Aeta Math. Hung.

44 (3--4) (1984), 409---417

SOME THEOREMS ON BEST L1-APPROXIMATION OF CONTINUOUS FUNCTIONS A. KRO6 (Budapest)

In the last years the problems of unicity of best Ll-approximation of continuous functions have been widely investigated. The development of Ll-approximation theory was inspired by the fact that the unicity of best Ll-approximation does not impose strong restrictions on the approximating subspaces such as the uniqueness of Chebyshev approximation. Indeed, by the famous theorem of A. Haar, only the Haar subspaces can guarantee the unicity of best Chebyshev approximation of each continuous function. This excludes many important families, for example spline functions. Furthermore, Mairhuber's result on the absence of real Haar subspaces of functions of several variables shows that there is no possibility to achieve unicity of Chebyshev approximation in the multivariate case. Since the analogue of Haar's theorem is not true in case of Lx-approximation of continuous functions, the family of unieity subspaces is wider in this case. A remarkable illustration of this phenomenon was given by Galkin [2], Strauss [10], Carroll and Braess [1], Strainer [8] and others who verified the unicity of best L~-approximation of real continuous functions by different families of spline functions. In [6] we studied the unicity of multivariate L~-approximation and gave some examples of unicity subspaces of functions of several variables. In the present note we make an attempt to unify the study of best La-approximation considering vector-valued functions on convex bodies in R" (n~ 1). Applying our main result (Theorem 3) we shall obtain some known, as well as new results. In particular, Theorem 3 implies the uniqueness 0fbest L~-approximation of continuous complex-valued functions by different families of complex splines. This extends the results of [1], [2], [8], [10] to the complex case. Let us start with some notations and definitions. Let K c R " (n=>l) be a convex body, i.e. K is a convex compact subset of R" with nonempty interior. By Cm(K) (m= > 1) we denote the linear space of continuous functions f: K-~R", i.e., Cm(K)= = { f = ( f a , ...,f~): f. is real and continuous on K for each l~i~m}. Let W = = {w: w is measurable, bounded and positive on K} be the set of weight functions. Then taking an arbitrary wEW we denote by C~(K) the linear space Cm(K) endowed with the weighted L~-norm (1)

[]J'l]~,,= =

flfl~wdx (fEC=(K)).

K

Here I" [mdenotes the Euclidean norm in R m. Let Ube a finite dimensional subspace of

C"(K). We shall study the approximation of functions fEC,~(K) by elements of U in the norm (1). u0E U is called a best Lw,,~-approximation of fEC,~(K) if [If-uoll,~,,.<=llf-ullw,,. for each uEU. If 0 is a best Lw,,~-approximation o f f out of U then we say t h a t f i s Lw,,,-orthogonal to U. The subspace U is called a unicity A c t a M a t h e r n a t i c a H u n g a r i c a 44, 1984

410

A. KROO

subspace of C~(K) if each fEC~(K) possesses a unique best L,,,,,-approximation out of U. In what follows (., .) will denote the inner product in Rm; for fEC~(K) we set Z ( f ) = {xEK: f(x)=0},

sign,,f=ff/[flm,

K\Z(f)

on

l 0,

Z(f).

on

REMARK 1. In the case when m = l we shall omit the index m, i.e. CI(K)= =C(K), C~(K)=Cw(K), I" 11=I:1, signlf=signf, [].Hw,~=ll.[lw, etc. First of all we shall give a characterization theorem, which is an extension of a known criterion to the vector-valued case (see Singer, [7], p. 46). For the sake of completeness we include the proof, which is based on standard variational arguments. THEOREM 1. Let U be a finite dimensional subspace of C"(K), wEW, fEC,~(K). Then the following statements are equivalent: (i) f is Lw,m-orthogonal to U; (ii) we have for any uE U

r',,f:) (sign" f' u) wdxl <=zm f [ul'wdx"

(2)

PROOF. (i)~(ii). Consider an arbitrary uEU and t>O. Since 1 ([if+ tun~,,,-I[fU~,,,)= +

f

K\Z(:)

(If+ tu[,~-If Ira)wdx+

+ f iut=wdx = f 2(f u)+tiul~ wdx + f [UImwdx, zo') K\zc:) If + tul,, + ifl~ z(s) we obtain by the Lebesgue dominated convergence theorem

(3)

f

li+mo +(llf+tull.,,.,-Ilfll.,:.,)=

(sign,,fu)wdx+ S [u[mwdx.

KNZ( f )

Z(f)

It follows from (i) that the quantity on the left hand of (3) is nonnegative for any uE U. This immediately implies (ii). (ii)=,(i). We have for any uEU

Ilfll~,m = =

f

K\Z(f)

f ifl,,wdx = f (signmfif)wdx = xc\z(f) KNz(D

(sign,.fif--u)wdx+

f

K\z(f)

(sign.f, u)wdx~

<= f If-ul,.wdx+ f lul,.wdx Ilf-ull..,.. =

K\Z(S)

Z(y)

The proof of the theorem is complete. Our next theorem characterizes the unicity subspaces of C~(K). This characterization is based on inner properties of approximating subspaces. In the real case this result was given by Strauss [12]. For the complex functions a similar statement was Acta Ma~hematica Hungar~ca 44, 1984

411

ON BEST LI-APPROXIMATION OF CONTINUOUS FUNCTIONS

proved in an earlier paper of Havinson [3]. Our theorem generalizes the above mentioned results for the vector-valued functions. For UcC"(K) we set U*={u*ECm(K): there exists u'EU such that u*(x)= =u'(x) o r - u ' ( x ) for any xEK}. THEOREM 2. Let wEW and let U be a finite dimensional subspace of C"(K). Then the following statements are equivalent: (i) U is a unicity subspace of C~(K); (ii) the only element of U* which is L,,,"-orthogonal to U is zero. PROOF. (i)~(ii). Assume the contrary. Then by Theorem 1 there exists u*E U*\{0} such that

{ f

(4)

(sign"u*,u)wdx <- f lUlmWdX

K\Z(u*)

Z(.*)

for any uE U. By definition of U* there exists u'E U",,{O} such that u*(x)= +__u'(x) for each xEK. Set ~=u*+~u', where 0<~<1. Evidently Z(~)=Z(u')=Z(u*). Moreover, since u'(x)=~(x)u*(x), where 7(x)ER, lT(x)]=l, we have for any

xEK\Z(fO u*+eu' (1 +~?)u* sign"g = 1~{---~ = fu.+~u,[" = ii+e?l lu.l" = sign,,u*. Thus it follows from (4) and Theorem 1 that fi is Lw,"-orthogonal to U, i.e. u*EC~(K) has two distinct best Lw,"-approximations, 0 and -~u'E U',..{0}. (ii)=*(i). Assume again the contrary. Then there exists fEC~(K) and Ul, u~E U, ul#u2 such that Ul and u2 are best Lw,"-approximations of f. Since (u~+u~)[2 is also a best Lw, ~-approximation we have

2/{f-

f

u~+2u2 ~ wdx = t( ( ] f - Ull, + ]f--u2,")wdx.

Thus by continuity of the functions f, u~ and u~ we obtain that (5)

l(f-- Ul) "~-(f-- u2)]" = If-- Ull" . -~ I f - u~l"

for any xEK. Therefore (6)

( f - u l , f - u2) = I f - UalmIf-- U21" (xEK). Set f * = f -

2

ECw(K),u=u~-u2EU~{O}. By(5), Z(f*)cZ(u). Consider an arbitrary 2EK",,Z(u). (6) implies that at 2, (f-uO=~z(f-u.~)or ( f - u 2 ) - = ~ ( f - u x ) , where ~ER, ~ 0 . Moreover, since 2EK\Z(u), ~ 1 . Assume, e.g., Zl 1 - - ~ U 2

that (f-uO=o~(f-u~). Then ]-- I Z ~ and we have by easy calculations -

I

_~

= -ff { ( f - u ~ ) + U ' - . ~ ) }

=

1+

(f-u~)O+~)

-

20-~)

(U 1

-

-

U2) ,

i.e.

(7)

f* = c(ut--u~)

(cER, c # O) Acta Mathematica Hungarica ~4, 198~

412

a.

KRO6

at ~EK\Z(u). The relation (f-u2)=a(f-ul) also leads to (7), Thus (7) holds for any xEK\Z(u) with a constant c depending on x. This implies that (8)

sign,, f * = ! sign" ( u l - u2) = 4- sign,, u

for any xEK\Z(u). Set u*= [uf,. sign,,f*. Since u is continuous and sign,.f* has discontinuities only at Z(f*)cZ(u), u*EC"(K). Moreover (8) implies that for any xEK\Z(u) u* = [ul" sign,,f* =___[u[,. sign"u =:ku. Hence u*=+_u for each xEK, where uEU\{0}. Therefore u*EU*\{0}. By our construction f * is Lw,,,-orthogonal to U, hence (2) implies that

[ f

(sign"f*,u>wdxl<=

Furthermore, for any

xEK\Z(u)=K\Z(u*)

(9)

K\zO'*)

sign"u* = ~ u*

(10)

f lul,.wdx (u
zfy*)

holds. Thus (9), (10) and relation

=

lu[,.sign,.f* = sign,.f* lul~

Z(f*)cZ(u)=Z(u*) imply that

I f (sign,.u*,,,>wdx = I f w&1_-< K\Z(u*)

< =

K\z(u*)

l,,\zcsf.,wdxI+ Iz,..,\f s.,
f

z(u*)\z(f*)

u>wdx[ <=

;ul,.wdx= f lul"wdx z(,*)

for each uE U. Finally, it follows from Theorem 1 that u*E U*\{0} is L,~,,.-orthogonal to U, a contradiction. Theorem 2 shows that the property of U to be a unicity subspace of C~(K) is closely connected with the weight w. Now we areinterested in such U, which are unicity subspaces of C~(K) for each wEW. This property of U is already independent of the weight. Set m = l and let U be a finite dimensional subspace of C(K). DEFINITmN. We say that U satisfies the Property A (is an A subspace of C(K)) if for any u*EU*\{0} there exists u'EU'~{0} such that a) u ' = 0 a.e. on Z(u*) b) u'sign u*=[u'] on K\Z(u*). In the particular case when n = 1 and elements of U have finite number of separated zeros the notion of A subspace was introduced by DeVore and Strauss (see [11]). A further investigation of Property A was given by Sommer [8], [9]. Let us give some important examples of A subspaces. EXAMPLE 1. Let n----l, K=[a,b] and let U be a Haar subspace of C[a,b]. Then by the properties of Haar subspaces we can easilyprove that U is an A subspace

of C[a. b]. Ae$ct M a t h e m a t i c a H u n g a r i c a 44, I984

413

ON BEST /.x-APPROXIMATION OF CONTINUOUS FUNCTIONS p

.

k

K=[a,b]; U=Sp, k={~a,x'+2%+,(X--X,y+, where a
Set

n=l,

--

i=1

is the set of splines of order p with k fixed knots. Then Sp,k is an A subspace of C [a, b]. (For the proof see Somrner [9].) EXAMPLE 3. Let n = l , K=[a,b], a=Xo
[

f

tulmw*ax

(signmu*, u) w*dx I <= f

K\Z(u*)

z(u*)

for any uE U. Since u*=(u~, ..., u*)EU*\{0}, there exists u=(ul, ..., u,,)EU',,.{0} such that u*(x)=+_+_u(x) for any xEK. Therefore u~(x)=+_u~(x) for any xEK and l<=i<=m. But uiEUi, hence u~EU* (l<=i<-m). Moreover, for some l<=j<=m, u*E U*~{0}. By the A property of U~ there exists u'E Uj\{0} such that u ' = 0 a.e. on Z(u~) and lu'l=u sign u* on K"xZ(u~). Furthermore (11) implies that

ujw*dx <= f lulw*dx K\z(u*)

Z(u*)

for any ujEUj. Setting uj=u' in (12) and using that Z(u*)cZ(u*) we have

0=

f K~Z(u*)

u'w*ax=

f lu*b. lull lu'[w*dx" a, Kx,~Z(u j)

Therefore lu .ltu'lw*=O a.e. on K",,Z(u*), i.e. u ' = 0 a.e. on K\Z(u*). But u' is also zero a.e. on Z(u~), hence we obtain that u' is identically zero, a contradiction. The theorem is proved. Ar

M a t h c m a t ~ c a H u n g a r i c a 44~ 198a~

414

A. K R O 0

This theorem implies a number of interesting corollaries. E.g., using Example 1 we obtain that a Cartesian product of Haar subspaces of C[a, b] is a unicity subspace of C~[a, b]. This Jackson--Krein type theorem was proved in [5] by a different method. In a special case (complex functions) it was verified earlier by Kripke and Rivlin [4]. Moreover, we can extend the results on unicity of best Ll-approximation by real splines given in [1, 2], [8--10] to the case of approximation by Cartesian products of families of spline functions. In particular, this implies the unicity of best Ll-approximation of complex valued continuous functions by certain families of complex splines. Let us give a typical result of this kind. Let C[a, b] be the space of complex valued continuous functions on [a, b] P 2 / P endowed with the Lw-norm on [a, b]. (wE W is arbitrary.) Set Sp, k={/==0cixk + -TC,+,(X--XOP+, where a
arbitrary complex number~~}. This is the . set of complex splines of order p with k fixed knots. Then we have the following COROLLARY 1. Each fE C[a, b] possesses a unique best L,~-approximation out

of Sp, k. PROOF. Evidently, this approximation problem may be considered as approximation in C~[a, b] by elements ofSp,gXSp k. Since Sv,k is an A subspace of C[a, b]. Theorem 3 immediately implies the need~'~J statement. By Theorem 3 and Example 4 we obtain that U~ ... X U ~ is a unicity subspace of nt

C,~(K), where K is a convex body in R" and wEW is arbitrary. For r e = l , w - i this result was given in [6]. EXAMPLE 5. Set m = l , n = 2 , w = l , i=[0, 1] and consider the space C1(12). It is proved in [6] that the product of a 2-dimensional and a k-dimensional Haar subspace is a unicity subspace of CI(P). For example, I

k--i

i=0 j=0

the set of algebraic polynomials of degree 1 and k - 1 in x and y, resp., is a unicity subspace of Cx(I2). On the other hand, it can be easily shown that Pl,k-~ is not an A subspace of C~(I2). By Theorem 3, the A subspaces are unicity subspaces of Cw(K) for any wE W, while Example 5 shows that in general for a given weight wE W, A subspaces are not the only unicity subspaces of C~(K). This leads us to the following problem: is it true, that only A subspaces of C(K) have the property to be unicity subspaces of C~(K) for each wE W? Now we shall verify that the above conjecture is true in case of approximation by one dimensional subspaces. THEOREM 4. Let ~oECm(K), whereKisaconvexbody in R'L Set U={c~p: cER}. Then the following statements are equivalent: (i) U is a unicity subspace of C~,(K) for each wEW; (ii)/r is connected; (iii) U* = U. Acta Mathernatica Hung'ar~ca 44~:1984

O N BEST L 1 - A P P R O X I M A T I O N O F C O N T I N U O U S F U N C T I O N S

415

PROOF. (i)~(ii). Assume the contrary, i.e. (i) holds and K\Z(~o) is disconnected. Then K\Z(~o)=AUB, where A I ~ B = 0 and A and B a r e open ( i n K ) n o n e m p t y subsets of K. Since the Lebesgue measures of A and B are positive and q0 does not vanish on A and B, a = and b : Set

f >l=dx>O

l=dx>O. fI

A

(13~

B

,

1/a, ff~={l/b, [

1,

on on on

A B Z(~o),

{

qo, on qo*=-9, on 0, on

A B Z(rp).

Clearly, mEW. Furthermore, ~p*(x)=+_cp(x) for any xEK. Let us prove that (p*EC~(K). Let xoEK and xi~xo, xiEK. If e.g. x0EA, then using that A is open in K we get that xiEA for i>=io. Hence (13) and the continuity of (p imply that ~o* is continuous at x 0. Assume now that x0EZ(~o), i.e. q0*(x0)=0. Since [(p*(x,)]~= = [(P(Xi)l~, the continuity of(p at x0EZ(~p) implies that (p* (xi)-~0. Thus ~0*ECm(K) and therefore ~o*EU*\{0}. Further, using (13) we have

f

K\Z(~*)

(sign~o*,~p)~dx = 1 - f[q~tmdx--~ a .4

l~oLJx=0.

Hence it follows from Theorem t that (p* is L~,,,-orthogonal to U. Since ~o*EU*\{0} this contradicts Theorem 2. (ii)~(iii). Take an arbitrary u*EU*\{0}. Then u*(x)=co~,(x)cp(x), where coER\{0}, 7 (x)= _+1 (xEK). Consider an arbitrary xoEK\Z(qO. Set _<= "-<:

q~= ((Pl, ~P2.... , (Pro),where ~p~EC(K) (1 = z= m). Since Z(cp)= fi Z(qh), xoEK\Z(~pj) i=l

for some l<=j<=m. Setting u* =(u~, u~, ..., u*) we have u~(x)=co~,(x)~pj(x). But ~pj(x0)#0, hence 7(x)=u~(x)/(Co(pj(x)) in a n e i g h b o u r h o o d o f x0. Thus y is continuous at xo, i.e. 7 is continuous on K\Z((p). Moreover, 7 = 1 or - 1 on K\Z(~o) where K\Z(~p) is connected. Then for any xEK\Z(q~), ~(x)=?0, where y0=l or --1, i.e. u*=co~oqoEU. (iii)~(i). Take arbitrary wEW, u*EU*\{0}. Since U*=U, u* is not L .... orthogonal to U. Hence Theorem 2 yields that U is a unicity subspace of Cm(K). The proof of the theorem is complete. COROLLARY 2. Let m= 1, Uc-C(K), dim U = 1. Then the following statements are equivalent: (i) U is a unicity subapce of C,~(K) for each wEW; (ii) U is an A subsTace of C(K). PROOF. (i):~(ii) follows from (i)~(iii) of Theorem 4. (ii)~(i) is a consequence of Theorem 3. We now apply Theorem 2 to the study of best simultaneous Ll-approximation of continuous functions. Let fl,f~ .... ,fmEC(K) and let U be a finite-dimensional sub space of C(K). In [13] Carroll and McLaughlin studied the question of uniqueness of the element fie U for which m

(14)

/

i=1 ~ ]~-~[dx = .~vinfd f i__~ 1"= [fru[dx. A c t a Matherna~iea H u n g a r i c a 44, 1984

416

A. KRO6

This problem of best simultaneous Ll-approximation was considered in [13] in case when K=[a, b] and U is the set of algebraic polynomials of order at most k. It was shown by the authors that (14) has a unique solution if m is odd. We now present a different approach to the question of best simultaneous L1approximation. For given fl,f~ .... , f , EC(K) we are looking for the solution of

(15)

f(

K

"=

(f,- )91/ ax=inf uEU

( .= ( -u)91 ax

The existence of a solution of (15) follows by standard compactness arguments. Moreover, it can be proved that if U is an A subspace of C(K), then for any m ~ l and f~ ..... fmEC(K) the solution of (15) is unique. For a given UEC(K) we set U(,,)={(u . . . . . u ) E C m ( K ) : uEU}. Then U(,) is a finite dimensional subspace of Cm(K) and the approximation problem (15) is equivalent to the Lx, m-approximation of f = ( f ~ , f2, ..., f,)ECT(K) by elements of U(,) (here w~-l). We have the following THEOREM 5. Let U be an A subspace of C(K). Then for any m>=l, U(m) is a unicity subspace of C'~(K). This theorem immediately implies the needed statement. COROLL~Y 3. Let U be an A subspace of C(K). Then for any m>-i and ]'1, ...,f, EC(K) there exists a unique solution of (15). PROOF. Assume that the theorem is false. Then by Theorem 2 there exists

U,nEU(,)\{O} which is L~,~-orthogonal to U(,), i.e. by Theorem 1

[ f

(16)

(sign.u*, u('))dx ~

f. [u('l[,dx

x\z(~*,*.)

Z(u,,,)

for any u(~)EU(m). Furthermore, u*,(x)=7(x)u(om)(x) for some u0(")EUo,)\{0}, where 7(x)--- +1 (xEK), u0~')=(u .... , u) (uEU\{0}), u*~=(~l, ..., ~,). Therefore ~i(x)=?(x)u(x), l<-i<=m, hence ul=u2 . . . . . ~ , = u * E U * \ { 0 } . Property A of U implies that for given u*E U*\{0} there exists u'E U \ { 0 } such that u'=0 a.e. u' sign u* = [u'l

(17)

on on

Z(u*) IC'.,,Z(u*).

Set u'=(u', ..., u')EU(,). Since Z(u*)=Z(u*) we have by (16) and (17)

0=

f

dx =

K\Z(u*)

=

f

f

u dx =

/C\Z(u*)

I/-mu" signu*dx= ~

g\Z(u*)

f

lu'ldx.

K\Z(u*)

Thus u ' = 0 a.e. on K\Z(u*). This and (17) imply that u' is the zero function, a contradiction. A c t a Maf;hematica H u n g a r i c a ~4, 1984

ON BEST Lx-APPROXIMATION OF CONTINUOUS FUNCTIONS

417

References [1] M. P. Carroll, D. Braess, On uniqueness of Ll-approximation for certain families of spline functions, J. Approx. Th., 12 (1974), 362--364. [2] P. V. Galkin, The uniqueness of the element of best mean approximation to a continuous function using splines with fixed knots, Math. Notes, 15 (1974), 3--8. [3] S. Ja. Havinson, On uniqueness of functions of best approximation in the metric of the space L 1, Izv. Akad. Nauk SSSR, 22 (1958), 243--270. [4] B. R. Kripke, T. J. Rivlin, Approximation in the metric of L~(X, lt), Trans. Amer. Math. Soc., 119 (1965), 101--122. [5] A. Kro6, Best L~-approximation of vector valued functions, Acta Math. Acad. Sci. Hungar., 39 (1982), 303--310. [6] A. Kro6, Some theorems on unicity of multivariate Ll-approximation, Aeta Math. Aead. Sci. Hungar., 40 (1982), 179--189. [7] I. Singer, Best approximation in normed linear spaces by elements of linear subspaees, Springer (Berlin--Heidelberg--New York, 1970). [8] M. Sommer, Ll-approximation by weak Chebyshev Spaces, in Approximation in Theorie und Praxis (G. Meinardus, ed.), 85--102, Bibliographisches Institut (Mannheim, 1979). [9] M. Sommer, Weak Chebyshev spaces and best L~-approximation. Preprint No 041, 1978. [10] H. Strauss, L~-approximation mit Splinefunktionen, in Numerische Methoden der Approximationstheorie (L. Collatz, G. Meinardus, eds.), ISNM 26, Stuttgart, Birkh/iuser-Verlag (1975), 151--162. [11] H. Strauss, Best L~-approximation, Bericht 035 des Institut ftir Angewandte Mathematik I der Universit~it Erlangen-NfiJmberg, 1977. [12] H. Strauss, Eindeutigkeit in der Ll-approximation, Math. Zeitschrift, 176 (1981), 63--74. [13] M. P. Carroll, H. W. McLaughlin, L~ approximation of vector-valued functions, J. Approx. Th., 7 (1973), 122--131. (Received March 3, 1983) MATHEMATICAL INSTITUTE OF T H E H U N G A R I A N ACADEMY OF SCIENCES BUDAPEST, R E A L T A N O D A U. 13--15 H--1053 H U N G A R Y

AcSa M a t h e m a $ i c a H u n g a r i c a 44, 1984