Nearly optimal meshes in subanalytic sets

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≤

C

Numer Algor (2012) 60:545–553

ORIGINAL PAPER DOI 10.1007/s11075-012-9578-6

Nearly optimal meshes in subanalytic sets

Wiesław Ples´niak

Received: 12 December 2011 / Accepted: 12 April 2012 / Published online: 26 April 2012

AbstractWe prove that any fat, subanalytic compact subset ofR^Npossesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic)setslikeN-dimensionalpolynomialoranalyticgraphdomainsor

polynomialandanalyticpolyhedrons.(Hereagoodsourceofreferencesisthe

recentpaper(PiazzonandVianello,EastJApprox16(4):389–398,2010).)We also show that an infinitely differentiable mapffrom a compact setQinR^Nonto a Markov compact setKinC^l(lN)transforms a (weakly) admissible mesh inQonto a (weakly) admissible mesh inK,which extends a result of PiazzonandVianello(EastJApprox16(4):389–398,2010)foranalyticmaps in caseQis a subset ofR^N.Versions for^kmaps with sufficiently largekarealsogiven.

KeywordsAdmissible polynomial meshes·Optimal meshes·Subanalytic geometry·Hironaka rectilinearization theorem·Bernstein-Walsh-Siciak theorem·Jackson theorem

AMS 2000 Subject ClassificationsPrimary 41A10; Secondary 32B20·

32U35·41A17·41A63·65D05

LetKbe a compact subset of theN-dimensional complex spaceC^N. LetPd= Pd(C^N)be the set of all polynomials onC^Nof degree at mostdand letP=

W. Ples´niak (B⁾

WydziałMatematykiiInformatyki,UniwersytetJagiellon´ski,ul.StanisławaŁojasiewicza6,30- 348 Kraków, Poland

e-mail:Wieslaw.Plesniak@im.uj.edu.pl

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{ } ∈

∈

=

d

N N+d N

∈ =

≡

∞

=

∈

⊂ ≥

⊂

= →∞

≥

∈

⊂ = + ⊂

≥

∈ ∩

∞d

1P^d.Afamily(A(d))^∞_{d 1}of finite subsetsA(d)ofKis said to be aweakly adm=

issiblemeshifthecard=inalityofA(d)growspolynomiallywhend , i.e.#A(d)

O(d^α),f o r s o m e α>0,a n d t h e re e x i s t s a p o l y n o m i a l l y g r o w in g sequence C(d)of positive constants such that foreachd Nand

forallP Pdonehas

IPIK≤C(A(d))IPIA(d). (1)

HereIhISstandsfortheuniformnormsup|h|(S).IfmoreoversupC(A(d))<

, then(A(d))is said to be anadmissible mesh. Suppose thatKisP- determining, i.e. for eachPP,P0 onKforcesP(z)0. Then by the mul- tivariateLangrangeinterpolationformula(seee.g.[13,15])thereisaweakly

admissible mesh(A(d))onK,whereA(d)is a set{t1,...,tmd}of Fekete- Leja type extremal points ofKof ordermd:=dimPd= =O(d). IfK

is a Markov compact subset ofC^N, i.e. a compact set that admits a Markov inequality

I∇PIK≤Md^rIPIKforallP∈P^d (2) with positive constantsMandrdepending only onK, then following [3] one can construct an admissible mesh(A(d))onKwith #A(d)=O(d^2rN)(and

withO(d^rN)cardinality, ifKR^N∼R^Ni0C^N). Observe thatr1 if

KC^Nandr2 for any compact setKR^N(cf also Example 7) and for computational reasons one would like to construct meshes with moremodest cardinalities.Ontheotherhand,foranydN ,A(d)mustbePd-determining, whence

#A(d)md.This leads to the notion ofoptimalpolynomial meshes:

anadmissiblemesh(A(d))issaidtobeoptimal,if#A(d)O(d^N)asd . If#A(d)O((dlnd)^N),itiscallednearlyoptimal.Themainpurposeofthis

noteistoshowthatnearlyoptimalmeshescanbeconstructedonfat,compact

subanalyticsubsetsofR^NthatareknowntoadmitMarkovinequality(2)(see [8]). Let us first recall some basic notions of subanalytic geometry that was developedmainlybyŁojasiewicz,GabrielovandHironaka.

A subsetEofR^Nis said to be semianalytic if for each pointxR^None can find a neighbourhoodUofxand a finite number of real analytic functionsfijandgijdefined inU, such that

E∩U=1n

{fij>0,gij=0}.

The projection of a semianalytic set need not be semianalytic (cf [2,7]).

The class of sets obtained by enlarging that of semianalytic sets to include images under the projections has been called the class of subanalytic sets.

More precisely, a subsetEofR^Nis said to be subanalytic if for each pointxR^Nthere exists an open neighbourhoodUofxsuch thatE Uis the projection of a bounded semianalytic setAinR^N+M,whereM0. IfN3,

→∞

i j i j

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Numer Algor (2012) 60:545–553

theclassofsubanalyticsetsisessentiallylargerthanthatofsemianalyticsets, theclassesbeingidenticalifN≤2.Theunionofalocallyfinitefamilyandthe

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: i→

1

⊂

intersectionofafinitefamilyofsemianalytic(resp.subanalytic)setsissemi- analytic (resp. subanalytic). The closure, interior, boundary and complement ofasemianalytic(resp.subanalytic)setisstillsemianalytic(resp.subanalytic), the last property in the case of subanalytic sets being a (non-trivial) theorem of Gabrielov. For an excellent survey on subanalytic geometry, the reader is referred to [2]. In particular, one can find there an elegant proof of a crucial for this theory Hironaka Rectilinearization Theorem which (in a scalar space version)readsasfollows.

Theorem 1Let E be a subanalytic subset ofR^N.Let K be a compact subsetofR^N.Then there are f initely many real analytic mappingsϕjR^NR^Nsuchthat:

(1) There is a compact subsetKjofR^N,for eachj,s u c h that

jϕj(Kj)isaneighbourhood of K inR^N. (2) ϕ⁻_j¹(E)isaunionofquadrantsinR^N.

With the aid of the above theorem one can prove (see [8]) the following Theorem 2Let E be a bounded, subanalytic subset ofR^Nof pure dimension

N. Then there are f initely many real analytic maps fjR^NR^Nsuch that foreach j,

fj(J^N)⊂E and fj(I^N)=E,

j

where J^N:= {x∈R^N: |xi|<1,i=1,...,N}, and I^N:= {x∈R^N: |xi|≤

1,i=1,...,N}.

Subanalytic geometry methods have appeared very useful in polynomial approximation, since they provide tools for investigating regularity of the pluricomplexGreen’sfunction(seee.g.[8,9,12,13]).Asanexample,werefer the reader to an important application of Hironaka’s theorem (in version of Theorem2)whichisthefollowing Corollary 3[8]If K is a fat(i .e. K

intK)compactsubanalyticsubsetofR^N,thenitadmitsMarkov’sinequality(2).

Actually, in [8], it has been shown essentially more, namely that the setKoftheabovecorollaryisUPC,i.e.itisuniformlypolynomiallycuspidaland

consequently,itspluricomplexGreenfunctionisHöldercontinuousinC^N.

We shall need a multidimensional version of the well-known Bernstein- Walsh theorem which is due to Siciak [15].

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/

= :={∈ :| | ≤ II ∈}

| | ≤^NIIK

≤

N

n

−

M

Theorem 4Let K be a compact subset of the spaceC^N.Assume that K ispolynomiallyconvex,i.e.K Kˆ z C^N p(z)

pKforallp P.If

f is a holomorphic function in an open neighbourhood of K then lim supⁿdistK(f,Pn)<1.

n→∞

One can also easily prove the following

Lemma 5(cf [13])If K is a Markov compact set inC^Nthen for every polyno-mial P∈Pd(d=1,2... ),

P(z) e P if dist(z,K) 1

Md^r, (3)

whereMandraretheconstantsofinequality(2).Now w e c a n s t a t e t h e m a i n r e s u l t o f t h i s p a p e r .

Theorem 6Let K be a fat, compact subanalytic subset ofR^N. Then one canconstruct an admissible mesh(A(d))on K such that#A(d)=O((dlnd)^N)asd→ ∞.

ProofLet

fj=(fj,1,...,fj,N):R^Ni→R^N(j=1,..., m)

be real analytic functions of Theorem 2 forE=K.LetP∈Pd.Choose a pointw∈Ksuchthat|P(w)|

=IPIK.Thenthereisj∈{1,...,m}suchthatw∈fj(I).Nowchoosex∈Isuchthat w=fj(x).Sinceanycompactsetin

Ris polynomially convex, by Theorem 4 there exist polynomialsPn,k∈Pn, n=1,2,..., and constantsL>0 anda∈(0,1)independent ofnsuch that

Ifj,k−P^n,kI ≤Laⁿ=:

εn (4)

fork=1, . . . ,N.SetPn√=(P^n,1, . . . ,Pn,N).Letw=Pn(x).ThenIw−

wⁿI = Ifj(x)−Pn(x)I ≤ Nεn.Let(A(d))^∞_d1beanoptimaladmissiblemesh inthecubeI.(Itiswell-knownthatsuchme=shesexist;e.g.onecantakethe Cartesian product of a one dimensional meshY(d)on [ 1,1]

with#Y(d)O (d),constructedin[4],chap.3,sec.7,Lemma3.)Bythemeanvaluetheorem, Lemma5andMarkov’sinequality(2),wehave

n n N r

|P(w)−P(w)| ≤I∇PI[w,wⁿ]Iw−wI ≤NeMdIPIKεn, provided√

Nεn≤¹r. Hence, settingϕ(d,n):=Ne^NMd^rεngives IPIK= |P(w)|≤|P(w)−P(wⁿ)|+|P(wⁿ)|

≤ϕ(d,n)IPIK+CIPIPn(A(dn)) (5)

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( ) m j=

1

j ( )

) ε /

4

1

p

= − = p

= withC=C(A(d))≥1,as√

Nεn≤1/Md^r.Byasimilarway,weshallnowestimateIPIP

n(A(dn)). Letz∈Pn(A(dn))be such that|P(z)|=IPIPn(A(dn)). Choosey∈A(dn)sothatPn(y)=z.Wehave

|P(z)|≤|P(Pn(y))−P(fj(y))|+|P(fj(y))|

≤ϕ(d,n)IPIK+CIPIfj(A(dn)). Hence by (5),

2

IPIK≤ϕ(d,n)IPIK+Cϕ(d,n)IPIK+CIPIA^t(dn), whereA^tdn:=f A dn,provided√

N ≤1Md^r. Now, it iseasily seenthatthereisa√sequencen(d)=O(lnd)ofpositiveintegerssuchthat

ϕ(d,n(d))≤^Cand

Nεn≤1/Md^r.Then

IPIK≤2C²IPIA^t(dn(d)). Onealsoverifiesthat#A^t(dn(d))=O((dlnd)^N).

nu

In general, Theorem 6 gives better estimates of the cardinality of accessible meshes in subanalytic sets than those yielded by [3, Theorem 5]. This is seen by thefollowing

Example 7Consider the set

K=x=(x1,x2)∈R²:0≤x1≤1,0≤x2≤g(x1) ,

wheregis an analytic function in an open neighbourhood of [0,1] such that 0<g(x₁)≤x for somep∈N. ThenKis a semianalytic set, whenceby Corollary 3 i s Markov. Its Markov exponentrhas to be greater thanM1 ^p

ti _lnp

forpsufficiently large, which can be easily seen by considering the polynomials P(x1,x2)x2(1x1)^p. (Actually, ifg(x1)x₁, then by Goetgheluck [5]r

2p.) Thus Markov’s exponent ofKcould be as large as we want. By Theorem 6onecanconstructanadmissiblemesh(A(d))inKwith#A(d)=O((dlnd)²), asd→

∞, while by [3, Theorem 5] we know only that there exists an admissible mesh(A(d))inKwith #A(d)=O(d^2r).

The idea of applying Markov’s inequality and the mean value theorem to constructingadmissiblemeshesgoesbacktoCheneyandithasbeendescribed

inhismonograph[4]inthecaseofunivariatepolynomialapproximation.Inthe proofoftheabovetheoremwealsoexploitthepossibilityofrapid(geometric)

approximationofanalyticmapsbypolynomials.Suchamethodhasalsobeen used by the authors of the recent interesting paper [10], where they provethe following

Theorem 8Let K be a Markov compact subset ofC^Nand let Q be aP- determiningcompactsetinC^NsuchthatK=f(Q),wherefisananalyticmap inanopenneighbourhoodofthepolynomialhullQˆofQ.Let(A(d))be a

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≤

=

≤

=

: i→

≤

=[ ] ≤ ≤ ≡ ≡ ≤ ≤ ≤

∈

(weakly)admissiblemeshforQ.Thenthereexistsasequencej(d)=O(lnd)ofna turalnumberssuchthat(A^t(d)):=((f(A(dj(d))))isa(weakly)admissible mesh for K with C(A^t(d)):::C(A(dj(d)))and#A^t(d)≤#A(dj(d)).

Observe that in the above theorem we are able to letfhave values in the spaceC^lwithlN.Let us also note that we cannot directly apply Theorem 8 in the proof of Theorem 6, since we do not know whether the setsfj(I)are Markov. We only know, by [1], that this is the case if det[f_j^t(x)]/=0 ateveryp o i n t x∈I.

Remark 9In a recent paper [6], Kroó constructs admissible meshes in graph domains inR^Nthat are sets of the type

Kg:= {(x1,...,xN)∈R^N:fk(x1,...,xk−1)≤xk≤gk(x1,...,xk−1), (x1,...,xk−1)∈I^k−1,1≤k≤N},

whereI^k 0,1^k,1 k N,f10,g11and0 fk(x) gk(x)

1,xI^k−1,2 k N.(Such domains are

also called “normal domains”i n textbookson multiple integrals.) He shows (Proposition 1) that in case the functionsfkandgkare algebraic polynomials the domainKgpossesses ano p ti ma l

polynomialmesh.Actually,itimmediatelyfollowsfromthefactthatanygraph setKgissimplytheimageofthecube[0,1]^Nbythemap

F(t1,...,tN):=(t1, (1−t2)f2(t1)+t2g2(t1), ...,

(1−tN)fN(t1,...,tN−1)+tNgN(t1,...,tN−1)).

Indeed, if(A(d))is an optimal mesh inI^NandF=(F1,...,FN):R^Ni→R^Nis a polynomial map of degreesmax1≤k≤NdegFk, then for any polynomialPinR^Nof degreedoneh a s

IPIF(I^N)=IP◦FII^N≤CIP◦FIA(sd)≤CIPIF(A(sd))

with #F(A(sd))#A(sd)Ms^Nd^N.The same holds true ifKis a finite union of the imagesF^j(I^N)of the unit cubeI^Nby polynomial mapsF^jR^NR^N, in particular ifKis a polytope.

If the functionsfkandgkare traces onI^k−1of real analytic functions then the corresponding graph domainKgis clearly a (global) semianalytic set.

Then by Theorem 6 one can construct inKgan admissible mesh(A(d))with

#A(d) O((dlnd)^N)which is better than the

estimate#A(d)O(d^NlnN(N−1)d)yieldedinsuchacaseby[6,Theorem1].Letusaddthatinthe analytic case the cardinality result#A^t(d)

O((dlnd)^N)forKgalsofollowsfromCorollary3andTheorem7.

Other typical sets fulfilling the assumptions of Theorem 6 are analytic polyhedrons, i.e. compact subsetsKof a domainQinR^Nof the type

K:= {x∈Q: |hj(x)|≤1,j=1,...,m}, wherehjare real analytic functions inQ.

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j

C

≤

=

:=

| | = I I

I− I≤ ∈ ∈

=

n

d

4

=

Now we are going to show that in caseQis a subset ofR^NTheorem 8 is also valid for^∞maps and even for^kmaps with sufficiently largekdepending on Markov’s exponentrof inequality (2) and the growth of thes e q u e n c e {C(A(d))}.

Theorem 10Let Q be a compact set inR^Nand let f(f1,...,fl)be a mapdef ined on Q, with values inC^l(lN),whose components fjare traceso f

∞-functions onR^N.Suppose that the set K f(Q)is Markov.

Let(A(d))bea(weakly)admissiblemeshinQ.Thenthereisapositiveintegerm suchthat(f(A(md²)))isa(weakly)admissiblemeshinK.

ProofBy the multivariate Jackson theorem (applied to a cubeI⊃QinR^N),onecanfindpolynomialsPj,n∈Pnsuchthatthesequenceε^j,n:=

Ifj−Pj,nIQis rapidly decreasing, i.e. for eachk>0,n^kεj,n→0 asn→ ∞ forj=1,..., l(see[13,16]).LetPn=(P^1,n,...,Pl,n)andεn=maxε^j,n.W ehavef Pn Q

√lεn. Take a polynomialWPd(C^l)and choosewK f(Q) so thatW(w)WK. Then, by a similar argument to that of the proof of Theorem 6 (cf also the proof of Theorem 7 in [10]) we arrive at the estimate

IWIK≤ψ(d,n)IWIK+C(A(dn))ψ(d,n)IWIK

+C(A(dn))IWIf(A(dn))

withψ(d,n)Mle^ld^rεn,provided√

lεn1/Md^r.Observethatforeachk>0 we have

kd^r _k d^r d^r

ψ(d

,n)=Const.nεn

n^k≤Const.sup(nεn)

n^k=C(k) n^k. Consider now two cases.

1^◦C:=supC(A(d)) <∞, that is the mesh(A(d))is admissible. We may assume thatC≥1. Then, settingk= [r]+1, where[r]denotes the entire

part ofr, one can find a positive integermsuch thatCψ(d,md)≤¹and εm

d

4

≤1/Md^r.Consequently,

IWIK≤2CIWIf(A(md²),

and if#A(d) O(d^α)for someα>0, we get #f(A(md²) O(d^2α).T h u s (f(A(md²)))is an admissible mesh inK.

2^◦SupposeC(A(d))=O(d^β)for someβ >0. Then again, settingk= [β+

r]+1, we can find a positive integerm^tsuch thatC(A(m^td²))ψ(d,m^td)≤

1andεmd≤1/Md^r.Thisyieldstheinequality

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552 Numer Algor (2012) 60:545–553

IWIK≤2C(A(m^td²))IWIf(a(m^td²)).

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n

{ }

C

Moreover,if#A(d)=O(d forsomeγ> 0,then#f(A(md ))=O(d .Thismea nsthatthemesh(f(A(m^td²)))isweaklyadmissible. nu Remark11By aversionofthemultivariateJacksontheoremin[16],ifamapf=(f1,..., fl)defined onQextends to aC^k+1map fromR^NtoC^l,then for eachj∈{1,...,l},

supn^kεn≤C(k)ID^αfjII≤D(k,f),

|α|≤k+1

whereIis a compact cube inR^Ncontaining the setQ. Then, if the mesh(A(d))is admissible, Theorem 10 holds iffis aC^[r]+2map, and ifC(A(d))=O(d^β)(β >0), then Theorem 10 is valid for anyC^[β+r]+2mapf.

Remark 12By a non-trivial result of [11], bounded, fat and definable sets in some polynomially bounded o-minimal structures generated by special classes ofC^∞functions inR^Nare uniformly polynomially cuspidal, whence by [8] they are Markov. This is e.g. the case of the Rolin-Speissegger-Wilkie structure (cf [14]) generated by the Denjoy-Carleman classes of quasianalytic functions with partial derivatives tempered by a strongly logarithmically convex sequenceMp. In [11], Pierzchała has proved a version of Theorem 2 for such a structure. Thus it should be possible to extend Theorem 6 to the case of definable sets in the Rolin-Speissegger-Wilkie o-minimals truc tu re.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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