On the $L_p$-error of approximation of bivariate functions by harmonic splines

0.13 MB·English

by  Yuliya Babenko

BITmanuscriptNo.
(willbeinsertedbytheeditor)
On the L -error of approximation of bivariate functions by
p
harmonic splines
YuliyaBabenko TatyanaLeskevich
·
1
1
Received:date/Accepted:date
0
2

n Abstract Interpolationbyvarioustypesofsplinesisthestandardprocedureinmany
a applications. In this paper we shall discuss harmonicspline “interpolation”(on the
J
lines of a grid) as an alternativeto polynomialspline interpolation(at vertices of a

4
grid).Wewilldiscusssomeadvantagesanddrawbacksofthisapproachandpresent
1
theasymptoticsoftheL -errorforadaptiveapproximationbyharmonicsplines.
p

]
A Keywords interpolation adaptive harmonicspline optimalerror asymptotics
· · · ·
N
MathematicsSubjectClassification(2000) 41A15 41A60 68W25 97N50
. · · ·
h
t
a
m
1 Harmonicsplinesasanalternativetopolynomialsplines
[

In [5] authors investigated the question of adaptive approximation by polynomial
1
v splinesonboxpartitionsin Rd.Theobtaineda generaldescriptionaswellassharp
6 constantinseveralpopularcases.However,polynomialsplinesmightnotbethebest
9 option when it comes to interpolating a function over box partitions. There is an
6
alternative-harmonicsplines- whichcanbeviewedasa directmultivariategener-
2
alizationoflinear splinesin onedimension.Thispaperaddressesapproximationof
.
1
bivariatefunctionsand, therefore,fromnow on we shall considerharmonicsplines
0
oftwovariables.
1
1 The following several observations help to see the analogy between univariate
: linearsplinesandbivariateharmonicsplinesonrectangularpartitionsmoreclear:
v
i
X
Y.Babenko
r DepartmentofMathematicsandStatistics,KennesawStateUniversity,Kennesaw,GA,USA,30344
a E-mail:[email protected]
T.Leskevich
DepartmentofMathematicalAnalysisandTheoryofFunctions,DnepropetrovskNationalUniversity,Dne-
propetrovsk,UKRAINE,49050
E-mail:[email protected]
2 YuliyaBabenko,TatyanaLeskevich
1. Linear univariate splines interpolate the given function on the whole boundary
of the mesh which is the points of the partition of the domain. While bilinear
splines defined on rectangular partitions interpolate the given function only at
theverticesofthemeshelements,notonthewholeboundary,harmonicsplines
are constructed to coincide with the original function on the boundary of the
meshes-onthelines(asopposetojustatthepoints).Litvinin [10]introduces
differentterminologytodistinguishbetweenthese typesofinterpolationandhe
says thatharmonicsplines “interlineates”functioninstead of just interpolating.
Using harmonic splines has advantagesand drawbacks and the choice depends
on a particular problem. The strong advantage is that it uses more information
aboutthefunction.However,italsoimposesrestrictionsonthemesh.Inthecase
of square domain (considered in this paper) the meshes will consist of mostly
squares.Inthecaseofarectangulardomain,theelementsofthemeshwillhave
the same shape (same width to length ratio) as the domain, and only size will
adapttothelocalbehaviorofthefunction.
2. Both univariatelinear spline P(x) andbivariate harmonicsplineU(x,y) satisfy
1
correspondingoperator equations. Indeed, if we denote by D = d2 and D =
1 dx2 2
¶ 2 ¶ 2
+ ,thenclearly
¶ x2 ¶ y2
D P(x)=0 and D U(x,y)=0.
1 1 2
3. Aunivariatelinearinterpolatingsplineandbivariateharmonicsplineattainmax-
imumandminimumvaluesontheboundaryoftheelement.
4. Ifthevaluesoftheunivariatelinearsplineatinternalnodesaredeterminedfrom
thecontinuityconditiononthefirstderivative,splinedegeneratesintoasegment
ofstraightline.Ifthevaluesofharmonicsplinearedeterminedfromthecondition
ofcontinuityofthenormalderivativetotheinteriorboundaryofthemesh,then
thesplinedegeneratesintoaharmonicfunctionontheunionofdomains(which
is,bytheway,farlessrestrictive).
5. Bothhaveverysimilarlocalestimatesoftheerrorofapproximation [8].
For more discussion of properties of harmonic and polyharmonic splines (or
polysplines)seethemonographs [9,10]andreferencestherein.
To approximate the bivariate function (so that the approximant also coincides
with the given function on some lines) polynomial splines, in particular, blending
splines, can also be used. However,there exists a broad class of functionsthat can
be well approximatedby harmonicsplines, butare notatall approximatedby such
polynomialsplinesunderthesamerequirementsconcerningtheorderofsmoothness.
Letuspresentanexampleofsuchafunction.
Example([8]).Letusconsiderthefunction
¥
f(x,y)= 4 (cid:229) 1 (sinh(a (y b))+sinh(a b)
p (2k+1)a 2sinh(a b) k − k
k=0 k k
sinh(a y))sin(a x),
k k
−
0 x a, 0 y b, a =(2k+1)p /a.
k
≤ ≤ ≤ ≤
OntheLp-errorofapproximationofbivariatefunctionsbyharmonicsplines 3
Forthisfunction
¶ 2f(x,y)
lim =¥ ,
x 0,y 0 ¶ x¶ y
→ →
and,hence,itcannotbeapproximatedbyblendingpolynomialsplines.Ontheother
hand, D f(x,y)= 1 which implies that f can be well approximated by harmonic
−
splines.
Inunivariatecasetheideaofusingsplinesconstructedbasedonsolutionofsome
differentialoperatorequations,calledL-splines,hasbeenmoreorlesswell-developed
(see, for example, [15] and referencestherein).Therehas been severalattempts to
generalize this concept to multivariate case. For example, Litivn in his monograph
[10]introducesconceptsof“interlineation”and“interflatation”.Theseconceptsare
naturalanddirectgeneralizationsofinterpolationtothecasewhenoneneedstore-
constructthe given functionbased on knowing values of function(and possibly its
normalderivativesuptosomeorder)ononeorseverallines(“interlineation”)orm-
dimensional(1<m<d)linearmanifolds(“interflatation”)inRd.
As already mentioned earlier, in [10] for the given on R2 function Litvin sug-
gestsandcomparestwomethodsofpolynomialandnon-polynomial“interlineation”
on several lines. The first (polynomial)is a generalization of Hermite polynomials
whichpreservesthesmoothnessorderoftheoriginalfunction.Thesecond(notpoly-
nomial)operatorisageneralizationofD’Alambertoperator,whichisthesolutionof
homogeneouswaveequation,tothecasewhendataisgivenasvaluesofthenormal
derivativesofallordersuptoandincludingN (N>1).
Thereexistmanyapplicationsinwhichdataisgivennotatadiscretesetofpoints
butonsomelinearmanifolds(lines,planes,etc.).Oneofmanyexamplesiscartogra-
phy.Themeasurementandanalysisofbathymetricmeasurementsisoneofthecore
areasof modernhydrography,and a fundamentalcomponentin ensuringthe safety
of water transportation, efficiency of offshore resource development,precision and
effectivenessinrecoveringunderwaterobjectsaswellassearchandrescuemissions.
Thedatausedtomakebathymetricmapstodaytypicallycomesfromanechosounder
(sonar)mountedbeneathoroverthesideofaboat,“pinging”abeamofsounddown-
ward at the seafloor or fromremote sensing LIDAR or LADAR systems. In all the
cases,thedataobtainedareunivariate-alongthelines,curvesor,moregenerally,a
gridcomposedofthem-dependingonthecourseofthesurveyorvessel(whichcan
beadaptedwhenneeded).Thegoalistorecoverthefunctiondescribingthereliefof
the seafloor based on the “traces” of this function on the lines which described the
courseofthevesselwithsonar.
Similar technology is used for mapping surfaces of cosmic objects. Only in-
steadofsonar,whichusessoundwaves,radarsareused.Radarsarebasedonhigh-
frequencyelectromagneticradiationwhichcanpenetratethesurroundingsofobjects,
forinstance,thecloudscoveringVenus.
Another example of applications to surface reconstruction based on data given
onlines,curves,orhyperplaneswouldbemanufacturingofcar,aircraftetc.bodies.
Currently,mostpopulartoolusedinthesemodelsispolynomialsplines.Beingeasy
tomanipulate,theyhavemanydrawbacks:lackofglobalsmoothness(ifworkingwith
convenientlowdegreepolynomialpieces)andlackofflexibilityinchoosingthemesh
(forinstance,itmightbenecessarytobaseapartofthebodyonhexagonorpentagon,
4 YuliyaBabenko,TatyanaLeskevich
notonly triangleor rectangle).Harmonicand their generalization- polyharmonic-
splineshavegreatadvantageinbothdirections:theypreservehighersmoothnessand
theycanbeconstructedonany(moreorlessreasonable)domain.
2 Notation,statementofthemainproblem,andthemainresult
LetinthespaceR2 ofpointsx=(x ,x )theunitsquare[0,1] [0,1],begivenwith
1 2
interiordenotedbyI2,andtheclosuredenotedbyI¯2.By(cid:3) w×eshalldenoteapar-
N
tition of I¯2 whose number of elements has order N as N ¥ , and is so that the
majorityofelementsaresquares,howeversomesmallnum→ber(o(N)asN ¥ ) of
rectanglesisalsoallowed.Theinteriorofanarbitraryelementofthepartition→(cid:3) we
N
shalldenotebyW N,theclosurebyW N,andtheboundaryby¶W N.
Weshallneedthefollowingstandardnotation:C(W )denotesthespaceofcontin-
uousinsomeregionW functions,C2(W )denotesthespaceoftwicedifferentiablein
W functions,andL (W ), 1 p<¥ denotesthespaceofmeasurableandintegrable
p
inpower pfunctions f :W ≤Rwiththenorm
→
1
p
kfkLp(W )= |f(x)|pdx , 1≤p<¥ .
WZ
 
Inordertointroducethesetoffunctionsthatwewilluseasanapproximationtool,
letusdenotebyH(W )thesetofharmonicfunctionsinboundeddomainW R2,i.e.
⊂
H(W )= u(x) C2(W ): D u(x)=0 ,
∈
whereD = ¶ 2 + ¶ 2 istheLap(cid:8)laceoperator. (cid:9)
¶ x2 ¶ x2
1 2
Now we shall consider the set S((cid:3) ) of continuous on I¯2 functions such that
N
whenrestrictedtoanyW N fromthepartition(cid:3) areharmonicfunctions,i.e.
N
S((cid:3)N)= g C(I¯2): gW N =ui W N, ui H(W N) W N (cid:3)N .
∈ | | ∈ ∀ ∈
Fromnowonwesh(cid:8)allrefertothefunctionsfromsetS((cid:3) )asharmonic(cid:9)splines.
N
Let now a function f C2(I¯2) is given. We will approximate it by a harmonic
splines(f,(cid:3) )insuchaw∈aythattheirvaluescoincideontheboundaryofelements
N
W N. In other words we shall require the spline s(f,(cid:3) ) to “interlinate” the given
N
functionon the boundaryof all partitionelements. Hence, the functions(f,(cid:3) ) in
N
eachW N shallsatisfytheequation
D s(f,(cid:3) )=0
N
andtheboundarycondition
s(f,(cid:3)N)¶W N = f ¶W N.
| |
Inotherwords,functions(f,(cid:3) )ineachdomain(cid:3) mustbeasolutionofDirichlet
N N
problemforLaplaceequation,whichisunique(see,forexample,[13])andtherefore
harmonicsplines(f,(cid:3) )foreach f C2(I¯2)iswelldefinedonI¯2.
N
∈
OntheLp-errorofapproximationofbivariatefunctionsbyharmonicsplines 5
The main goal of this paper is to study the minimal L -error (1 p <¥ ) of
p
approximation of a given function f by harmonic splines s(f,(cid:3) ) for≤all possible
N
partitions(cid:3) whichweshalldenoteby
N
RN(f,Lp)=i(cid:3)nfkf−s(f,(cid:3)N)kLp(I2).
N
The value of R (f,L ) we shall call the optimal error. The partition (cid:3) on which
N p N
theoptimalerrorisachievedwillbecalledoptimalpartition,i.e.
e
RN(f,Lp)=kf−s(f,(cid:3)N)kLp(I2).
Foranarbitraryfunction f itisimpossible(exceeptforsometrivialcases)tofindthe
optimalpartition(cid:3) andexplicitlycomputethevalueoftheoptimalerror.Therefore,
N
weareinterestedinthefollowingtwonaturalquestions:firstofall,howtheoptimal
errorbehavesasneumberofpartitionelementsincreases(N ¥ ),andsecondly,how
toconstructasequenceofpartitions (cid:3) ¥ ,thatisasym→ptoticallyoptimal,i.e.
{ ∗N}N=1
lim kf−s(f,(cid:3)∗N)kLp(I2) =1.
N ¥ RN(f,Lp)
→
Inordertostatethemainresultofthispaper,weneedthefollowingnotation.By
GW (x;v)withx,v W weshalldenotetheGreen’sfunctionoftheDirichletproblem
for the domain W ∈(for the detailed definition, see, for example, [13]). In addition,
denoteby
I(x):= GI2(x;v)dv, x I2. (2.1)
∈
IZ2
Themainresultofthispaperisthefollowingtheorem.
Theorem2.1 Foranarbitraryfunction f C2(I¯2)thereexistsasequence (cid:3) ¥
andcorrespondingsequenceofsplines s(∈f,(cid:3) ) ¥ ,suchthat { ∗N}N=1
{ ∗N }N=1
Nlim¥ Nkf−s(f,(cid:3)∗N)kLp(I2)=kIkLp(I2)kD fkL p (I2).
→ p+1
Remark1.Notethatthemajorityofthepartitionelementsfrom(cid:3) aresquares.
∗N
This is rather restrictive conditionon the mesh. However,it is somewhatnaturalto
expect since we are using more information about the function than in the case of
interpolationbypolynomialsplines.
Remark 2. In the case when the domain of the functionis rectangle, the mesh
elementsmusthavethesameshapeasthedomain,i.e.thesamewidthtolengthratio.
Comparing this result with analogous results in the case of approximating the
givenfunctionfromC2(I¯2)bybilinearpolynomialsplines([3,5]),weseethattheor-
deroftheerrorofapproximationisthesame.Therefore,dependingontheparticular
problemonecouldchooseeitherpolynomialorharmonicsplines,thelatterproviding
interpolationonalargerset,butallowinglessflexiblepartitions.
6 YuliyaBabenko,TatyanaLeskevich
3 Ideaoftheproofandauxiliaryresults
Themainideaoftheconstructionofeachpartitioninthesequenceistousefirst“in-
termediateapproximation”ofthegivenfunction f whichneedstobeapproximated
by splines. As an intermediate approximation we shall use the piecewise function
“glued”fromseconddegreeTaylorpolynomialsof f.Thentheintermediateapprox-
imationisusedtobuildtheasymptoticallyoptimalsequenceofpartitionsofthedo-
mainand,consequently,thesequenceofcorrespondingharmonicsplines.
Theprocessofconstructingthissequenceofpartitionsconsistsoftwosteps:
1. ThedomainI¯2isfirstdividedintosomenumber(smallcomparingtoN)ofequal
sizesubdomains, and on each instead of the original function f we consider its
seconddegreeTaylorpolynomialconstructedat,say,centerofthecorresponding
subdomain.Thenumberofthesesubdomainsischosensothattheabsolutevalue
ofthedifferencebetweenthefunctionanditsTaylorpolynomialissmallenough
oneachcorrespondingsubdomain.
2. Next, each subdomain is refined further, depending on the Taylor polynomial
(and therefore the original function) on this subdomain. The final partition (cid:3)
∗N
will consist of the squares and possibly some rectangles(their number is small
comparingtoN).Thetotalnumberofpartitionelementsisdeterminedfromthe
followingtwoconditions:firstofall,thetotalnumberofalltheelementsin(cid:3)
∗N
isapproximatelyequaltoN, andsecondly,theglobalerrorofapproximationof
piecewisequadraticfunction(consistingofTaylorpolynomialsbuiltonthepre-
viousstep),bycorrespondingharmonicsplinesisminimal.
Theproblemoffindingthiserrorofapproximationon“smallest’subdomainele-
mentsisreducedtoapproximatingthefunctionQ(x)=Ax2+Bx2,whereAand
1 2
B are constants, since the rest of the terms in second degree Taylor polynomial
constituteaharmonicfunction.
Theconstructedinsuchawaypartition(cid:3) foreachfixedN isusedintheproof
∗N
ofthemainTheorem2.1.
Theideaofthisconstructionbasedonintermediateapproximationbypiecewise
quadraticfunctionshasbeenalreadyusedinpapers [1], [2], [3], [4], [5]byauthors
and co-authorsto build asymptoticallyoptimalsequencesof polynomialsplines on
triangulations, rectangular partitions, and their generalizations in various settings.
Moreonthehistoryofadaptiveapproximationandasymptoticallyoptimalsequences
ofsplinesinthecaseofpolynomialsplinescanbefoundin [2], [3].
Let us turn now to the auxiliary results that we shall need to prove the main
theoremofthispaper.
Forafunction f C2(I¯2)weshalldefineamodulusofcontinuityasfollows
∈
w (f,d )=sup f(x) f(x) : x x d , x x d , x,x I¯2 .
{| − ′ | | 1− ′1|≤ | 2− ′2|≤ ′∈ }
Thenweconsider
w (d )=max w (f ,d ),w (f ,d ),w (f ,d ) . (3.1)
xx yy xy
(cid:8) (cid:9)
OntheLp-errorofapproximationofbivariatefunctionsbyharmonicsplines 7
Lemma3.1 Letfunction f C2(I¯2)begivenandletP(x)=P(f;x;x )beitssec-
2 2 0
∈
ondorderTaylorpolynomialtakenatthepointx ,thatisthecenterofsomesquare
0
D I¯2 withsidelengthh.Then
h
⊂
h2 h
f(x) P(x) w , x D . (3.2)
2 h
| − |≤ 2 2 ∈
(cid:18) (cid:19)
Fortheproofofthislemmasee,forexample, [1].
Tostatethenextlemmaweshallneedthefollowingnotation.
LetGW (x;v)beaGreen’sfunctionoftheinteriorDirichletproblemforbounded
regionW R2. By W +d we denote the regionobtainedby the shiftof the region
W by a ve⊂ctord R2, and by aW we denote the region obtainedby scaling W by
coefficienta . ∈
Lemma3.2 ThefollowingpropertiesoftheGreen’sfunctionholdtrue:
x v
GaW (x;v)=GW a ;a , x∈aW ,v∈aW , (3.3)
(cid:16) (cid:17)
GW +d(x;v)=GW (x d;v d),x W +d,v W +d. (3.4)
− − ∈ ∈
Proof. In the proof of this lemma we shall use the well-known (see, for in-
stance, [13]) formulafor the Green’sfunctionofthe interiorDirichletproblemfor
theboundedregionW .Forx W , v W theGreen’sfunctioncanbewrittenas
∈ ∈
1 1
GW (x;v)= 2p ln x v +g(x;v),
| − |
wherefunctiong(x;v)is harmonicin bothargumentson W , continuousin v onW ,
andischosensothattheGreen’sfunctionhaszerovalueontheboundary.
Forx aW , v aW
∈ ∈
x v 1 1 x v 1 a x v
GW a ;a = 2p ln x v +g a ;a = 2p ln x v +g a ;a =
(cid:16) (cid:17) a −a (cid:16) (cid:17) | − | (cid:16) (cid:17)
(cid:12) (cid:12)
= 1 lna +ln(cid:12) 1 (cid:12) +g x; v = 1 ln 1 +g˜(x;v),
2p x v a a 2p x v
(cid:18) | − |(cid:19) (cid:16) (cid:17) | − |
wherefunctiong˜(x;v)=g x; v + 1 lna isclearlyharmonicinbothargumentsin
a a 2p
aW andcontinuousinvonaW .Besidesthat,sincetheboundaryoftheregionW is
(cid:0) (cid:1)
mappedtotheboundaryoftheregionaW ,andtheGreen’sfunctionhaszerovalue
on¶W ,thentheexpressionabovewillbezeroon¶ (aW ).Hence,(3.3)isproved.
LetusconsidernexttheshiftoftheregionW .Forx W +d, v W +d
∈ ∈
1 1
GW (x−d;v−d)= 2p ln (x d) (v d) +g(x−d;v−d)=
| − − − |
1 1
= ln +g˜(x;v).
2p x v
| − |
8 YuliyaBabenko,TatyanaLeskevich
Sinceallthenecessaryconditionsforthefunctiong˜(x;v)=g(x d;v d)aresatis-
fied,thenthe obtainedexpressionis theGreen’sfunctionforth−e regio−nW +d,and
hence(3.4)isproved.(cid:3)
NextweshallconsiderasquareregionW R2 withsidesparalleltothecoordi-
⊂
nateaxis,andonwhichthefollowingfunctionsisdefined
Q(x)=Ax2+Bx2, (3.5)
1 2
whereAandBareconstants.
WeshallfindtheL -errorofapproximationofthefunctionQ(x)byaharmonic
p
function u(Q;x) on W such that the values of u(Q;x) coincide with the function
Q(x) on the boundaryof W . In other words, for the function u(Q;x) the following
conditionsmustbesatisfied
D u(Q;x)=0, x W (3.6)
∈
and
u(Q;x)¶W =Q(x)¶W . (3.7)
| |
Lemma3.3 ForthequadraticfunctionQ(x)=Ax2+Bx2suchthatD Q(x)=2(A+
1 2
B)=0,wehave
6
kQ−u(Q)kLp(W )=2|A+B|·|W |1+1pkIkLp(I2), (3.8)
where W denotestheareaofW ,andI isdefinedin(2.1).
| |
Proof.Weshallusethefollowingrepresentationofanarbitraryfunctionu(x)
C2(W ) C(W )ontheregionW R2withsmoothenoughboundary: ∈
⊂
T u(x)=− ¶ GW¶ (n¯x;v)u(v)dv− GW (x;v)D u(v)dv, x∈W , (3.9)
¶WZ WZ
where ¶ GW (x;v) is itspartialderivativein theouternormaldirectionto ¶W (see,for
¶ n¯
example, [13]).
ThenforthefunctionQ(x),takingintoaccount(3.9)andthefactthatD Q(x)=
2(A+B),wehave
Q(x)=− ¶ GW¶ (n¯x;v)Q(v)dv−2(A+B) GW (x;v)dv, x∈W . (3.10)
¶WZ WZ
Inaddition,withthehelpof(3.9),thesolutionoftheproblem(3.6)–(3.7)canbe
writtenas
u(Q;x)= ¶ GW (x;v)Q(v)dv, x W . (3.11)
− ¶ n¯ ∈
¶WZ
From(3.10),(3.11),andthefactthattheGreen’sfunctionisnonnegative,itfol-
lowsthat
Q(x) u(Q;x) =2A+B GW (x;v)dv, x W ,
| − | | | ∈
WZ
OntheLp-errorofapproximationofbivariatefunctionsbyharmonicsplines 9
and,hence,
p 1
p
kQ−u(Q)kLp(W )=2|A+B|  GW (x;v)dv dx . (3.12)
WZ WZ
   
Ournextgoalistotransformtheintegralintheright-handsideof(3.12)sothatitex-
plicitlydependsontheareaofW .InordertodosoweneedtheresultofLemma3.2.
Let I2 be the square unit region such that when centered at one of its vertices
andstretchedbya itbecomesW . Thenthevolumesoftheseregionsarerelatedby
W =ae2 I2 ,fromwherewehavea = W .
| | | | | |
Therefore,wehavethefollowingequality
p
e
p 1p p 1p
x v
  GW (x;v)dv dx =  GI2 a ;a dv dx .
WZ WZ aZI2 aZI2 (cid:16) (cid:17)
      e  
   
Changingvariableswearriveat e e
p 1 p 1
p p
  GW (x;v)dv dx =a 2+2p  GI2(x;v)dv dx . (3.13)
WZ WZ IZ2 IZ2
      e  
Itislefttonoticethatduetoproperty(3.4)eoftheeGreen’sfunction,thelastequal-
ity will not change after the shift of the region I2 by an arbitrary vector from R2.
Therefore,asI2 we maytakeanarbitraryrectangularregionofunitareawithsides
paralleltothecoordinateaxis,inparticular,wemeaytakeI2.
Takinginteoaccount(3.13),notation(2.1),andhavingrecalledthatforthegiven
regionW thevaluea = W ,equality(3.12)becomes
| |
kQ−pu(Q)kLp(W )=2|A+B|·|W |1+1pkIkLp(I2).
(cid:3)
NowletusturntotheproofofTheorem2.1.
4 Proofofthemainresult
We shallbegintheproofbyconstructing(foragivenfunction f C2(I¯2)andfixed
numberN)thepartition(cid:3) ofI¯2 suchthatthesequence (cid:3) ¥ ∈isasymptotically
∗N { ∗N}N=1
optimal.TheideaofthisconstructionisgiveninSection3.
Firstofall,wedeterminethenumberoftheelementsoftheintermediatepartition.
Tothisend,foranarbitraryfixede (0,1)weset
∈
1 1 2 1 e
m :=min m N: w ,
N
( ∈ 2(cid:18)m(cid:19) (cid:18)2m(cid:19)≤ N)
10 YuliyaBabenko,TatyanaLeskevich
wherew (d )wasdefinedin(3.1).Itisclearthatm ¥ asN ¥ .
N
→ →
We shall take m2 as the number of elements of intermediate partition, and we
N
shallsubdividethesquareI¯2intoequalsquaresDl , l=1,...,m2 withthesidelength
N N
1 .Letusshowthatthenumberofelementsinintermediatepartitionisindeedsmall
mN
comparingtothetotalnumberofelementsN,i.e.
m2 =o(N), N ¥ . (4.1)
N →
Inordertodosoletusconsider N :
m2
N
N N 1 1 2(m 1)2
= w N−
m2N 2 (mN−1)2 (cid:18)2(mN−1)(cid:19)m2Nw 2(mN1 1)
−
(cid:16) (cid:17) 2(m 1)2
e N− . (4.2)
≥ m2w 1
N 2(mN 1)
−
(cid:16) (cid:17)
TasakNingi¥nt,oaancdchoeunncte(cid:16)(m4mN.1N−)1h(cid:17)ol→ds1truane.dw (cid:16)2m1N(cid:17)→0asN →¥ weobtain mN2N →¥
→
In order to obtain the intermediateestimates we define the functions f (x) and
N
Q (x),usingthenotationP(f;x;x )forTaylorpolynomialofseconddegreefor f(x)
N 2 l
atthecenterx ofeachsquareDN,asfollows:
l l
1. Forx DN set
∈ 1
f (x)=P (f;x;x )
N 2 1
and
1¶ 2f 1¶ 2f
Q (x)= (x )x2+ (x )x2.
N 2 ¶ x2 1 1 2¶ x2 1 2
1 2
2. When1<l m2 forx DN l 1¶ DN weset
≤ N ∈ l \∪i=−1 i
f (x)=P(f;x;x )
N 2 l
and
1¶ 2f 1¶ 2f
Q (x)= (x)x2+ (x)x2.
N 2¶ x2 l 1 2¶ x2 l 2
1 2
NextweshallfindthenumberofelementstoadditionallysubdivideeachDl , l=
N
1,...,m2. For that we shall cover the square Dl by a “mesh” consisting of squares
N N
ofthesamefixedarea,fixing,forconvenience,oneofthemeshverticeswithoneof
theverticesofthesquareDl .TheintersectionofDl withthismeshwillprovidethe
N N
necessarysubdivisionofthesquare.Itwillconsistofthesquaresoftheoriginalmesh
aswellas,possibly,somerectanglesalongtheboundaryofDl .
N
Weshallconsidertwocases:whenthepartitionconsistsonlyofsquaresandwhen
thepartitioncontainssomerectanglesalongtheboundary.
First,letusassumethatpartitionofDl consistsofsquaresonly.Weshallfindthe
N
area ofthe squaresnext.Thenumberofthe squaresin the partitionofDl we shall
N