Multivariate Rational Approximation - Scholar Commons

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1-1-1986

Multivariate Rational Approximation

Ronald A. DeVore

University of South Carolina - Columbia, [email protected]

Xiang Ming Yu

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Transactions of the American Mathematical Society, Volume 293, Issue 1, 1986, pages 161-169.
© 1986 by American Mathematical Society

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then Newma[n7] has shownthat rn( t ) decreases exponentially (as n oo) when

TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Voltlme 293, Number 1, January 1986

MULTIVARIATREiTIONAL APPROXIMATION

BY

RONALD A. DeVORE1 AND XIANG - MING yU2

ABSTRACWT.e estimatethe error in approximatinga function f by rational
functionsof degreen in the normof Lq(Q), Q:= [0,1]d.Amongotherthings,we
provethatif f is in the SobolevspaceWk(Q)andif k/d - l/p + l/q > O,thenf
can be approximatedby rationalfunctionsof degreen to an orderO(n-k/d).

1. IntroductionT. herearemanystrikingresultswhichshowthatrationalfunctions
can approximatemuchbetterthanpolynomials.For example,if rn( f ) denotesthe
errorin approximatingf E C[-1, 1]by rationalfunctionsof degreen in theuniform

norm and En(f) denotes the correspondingerrorfor polynomialapproximation,

f(x) = Ixl, but as we know from the resultsof Bernstein(see [6]) En(f) > cn-l,

n = 1, 2,.... In a slightlydifferentvein, Popov [8, 9] and Brudnyi[1] have given
classesof functionswhererationalapproximationis better than approximationby
polynomials.In fact,theyshowthatfor eacha > 0, the classof functionssatisfying

rn(f ) = O(n-a) is muchlargerthan the classwhichsatisfiesEnf( ) = O(n-a).For

example Popov [9] provedthat when a = 1, any f with f' E Lp, p > 1, satisfies
rn(f ) = Ofn-1). For polynomialapproximationw, e can only get such errorwhen
"roughlyspeaking"f ' E Loo.

We are interestedin provingmultivariateanaloguesof the resultsof Popov and
Brudnyi. We accomplishthis by extending the elementaryunivariateideas of
DeVore[3] to the multivariatesetting.The approachin [3]is to firstapproximatef
by a piecewisepolynomialof some small fixed degreeconsistingof n pieces and
then to use a partitionof unity of low degree rational functions to join these
polynomialpiecesintoa rationalfunctionapproximant.

In carryingthis approachover to severalvariables,we encountertwo main
difficulties.In the univariatecase,thepiecewisepolynomialis constructedby simply
"balancing"estimatesof theerrorof polynomialapproximationt;hatis constructing
n disjointintervalsI1,..., In wherethe errorsareaboutequal.In severalvariables,
such an approachwill not lead to cubes.Thereforewe replacethis firststageby an

adaptive approximationwhich generatescubes Q1.1..1 Qnon which the erroris

roughlybalanced.

Receivedby the editorsNovember19, 1984.
1980 Mathematics Subject Classification.Primary41A20,41A25,41A63.
Key wordsandphrases. Multivariatreationalapproximatione,rrorestimatesin Lq, Sobolevspaces.
1Thisauthoris supportedby NSF grantDMS8320562.
2This work was completedwhile this authorwas a visitingscholarat the the Universityof South
Carolina.

(C)1986American Mathematical Society
0002-9947/86 $1.00 + $.25 per page

161

162 R. A. DeVORE AND X.-M. YU

The secondproblemwe encounteris thatin generalwe cannotpiecetogetherlocal

Q1. 1Qnpolynomial approximantson such cubes when in the partition .1 large

cubes are allowed to neighborsmall cubes. For this reasonwe must modify our
adaptiveprocedureto insurethat this does not happen.These resultson adaptive
approximationareprovedin §3.

Ourresultson rationalapproximation(Theorem4.1) showthe sameimprovement
over polynomialapproximationas in the univariatecase. For example,it follows
fromTheorem4.1 thatif f iS in the Sobolevspace,Wpk(0) withQ:= [0,l]d the unit
cube in Rdand p > d/k, thent canbe approximatedin theuniformnormon g by
rationalfunctionof degreen withan errorO(n-k/d).

The Sobolev spaces are not satisfactoryfor describingresults about rational
approximation.For example,we are also interestedin the case wherethe orderof
approximationis O(n-a),wherea > 0 is not an integer.Thisrequiresthe introduc-
tion of smoothnessspaces of fractionalorder.But even more importantly,error
estimatesfor rationalapproximationlead naturallyto smoothnessspacesin Lpfor
p < 1. For example,in the univariatecase, the best resultson rationalapproxima-
tion state that when f has smoothnessof order a in Lp with a > l/p, then

cparn(n)= O(n-a). For this reason,we shall developour resultsabout rationalap-

proximationfor the smoothnessspaces which were studied by DeVore and
Sharpley[5]. These spacesconsist "roughlyspeaking"of functionst such that a

cpafractionalderivativef (a) iS in Lp.The precisedefinitionof involvesmaximal
cpafunctionsand is givenin §2 togetherwith the propertiesof thesespacesneededfor

our purposes. The spaces are a generalizationof the Sobolev spaces. For
example, Cpk= Wpkif 1 < p < so and k is an integer.Also COOis the generalized
LipschitzspaceLip a. Henceourresultscontaininformationaboutapproximations
of functionsfromthesemoreclassicalsmoothnessspaces.

2. PolynomialapproximationIf. t (k-1) iS absolutelycontinuousand f (k) E Lp(I),
I := [O1, ],thenfor eachx E I, theTaylorpolynomial

Tx(y):=f(X) + * * * +f(k-l)(x) (Y - X)

satisfies

(2.1) It(Y) - T (Y) |<IJI /PIIf(k(J)I)I
Jfor any interval whichcontainsx and y. We areinterestedin generalizationsof

thisresultto functionsdefinedon Q:= Id, d > 1, whichhavesmoothnessof ordera
(whichmaybe nonintegrali)n Lp,wherep maybe less thanone.

Therearewell-knownresultsforpolynomialapproximationof functionsof several
variableswhicharesomewhatweakerthan(2.1).For example,for any cube Q c Q,
thereis a polynomialPQof degree < k suchthat

(2.2) ll ll | Ik/d-l/pl t

1< p < % .

MULTIVARIATERATIONALAPPROXIMATION 163

Here WpkdenotestheusualSobolevspacewithseminorm

If | Wk(0) = ( | | kf | ) , kf E |Df 1

with derivativestakenin thegeneralizedsense.
Estimatesof the type(2.2)areknownas the Bramble-Hilberlet mmato workersin

finite elementsbut theygo backat leastto Sobolev[10]in his studyof embeddings.
The disadvantageof (2.2) when comparedwith (2.1) is that PQ depends on Q
whereasin (2.1) Txdependson x, but is independentof the interval.The reasonfor

this is that a functionin Wpk does not necessarilyhave ordinaryderivativesDaf,

1a<1k, particularlycontinuousones, so that the Taylorexpansionmay not make

sense.Howeveran analogueof (2.1)canbe givenif we replaceTaylorderivativesby
Peano derivatives.Any functionin Wpk 1 < p < % has Peanoderivativesof order

< k almosteverywhere(see [5,§5]).Moregenerally,suchresultshold for the spaces

cpwahichwe nowdefine.
cpa taThe idea behindthe spaces is to definea function whichtakesthe placeof

kf when k is not an integerand whichis meaningfulfor functionsin Lp when
p < 1. Thesefunctionsaredefinedby theirlocalpolynomialapproximationN. amely,
if f E Lp(u), p > Oand a > O,we define

(2.3) ( ) Q3t Xl-a/d
where

(2.4) E;a(f7Q)P= pnf (^QlJlf-P)l

is the "averaged"errorof approximationto f on Q by polynomialsP E PaSwhere
Padenotesall polynomialsof degree < a. Whenp = oo,the expressionon theright

llfside of (2.4)is replacedby - Plloo(Q)*
tapisWhena = k is aninteger, similarto kf. In fact,(see[5,p. 33])we have

(2.5) gkf(X) < fk,l(X) < M(§kf )(X)

whereM is the Hardy-Littlewoomd aximaloperator

(2.6) Mf(x=) a2sQuBpx-IjQQIIf1
cpaFor (x, p > O, we define the space as the set of all f E Lp(u) such that

tap E Lp(g) andwe defineits "seminorm"and"norm"by

IfIcp=lltapllp; llfIICPll=fIIPIflCp

Itlca ta1In [5], the definition°f
uses when p > 1, but this is equivalentto the

presentdefinition(see [5, Theorem4.3]).Hereand throughoutall normsare over Q

unless otherwiseindicated.It can be shown[5, p. 37] that Cpk is the Sobolevspace

Wpkwithequivalentnormsprovided1 < p < so and k is an integer.

R. A. DeVORE AND X.-M. YU

164If f E Cp, then f has PeanoderivativesD>v(x), lvl< a for almosteveryx in Q.
Welet

Pxf(y) =L

D>t(X)(Y-X)

TP(d2exh.n7febo)nyPteuxtfhsienisTgdathyeefloimnre-aPdxefiomarnaaoplolmopleyornsatotaomlrlMixalvl.lpw<W,aphee>nceaOvne,derteshtfiiemnceaodtrferoterhsgepaoEpnLpdripno(gxQdiem)rbiayvtaiotinovfesfexibsyt.
waLnEMitdMMhpfMftahA2pe.)1H(.xaL)rd<ety-sOLo,<itttphlee<wnPoqxofMamdenpaxdxi(sgiotm)ts:>a=anloOd|pMsseaart(tailisGtsfofilyer(so2)]/.d6).+ 1/q-1/p
> O. If f E Cp

ph(LfFqdwLe3QaiHatacitpamAth2omon1d.nhiraEohylha>ow.retlEotdddve8mricMerbiwnnoacstpieca)uiartthuepp3teheiteMpehbdiipdt.hebdapvfrob1iteaggrldroeAetieien.itwuyospaevoqeiasrinhaseafnnnniQtLep(te,barleaselQiligeeesv:mly[eoavpwerr0rstile=rsapaan)tie,thesOr1zNiretl.uem1iIiiee]ocLcazas,<mt.wohaxhiaItImetgnfcneinneLtirhmmcoottistdia§oehaofhqol<nmyagir4nsdiugtoNnbPtuese,yaahiipefw,peaoos.i2ttinid<raftnueelaxnG.Thya1rc1.Bs,s=tpmcihnt1ieeehswviinaa(wfetrrtoa1Qeuetnririvmrlooi,ntaetledlsa)hetinsiniataaetolasnfpah,lpsidmnuafobnoiaIcaopunstnJloooIonuelGnyclcQynusterdrstn|coettstQumilcotihinoSofttqsamicfhamlinosolmihsasrtnaQ,enlspenaa/io<axttiqtcdatonndamthiahOiil+rnahsnmtwaaccadtjplteLp-tta/aLthoee=1plcqtpnfkyIcplieo,irM[ycl(tvlmecE2tnoloQlihtehqny(]hxdmst(i=rmeiw)dsCheia<il,,tterdiatiayicnQ1pinnt=lcotqo,3l,ahguioa:nd.gn.im)Lto=n1nbE.n(oied)do.r-lep/yn,o[ilywd[nLtsstllO)efnOnowooiuiiyreon.et1l,[1f+c,iaslhni3.tf]Whq]fogtho,d]qoe1ithct.rtaEl,ifhe/Thl:rwpbqse-Tpa.enLohotahaftahhremerM-ofqenreieoelnZ.nptdyedrrp1iStgrlqetdycho/(eiosop.ufiowoeaso>>babnrmcfpio>oneenhr1f)lifesooe,gs.Ovr
(i) Q Q!E GandQ+ Q impliesint(Q)n
J 1t1 int(Q/)= 0
(i) s
Q n-l QE G,

(3 (1ii)i) IQ<I n-1 QE G
(iv) if QnQ+
(V) |G<I cn. 0withQ,Q'eG,then|Q'|<2dlQl,

saP=inRz=WdeOnle(nOetIl-)cFQlE..a.WnCIaoesnwssuiilmdlneerotfhiwrassttentlhe=ectpmgadorotwditicitouhnbmPeOsawofphoQicshiitnwivtioellinnentcedlgouesrpe.dLinceutobIue(rsQpQa)ro=tfitIeioqQnuGIar+l I

MULTIVARIATERATIONALAPPROXIMATION 165

and bad cubeswhichwillneedto be furtherdivided.We saya cube QE POis badif
eitherI(Q), £or I(Q'),E forsomeQ E POwhichintersectsQ.Wedenotetheset

of all bad cubesin POby Bo.All remainingcubesin POarecalledgood and we let

Godenotethe set of all goodcubes.
We now proceedinductivelyS. upposePk, Gk and Bk havebeendefinedas sets of

cubes of measure2-kd"-1. We definePk+l as the collectionof all cubeswhichresult
when the cubes of Bk are each subdividedinto 2d cubes of equal size. We say

QE Pk+l is bad if eitherI(Q), E or I(Q'),E for some Q!E Pk+l whichtouches
Q.The collectionof badcubesin Pk+l is denotedby Bk+l and the collectionof the

remainingcubesfromPk+1 is denotedby Gk+1.Welet G:= Uk a0 Gk.
Let us next observethat for some k, Bk = f so that this process terminates.

Indeed,fromHolder'sinequality,

(3.2) I(Q)= | 1f1< (| 1+17)IQI1r/>y

Since1-r/y > O,I(Q)< E providedIQisIsufficientlysmall.

We now verify(3.1).Properties(i), (ii) and(iii) areall obvious.To proveproperty

(iv) suppose Q, Q E G with QE Gk and Q E Gj. If j > k, (iv) is obvious.
Supposethen that j < k and let Qbe the cubein PJ+1whichcontainsQ.We will
now show that Q E Gj+1whichmeansQ= Qandproves(iv).Now if R is in PJ+
and R touches Q,then the parent Ro E PJof R must touch Q. Hence I(R) <

I(Ro) < E, as desired.

For the proof of (v), we introduceBJ'which consists of all QE BJ such that
I(Q), E. We now associateto eachcube QE Gj+1a cube QE BJ'in the following
way. If QOE Bj is the parentof Q then for a cube QE PJwhichis eitherQOor

touches QOwe have I(Q) > E. ClearlyQ E BJ.Also any R E BJ can be a Q for at

most Ad cubes Q E GJ+1with Ad dependingonly on d; namelyQ mustbe a first

generationdescendentof Q or of a neighborof Q.Hence

(3 3) IGJ+1< AalBJ1' , j = 0,1, ....

Now from(3.2) andthedefinitionof B>'w, e haveforeachQ E BJ',

Ey/r < I(Q)Y/r < (2_Jd"_l)-l+y/r| }+lY

Q

Summingthisoverall QE B>'andusingE = n-1 (and llfily = 1) gives

(3.4) |BJ'|An(2-j)r, :=d(-l+y/r)>O.

From(3.3),this givesl::1lGJ<l L0 IBy'<l cn. SinceIGol= n, we have(v). g

4. RationalapproximationI.f f E Lq(Q),1 < q < oo, t E C(Q) when q = oo, we
let

(4.1) rn( f )q= degi(nRf)<n ||t-R

eWqlkuiBvrakle'rnstrse<m1in, ofromr sBW.eshoevnpsp-ac1e, si.t follows fromthe embeddinWglk Crk for all

166 R. A. DeVORE AND X.-M. YU

wherethe infimumin (4.1)is takenoverall rationalfunctionsof degree < n, thatis
all R = P1/P2 with P1, P2polynomialsof totaldegree < n. Ourmainresultsare:

THEORE4M.1. If ot, p > O and 1 < q < oo satisfyo/d- 1/p + 1/q > O, then
thereis a constantc = c(ot,p, q, d) suchthatfor allf E Cp,wehave

(4.2) rn(f)qiclfIcpn-a/d, n=1,2,....

COROLLA4R.2Y. If k is a positiveintegerand 1 < p, q < oosatisfyk/d- 1/p +
1/q > O,thenthereis a constantc = c(k, p, q, d) suchthatfor allf E Wk, wehave

rn(f )q < clf | wkn-k/d, n = 1, 2,....
When 1 < p < so, Corollary4.2 followsfromTheorem4.1 since Wk= Ck with

r < 1. The latter can be deduced from Theorem7.1 of [5] and the embedding

The remainderof this section will be devoted to the proof of Theorem4.1.
Throughoutwe will use the notation c, cl, c2,... for constantswhich depend at
moston p, q, a and d; howeverthe valueof c may differat eachoccurrenceeven
withinthe sameexpression.

We introduce two auxiliarynumbers r and y which satisfy r < y < p and
o/d-1/r + 1/q > O. Certainlysuch numbers exist. Their exact value is not

importantbut they areto be fixedfor the remainderof the paper.If f E cp,we let

F:= Mp(fap). SinceMpmapsLpinto Lr for all r < p (becauseM mapsL1 into Ls
forall s < 1), thereis a constantA suchthat

(4.3) IIFII<YAl|taP||P=AlfICP

It will be enough to prove our theoremfor functionsf with IflC=a A-1. Hence

fromnow on, we assumethat

(4.4) 1F1II<Y1.

We now use the adaptivemethodof §3 to generatecubes (Qi)N which satisfy
Theorem3.1 for qi= F. Let xi E Qibe chosenso that

( ) ( 1 A Fr) < (n|Q |)-lXr, i= 1,...,N.

If Pi:= Pxt is the Peano-Taylorpolynomialas in §2, then for any Qi which

containsxl, we havefromLemma2.1,withX:= a/d + l/q,

(4.6) llf-Piliq(Q<I)ClQFiI(xl) < c(n|Qil) IQII.

We now use the partitionG and the points(xi)l to constructa partitionof unity
consistingof elementaryrationalfunctions.Let a:-[a] + 2 and

(4 ) + ( ) ( lx - x,l ) i1 N

MULTIVARIATERATIONALAPPROXIMATION 167

whereIYiIs theusualEuclideanlengthof thevectory. Thenf i is a rationalfunction
of degree < 2da. If we let t := ENfi, then Rl := f JA has the followingproper-
ties:

N xeQ,

(i) ERi(x)=l,

1

(4.8) {c, all x,

i ( lX Xil )

In fact fl > c on Qi,i = 1,..., N, andso t > c on Q.Hence(ii) followsfrom(4.7).

Our rational approximationto f is R:= ENPIRi- Since ENPiAl is a rational
function of degree < 2daotN and likewise t is a rational function of degree
< 2adN, it followsthatdeg(R) < cn.

To understandwhyR is a goodapproximationto f, we note that

ll(t-Pi)RI||q(Qi) < cn-T

becauseof (4.6) and the fact that IQi<l n-l. On the otherhand,awayfrom Qi,the

function(t-Pi)Ri is damperedby Ri sincethislatterfunctionis smallawayfrom

QiN. everthelessaddingup theseestimatesis a bit tricky.
We begin by estimatingllf- Rllq(Q)for an arbitrarycube QE G. Sincef- R

= L1(f-Pi )Ri, we have

N

(4 9) llt-R||q(Q) < ||(t-Pi)Rillq(Q) =L +,

1

whereL' is the sumoverall i suchthat Qin Q + 0 andL is theremainingsum.
To estimateL', we let Qibe the smallestcubewhichcontainsQand xi. Because

of (3.1)(iv),we have IQi<l C|Qainl dalso IQi<l CIQHI.encefrom(4.6)and (4.8),we

havewithX:= a/d + l/q

(t-Pi)Rillq(Q) < cllt-Pi||q(Q) < c||t-Pi||q(Qi)

< c(nlQI l)-l/rlQilT < C"-T

Here the last inequalityuses the fact that IQi<l n-l. Since the numberof terms

appearingin L' is less thana constantc (againbecauseof (3.1)(iv)),we have

(4.10) L' < Cn-T.

With a little morework,we can givea similarestimatefor L. In whatfollows,we

use the notation xQto denotethe cubewith the samecenteras Qand sidelengthX
timesthe sidelengthof Q.Now, for anyinteger- x < v < x, we let Ivdenotethose
indices i which appearin L and for which IQi=l 2vdlQ|, and we let g denotethat

portionof E whichis thesumoveri E I^.Furthermoref,oranyinteger- x < y < x,

168 R. A. DeVORE AND X.-M. YU

we let Iy,, denote the set of all i E Iv suchthat (1 + 2Z-1)Q, does not intersectQ
but (1 + 2Z)QIdoes intersectQ and we let ov,. be the sum of those termsin ov
whichhaveindex i E Iv. To estimateovMwe shalluse thefollowing

LEMMA4.3. F0r any,u,v, wehave

(i) I^,l=¢ if y< (-v)+-1,
(ii) Ql:= 4(1 + 2y)QI containsQforall i E IvMM

( ) (iii) cl2y|Q | / < |x -Y | < c22Z1Q}i , x E Q, y E Qi, i E IvMS

(iV) tI^UlA c2yd.

PROOFT. Oprove(i), we considertwocases.First,if v > Oand i E Iv,u,thenfrom
(3.1)(iv), any cube whichneighborsQl has measure > 2-dlQil.Since Q does not
neighborQl, (1 + 2-2)Qi cannottouchQ. Hencey > -1, as desired.

Secondly,we supposev < O.In thiscaseanycubewhichneighborsQ hasmeasure
> 2-dlQl= 2-d2-^dlQil.Hence (1 + 2-^-2)Qz does not touch Q and therefore
y > -V - z1. t1n1* SeStarD1-11l Sne1.S.

For (ii),we no.tethat(i) impliesthat IQI= 2-^dlQll< 2(Z+1)dlQ,Sl.ince(1 + 2Z)Q
touches Q, we have Q ' Qi. The upperestimatein (iii) follows from (ii) since Q,
containsx and y and hence lx-Yl < WlQ,ll/d. The lowerestimatein (ii) follows
fromthe fact that(1 + 2Z-1)Q,doesnot touchQ.

Finallyfrom(iii)we havethatforeachi E Iv, Q, is containedin theball B(z, p)
with center z the same as the center of Q and with radius p:= C22FIQII1/d=
c22y2^lQll/dS. incethe Ql aredisjointandall havemeasureIQil = 2^dlQl, we have

|IVUII< B(z, p) 1/2^dQlI < c2yd o

We can now estimateov. Let Q, be as in Lemma4.3. In view of (4.11)(iii),we
have (see (4.8))

|R (x ) l < C2-2yda X { Q i E I

Since Qi containsQ and xi, (4.6)giveswithX:= a/d + l/q

II(J-Pi)RIllq(Q) < 1lR,lloo(Q)1lf-P 11(Q )

< c2 Z ( n |Q, |) IQ. I

= c2 2yda"-1/r(2,ud) T | Q la/d-l/r+l/q

= cn-l/r(2^dlQ|) T l/r (2 ,(td) -2a + T
If we now sumoverall i E Ivy anduse (4.11)(iv),we get

(4 .12) av,, < cn 1/r(2^dl Q I) T l /r (2 yd ) -2a + T+ 1

Now, the exponent of 2yd on the right side of (4.12) iS negativebecauseof the
definitionof a (a := [o] + 2). Thereforet,o evaluatea^,we needto suma geometric
seriesfor,11> (-v) +.Thisgives

(4.13) av< cn 1/r[2^d|Q|]T l/r(2(-^)+d)-2a+T+1 -x < v < oo.

MULTIVARIATERATIONALAPPROXIMATION 169

If v > 0, then(-v)+= 0 andhence(4.13)gives

% < cn l/r[2^d|Q1]T-l/r V > 0.

Also, if M is the smallestnonnegativeintegersuch that 2MdlQl > l/n, then from
(3.1),we have Iv = 0 and g = Oif IJ> M. Hence

(4.14) M-1

QI]E (Jl,< cn l/r E [2^dl T-l/r

v>O v=O

< cn-l/rn-T+l/r= Cn-T

Similarly,(-v)+= lvl when v < 0. Since 2a-X-1 + X - l/r > 0 and IQ<I

l/n, we havefrom(4.13),

X, a,, < cn l/rlQl /r < cn-T

v<O

usingthiswith (4.14)givesE < cn-T.Hence,from(4.10)and(4.9),we have

(4.15) llt- R|lqlQ) < cn-T.

Whenq = x, (4.15)immediatelygivesTheorem4.1.Whenq < x, we have

1t1-R llq= , 1t1-R llq(Q) < cNn-Tq< cn-q/d
becauseN < cn. [1
QeG

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