f* is continuous at x' (l<-i<=m). PROOF. For arbitrary ~>0 we set B~={(xz,...,xi_l,xi+l,...,x,,)CI'-l: , ]f(x~ .... , xi-1, x', x~+~..... x~)]=<~}. It follows from (5) that (6)

/~,~-1(B~) ~ 0

(~ ~ 0).

Sincefis continuous on I ~, [f ( x ) - f (x*)[ <=~ for any x, x* EI" suchthat O(x, x*)<= <-0(~). (0(., .) denotes the Euclidean distance.) Therefore we have for [h]=<3(~) ]y~*(x')-y]* (x'+h)[ <_- f [sgnf(xl, ..., x,-1, x', x,+l ..... X,,)-Im-Z

-sgnf(xl,

..o~

&-l,

X t

=

+ h, xi+ 1,

f Ira-1

...~

xm) ]dxl . . . dxi-z dxi+l.., dx,, =

+f=f<-_21,m_l(ao). e

Be

B

This and (6) imply the statement of the lemma. Now we shall consider the special case of functions separating the variables. m

Set C~(I=)= {fECI(I=): f ( x ) = f ( x l , ..., x=)= 2J~(&)}. This is the subspace of i=1

separating functions in C1(I=). Let M,,,cCI(I) be nf-dimensional Haar subspaces of C1 (I), ni =>1 (1 <=i<=m). We assume that each 3/,, contains the constant functions. Set MN={=lqi(xi): q~EM,,, l<=i<-m}. Then MN is an N = =

ni-m+l

dimensio-

nal subspace of C~(I=).

LEMMA 3. MN is a Zm subspace of C* (I=). m

PROOF. Assume the contrary. Then for some q(xl .... , xm)= ~ ' qi(xl)EMN\{O}, i=l

~m(Z(q))>O. Since qCMN\{0} there exists l<=j<-m such that qj is not a constant Acta

Mathematica

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Scientiarum

Hungaricae

40, 1985

182

a.

KRO6

function. Let X be the characteristic function of Z(q). Then

O < l t m ( Z ( q ) ) = l f Xd#,,=p.~(/zdxi)dx,...dx~_ldxj+,...dxr,.

(7)

For a given (xl,--.;x~-l, x~+l,-..,x,,)EI m-l, X(xl .... ,21_l,x,~j+x, ...,2,,)=1 if and only if qj(x)= - __~q~(2~). Using that q~EM,s is not a constant function we obtain that the latter relation holds for at most nj - 1 values of x. Hence fzdx~=O for any (2~ .... ,.2j_~,2j+~,...,~m)EI'-L This evidently contradicts x (7). The lemma is proved. COROLLARY1. I f qE M s is not a constant function, then for any aER,/~{xEI'~:

q(x)=a}=O. The following property of Haar subspaces will play an important role in the present paper. Its proof can be obtained from a more general result proved in [7], p. 41. LEMMA4. Let U. be an n-dimensional Haar subspace of Cx(I). Then for any choice of points O=to
/ q~f~*dx~ = 0

(8)

in

for any qiEM,, (1-<-i<=m). Since pEMs\{0}, p (xx, ..., xm)= ~ p~(x0, where

pIEM,, (l<=i~m). We shall consider two cases. Case 1: for some 1-<_k, j<-m, kr /n

pj and Pk are not constant functions.

Then ,=.~=p, is not a constant function and applying Corollary 1 (for smaller dimension) we easily obtain that Pm-l{(xl, ..., xj-1, xj+~, ..., x~)EI"-~:p(x, ..... x~_~, x', xi+ x..... x.) = 0} = 0 for any x'EL Since Z ( f ) c Z ( p ) ~m-l{(X1 .... , Xj-1,

Xj+l .....

this implies that

Xrn)Clm-l:f(xl ..... Xj-1,

X", X j + l ,

. . . , Xm) -~- 0} ~--- 0

for any x'EL Thus by Lemma 2,f~ is continuous on L Moreover it follows from (8) A c t a M a t h e m a $ i c a A c a c l e m i a e Scien~iarurn H u n g a r i c a e 40, 1982

183

UNICITY OF MULTIVARIATE L1-APPROXIMATION

that f~- has at least ni zeros 0 < 21< x~<... < 2,~< 1 inside L (Otherwise by Lemma 4 for some q*EM, j\{0}, sgn q * = s g n f ~ / q - a . e , on 1, a contradiction to (8).) Therefore using that f ( x l , ..., Xm) = Z~J~(Xi) we have i=1

(9) o =f?(~3

=__fls g n f ( x l .....

X~-l, "2~,xj+l, ..., xm) dxl ... dxj_l dx~+l ... dxm = /'n

. . . , X j - 1 , X j + I , . . . , Xm)~Im-l:i=~afl (Xi) > - - f j ( X s ) } - -

= ,am-l{(X1,

m

--Pm-l{(xl .... , xj-1, xS+I .... , Xm)EI " - l : Z f ( x , ) < -f/(2~)}, i=1

m

Set f j ( x l .... , x~_l, xj+l,..., Xm)= Z~J~(Xi), fjEC~(Im-~).

where s = 1, 2,..., nj.

i=l

i~j

It is easy to see that the equation 12m_l{X~ I m - l : ? j ( X ) > a} = #._l{xE I m - l : f j ( x ) < a}

has at most one solution a = a o , (10)

where

e , " - ' f j (x). xmo_ 1fJ (x) =< a o <- ~max

Therefore by (9) we obtain (11)

fj(2~) = - a 0 ,

s = 1, 2 . . . . , nj.

Furthermore (10) implies that there exists x'C1 m-1 such that f j(x*)=ao,

i.e.

gtt

~J~(x*)=a0, where x'C1 ( l < = i ~ m , i r

Hence and by (11)

i=1

f(x~, Since Z ( f ) c Z ( p )

9

.., X j*_ I , Xs, x j +*~ ,

..

,x*)=0 . .

.

(l
n j).

we get tit

O=p(x~

. . . . , X j*- 1 , Xs, Xj+I, * ..., X*) = Z p i ( x * ) + p j ( 2 ~ ) i=1

(1 "<=S ~__nj).

This relations and the Haar property yield that pj is a constant function, but this contradicts our assumption.

Case 2: at most one o f t h e p , - s is not a constant function. Then p ( x l , ..., xm) = =P(Xk)EM, k\{O } (l<=k<-m). Let O<=x~
x ~ < x k < x s + 1, l<-_i<=rn, i ~ k } ,

where j = 0 , 1, ..., r; x ; = 0 , x ; + l = l . (If x [ = 0 or x ~ = l , then the first or, respectively, the last rectangle is empty.) Therefore s g n f = ? i for x~Bj (]?j[= 1, O<=j<-_r). Acta Ma~heraatica AcacIemiae Scientiarum Hungaricae 40, 1982

184

A. KRO6

By Lemma 4 there exists a qEMnk\{O} such that sgn q(x)=yj while x~O. We again arrived at a contradiction. The proof of Theorem 2 is complete. Let P, denote the set of algebraic polynomials on I of degree at most n. For a m

given n=(nl, n2, ...,nm)~Z'~ we set P*={i~=lp,,(xi): p,,EP,,, l<-i<-rn}. Then by Theorem 2 we immediately obtain the following COROLLARY 2. P~ is a unicity subspace of C~(Im). P~Mn~. The Chebyshev approximation of separating functions was studied by D. NnWMANand H. SHAPIRO[9]. They proved that a best Chebyshev approximation of a continuous separating function of two variables can be given by the sum of Chebyshev approximants of its component functions. Later in [4] it was shown that this is the unique best Chebyshev approximation of a separating function. In connection with Theorem 2 a natural question arises: is the best Ll-approximation of a continuous separating function equal to the sum of the best L~-approximants of its component functions? The following example shows that in general the answer is no. Set m = 2 and consider the function f(xl, x2)=fl(xl)+~(x2)CC'~(I~),

E~LE. where

fl(xx) =

1 f l "1 x x - T , x ~ [~-, 1];

3~(x2) =Xz-~,X2E[O, 1].

It can be easily verified that 0 is the best Lx-approximation of bothf~ andf~ on I by constant functions. Assume that the best/-a-approximation off(x1, x~) by constants is also zero. Since t h ( Z ( f ) ) = 0 , f sgnf(xl, x~) dxl dx~ = O, 19 i.e.

(12)

f l * ( X l ) d x I = O,

where f ~ ( x l ) = f sgnf(xl, xz)dxs. It can be derived by simple calculations that I

f~(Xl)=2fl(Xl).

But this contradicts (12).

w In this section we shall study the Ll-approximation of functions of two variables. Thus we set m = 2 and consider the space C1(1~), where I=[0, 1]. Theorem 1 in the previous section states that the subspace of linear polynomials L3= = {ax + by + c: a, b, c ER} is a unicity subspace of C1(I~). This is another illustration Acta Mathernatica Acaclemiae Scient~arum Hungartcae ~0,

1982

UNICITY OI~' MULTIVARIATE L r A P P R O X I M A T I O N

185

of the fact that the Haar property is not necessary for the uniqueness of best L1approximation. A theorem of this type can not hold for Chebyshev approximation. L. COLLATZ[2] investigated the unicity of restricted Chebyshev approximation problem on the plane. He verified the uniqueness of Chebyshev approximation of differentiable functions by linear algebraic polynomials of two variables. But later T. J. RIVLINand H. S. SnAl'mo [10] proved that linearity of the approximating polynomials is essential in this case. They showed that if the approximating algebraic polynomials are quadratic with respect to at least one of the variables, then there exists an infinitely differentiable function possessing more than one best Chebyshev approximation. In this section it will be shown that linear polynomials are not the only unicity subspaces of Cl(IZ). In particular, we shall verify the uniqueness of Ll-approximation when the approximating polynomials are linear with respect to only one of the variables. Let k, nCN, k<-_n and let U~' and U* be Haar subspaces of C~(I) of dimension k and n, respectively. Moreover, we assume that U~,jc * Un. * Consider an arbitrary continuous strictly increasing function r on I and set Un+k = {q(x, y) = ~o(y)qk(x)+ pn(x):qkE U~, pnE U* }. This is an n+k-dimensional subspace of C1(I~). REMARK.I f U~* ---- U~, * then U,+k = U~ can be considered as the tensor product of U* and the linear span of 1 and r The linear span of I and tp is a 2-dimensional Haar subspace, hence in this case U~ is a product of two Haar subspaces. TnEORI~M 3. U~+k is a unicity subspace of C1(I~). PROOF. First of all we shall verify that U~+k is a Z~ subspace. Take an arbitrary

q(x,y)=q~(y)qk(x)+p~(x)~U~+k\{O}. Let us prove that /~(Z(q))=0. Set E(~)= = {(x,y)CIz: x=2, yCI} (2EI) and let xi~I (l<=i<-s) be the all common zeros of qk and pn, s<=n--1. Then evidently E(xl)~Z(q). Further, if 2 ~ x l for each l = t = s and E(2)fqZ(q)~ f~ then qk can not vanish at 2. Therefore in this case E(2)NZ(q) consists of a single point {X=2;y=~-l(--p~(YO/qk(2))}, where q~-i denotes the inverse function of q~. Thus pl(E(2)NZ(q))=O for almost all 2EL hence p2(Z(q))=0. This implies that U.+k is a Z2 subspace. Assume now that the statement of the theorem is false. Then by Lemma 1 there exist fECI(I ~) and q*(x, y) = ~o(y)q~(x) +p*(x)E U,+k\{0} such that Z(f)cZ(q*) and for any q~U,+~. We shall consider several cases.

Case 1: q* is the zero function. Then q* (x, y)=p* (x), where p*~U~*'N{0}. Therefore Z(q*)= 0 E ( x i ) ,

where 0<=XI~<...
A c t a M a t h e m a t i c a A c a d e m i a e S c ~ e n t i a r u m H u n g a r i c a e 40, 198~'2

186

A. KRO6

x~_x
Thus considering ~ as an element of U~+~ we have on B~ ( l ~ i < = r + l ) , i.e. f ~ , s g n f d ~ > O . This contradicts (13).

Case 2: q~Uk*'X,{0}, q~

qk(x)+p~(x) is the zero function. Then q*(x,y)=


Bi =

xi_l
~-
Since Z(f)cZ(q*), f preserves sign on each A~ and Bi. Hence sgnf=~?i~

(14)

8i

where

on

Ai

on

B i,

I?il = ]e~l= 1, 1 <-i<=s+ 1. Furthermore 03) implies that

f p.(x) f *(x)dx = O (p.E U*),

(15)

I

where f*(x) =

1 f sgnf(x, y)dy. From (14) we easily derive that f l, (x)=~-(yz+e~)=]~i I

for x~=-l----O/A-a.e. on L But ~kEU~cU*, hence it follows from (15) that f(?/g=0 #l-a.e. o n / , i.e. f~'=0 /q-a.e. on I. Thus fl~=0 and therefore ~ = - e ~ for each l ~ i < = s + l . Using again Lemma 4, consider a polynomial qkEU~'~{0} such that sgnqk=e~ for

x ~ , l < x < x i ( l = ~ = s + 1). Furthermore, set

{

(1/

~t(x,y)=q~(y)qk(x)-q~ -~ qk(x)=

(11} = {-.,el on on

sgn~t(x,y)=sgnqk(x)sgn ~0(y)--tp~-

A,_{., on B~--e~

on

By

This and (14) imply that f ~ sgnfd/z~>0. Thus we again obtained a contradiction to (13).

Case 3: q ~ U ~ \ { 0 } , 9

q~(x)+p*(x)CU*\{O}. Let O=x~
...
be the all common zeros of q~ and 1)*, r<-k-1. As it was shown above if x#x2 for each l~i~-r, then E(x)OZ(q*) contains at most one point. Since Z(f)~Z(q*), E ( x ) ~ Z ( f ) also contains at most one point in this case. Thus applying Lemma 2 " ' (1 < we obtain thatf~(x) is continuous while x~_z
UNICITY O F MULTIVARIATE L,.APPROXIMATION

187

r+l

Q = U (x~_x, x~). (Otherwise f * changes sign at at most n - 1 points and this, in i=l

view of Lemma 4, contradicts (15).) Let xi, 1 ~ i ~ n - r , be any distinct zero of f * on Q. Since ~iE Q, xi is not a common zero of q~' and pn, * hence E ( ~ ) contains at most one zero off. On the other hand E(~i) should contain a zero offi because otherwise f~'(~) equals 1 or - 1. Thus E(Y~)AZ(f)= {x=~i; y=yi}, where, as it was shown above, (16) Yi = ~o-l(-P*(Xi)/q:)Y~,)), 1 ~ i ~ n-r. Moreover f must change sign on E(~i), hence sgnf(~,y)=fli while 0 < y < y i and sgnf(~i,y)=-fli while y i < y < l (lfl~l=l; l ~ i ~ n - r ) . Thus

0 =A*(~,) = f s g n f ( ~ ,

y) dy =/~,y,-/~i(1-y~) = Bi(2Y,-1).

1

1

Therefore we obtain y~=~- ( l ~ i ~ n - r ) .

(17)

9

From this and (16) we easily derive

qk(2f)+pn(2t) = O,

1 <-- i ~_ n--r.

Furthermore qk(xi)=p,(xl)=O * ' * ' (l<--i<=r), where x ~ 2 i for all i a n d j . This together with (17) imply that the polynomial ~p{1)q~(x)+p*(x)EU*\{O} has n distinct zeros on L Thus we arrived at a contradiction to the Haar property. The proof of the theorem is complete. Consider the set of algebraic polynomials on 12 of degree m with respect to x and k with respect to y

P,.,k =

ai, r xryi:ai, rER

(lc, mEZ+).

It can be easily seen that Po, k and Pro,0 are unicity subspaces of C1(I2). This can be proved by the arguments used in Case 1 of the Theorem 3. (Note that t'o,k=P~, Pm, o : P ~ . ) Moreover from Theorem 3 we can derive the following COROLLARY 3.

Pro,1 and Pl,k are unicity subspaces of C1(I~).

The question whether Pm,k is a unicity subspace of CI(F) if k, m->2, remains open. We have a feeling that the answer to this question is affirmative. Even the attempts to settle the case when k (or m) equals 2 were unsuccessful. We are able to prove only a weaker result. Set

P*m,2={Pm(X)--}-q2(y), where

PmEPm, q, EP,}.

Then we have the following TI-~OR]~M 4. For any mEN, P*,2 is a unicity subspace of Cl(F). PROOF. Evidently P*,2 is a Z 2 subspace. If the statement of the theorem is false, then by Lemma 1 there exist an fECI(I ~) and p*(x, Y)=Pm(*x)+q~(Y)E* P*m,~-\{0} Acl;a Ma~hemat~ca Acaclerniae S c i e n t i a r u m H u n g a r i c a e 40, 1982

188

A. KgO6

such that

Z(f)cZ(p*) and f p(x,y)sgnf(x,y)dydx

(18)

=

0

( p E / ' L*O .

In particular (19)

f p,.(x)A*(x)

dx = 0 (p,,CP,,,),

I

where

f*(x)=f sgn f(x, y)dy. I

Ifdeg q~'<2, then p* can be written in the form p*(x,y)=aly+ff,,(x) (ff,,EP,,,). In this case we can arrive at a contradiction analogously as in the proof of Theorem 3. Therefore we may assume, that degq*=2. Then the solution of the equation p*(x, y)=O is given by two curves (20)

,./y

= y0+

~I. [xE I',

. tY = Y0--

73. ~xE I ,

where p.EP,,,yoER, l"={xEI:~,,(x)>=O}. Since on T1 and V2.

Z(f)cZ(p*), f c a n vanish only

Case 1 :/~, is a constant function. Then f c a n change sign only on the line segments {xEI;y=yl} and {xEI;y=y~}, where y~=yo+l/ff-~,y2=yo-1/-~ and yl,y2ER. Hence one of the polynomials fl, e(y-ya), O(y-y~), 6(y-y~)(y-yz)E EP*,z (fl, e, 0, ~ = • has the same sign as f/z2-a.e, on I z. But this contradicts (18). Case 2: p~ is not a constant function. It follows from (20), that for any xEI; E(x) contains at most two zeros off. Thus by Lemma 2 f~* is continuous on 1. Moreover, it is easy to see that for any xEl, f*(x) can take only one of the following values: _ 1 ; _ ( 1 - 2 y 0 - 2 1 / ~ ) ; +__(1--2Y0+21/~); +__(1-4~). Since PmEP= is not a constant function we obtain that f** has at most 3m zeros on L Furthermore, by continuity off* and (19) we have that f1* has at least m + 1 zeros on L Let OO. Then i~,~>O for x~.zl-O and p=(x~'_l)=0 if i~m(X~_,)<0. Since i~= is positive and increasing for x~_~y~ on this interval. Set 5~(x)=max {0, min {I, ~(X)}}, i=0, 1. Clearly, ~, (~2) is continuous a n d increasing (decreasing) and 0~_9~<=,2~<_-1 for x*_,
~(x)
B 2 = {XL1 < x < x l ;

92(x) < y < 91(x)};

=

< x < x,;

Aeta Mathematt, ca A c a d e m i a e S c i e n ~ a r u m Hungar~cae 40, 1982

189

U N I C I T Y OF MULTIVARIATE L1-APPROXIMATION

(Some of these regions may be empty.) Since f does not vanish on Bi, f preserves sign there, i.e. sgnf=fl~ on B~ ([fl~I=l; 1<_-i<=3). Therefore we have for x*_l< <:X
1

7~(x)

A* (x) = , f sgnf(x, y) dy = ~~ s g n f ( x , y ) d y + h(f~) sgnf(x,y)dy+ ~Cx)

+ f

sgnf(x, y) ay =

=

0

[fll, if f l l = f l , . = f l a ; Jfll(1--29=(x)), if fll = fl~ =-fiB; = ]fl~(1--291(x)+292(x)), if /~1 =--/~2 =/~s;

if Since 91 (92) is increasing (decreasing) for x*_:
Acta Mathema~ica Acaclemiae Scient~arum Hungarieae 40, 1982