Bounded Quasi-Interpolatory Polynomial Operators

Abstract

We construct bounded polynomial operators, similar to the classical de la Vallée Poussin operators in the theory of Fourier series, which preserve polynomials of a certain degree, but are defined in terms of the values of the function rather than its Fourier coefficients.

Full text

Page 1

H?N?Mhaskar
?
DepartmentofMathematics? CaliforniaStateUniv ersity
Los Angeles
California? ?????? U?S?A?
J?Prestin
FBMathematik
Universit?atRostock
Universit?atsplatz?
?????
erio
Rosto
function?
ck?
and
German
for
y
Abstract
Weconstructboundedpolynomialoperators?similartotheclassical delaValle?e
Poussinop
erators
eratorsinF
the
ourierseries?
wing
whic
in
hpreservepolynomials
erties?
ofa
for
certain
in
de?
gree?butare de?nedintermsof thevaluesofthefunctionratherthanitsFourier
coe?cients?
AMSclassi?cation???A?????A??
Keywordsandphrases?Orthogonalpolynomials?weightedapproximation?delaVall?ee
Poussinmeans?
?Introduction
Letfbeacontinuous???pdicintegerm???s
?
m
?f?be them?th
partialsumof its trigonometricFourier series?It iswellknown?seee?g?????thatthede
laValle?ePoussinoperators
v
?
n
?f???
?
n
?n
X
k?n??
s
?
k
?f??n?????????
arelinearopwithfolloterestingpropwheretegern???
IH
n
denotestheclassofalltrigonometricpolynomialsofordernotexceedingn?Forall
integern???andcon tinuous???periodicfunctionsf?
?
Thisresearchwassupported?inpart?byNationalScienceFoundationGrantDMS????????AirForce
O?ceofScienti?cResearchGrantF????????????????andtheAlexandervonHumboldtFoundation?
?

Page 2

arestudiedthecon
and
max
???x??
jv
?
n
?f?x?j?cmax
???x??
jf?x?j??????
Theseoperatorsarenaturallyofgreatimportanceinthe studyoftrigonometricpoly?
nomialapproximation?Namely? it followsimmediatelyfrom?????that
max
???x??
jf?x??v
?
n
?f?x?j?cE
n
?f??
whereE
n
?f? denotestheb estapproximation in thesup?normby trigonometricpoly?
nomials ofdegreeatmostn?Letus?nallymentionhereasharperresultforstrong
approximationprovedbyLeindler?see???andtheliteraturecitedthere?
max
???x??
?
?
?
n
?n
X
k?n??
jf?x??s
?
k
?f?x?j
?
A
?cE
n
?f???????
Operatorssimilartothesealsointextofalgebraicpolynomial
approximation?where?instead ofthe trigonometricF ourierseries?onestudiestheFourier
serieswithrespecttosuitableorthogonalpolynomials????Suchoperatorsprov edtobe
indispensablein thetheoryofweightedpolynomialapproximation?cf?????????????????
??????Weobservethattheoperatorsv
?
n
are de?nedintermsoftheFouriercoe?cients of
f?whichinturn?involvetheevaluationofintegrals?Inmanyapplications?e?g???????itis
moredesirabletohaveoperatorswhichhavepropertiessimilarto?????? ??????and??????
butwhichare de?nedintermsofthevaluesofthefunctionf?
Inthecaseofperiodicfunctions?suchquasi?interpolatoryoperatorswerede?nedby
Bernstein???wherealsotheboundednessresultanalogousto?????isproved?In?????
SzabadosprovedsimilarresultsforcertainoperatorsbasedonthezerosofChebyshev
polynomials?Inthispaper?wegeneralizetheresultsintwoways?First?westudyop?
eratorsbasedonthevaluesofthefunctionatthezerosofcertaingeneralized Jacobi
polynomials?Second?westudysimilaroperatorsalsointhecaseofcertainFreud?typ e
weightfunctions?supportedonthewholerealaxis?
InSection??wegivethepreliminaryde?nitionsandestimates?Theseareapplied
tothecaseofgeneralizedJacobiweightsinSection?andthecaseofFreud?typeweight
functionsinSection??
?Preliminaries
Inthispaper?foreveryrealnumberx???wedenotetheclassofallalgebraicp olynomials
ofdegreeat mostxby?
x
?Thissomewhatunusualconventionwillactuallysimplifyour
notationslaterwhenweneedtodiscusspolynomialsofdegreenotexceedingcnforsome
?

Page 3

If?hasat leastNpointsof increase?then thereexistsauniquesystem ofpolynomials
p
n
?d??x? ???
n
?d??x
n
??????
n
?d?????n?????????N???
suchthat
Z
p
m
?d??t?p
k
?d??t?d??t??
?
??ifk?m?
??otherwise?
?????
IffisBorelmeasurablefunctiononIR?wewrite
a
k
?d??f? ??
Z
f?t?p
k
?d??t?d??t??k???????????N???
whenevertheseintegralsarewellde?ned?ThepartialsumoftheFourier?orthonormal
expansionof fisthengivenby
s
m
?d??f???
m??
X
k??
a
k
?d??f?p
k
?d???m?????????N?
Using??????weobtaintheintegralrepresentation
s
m
?d??f?x??
Z
f?t?K
m
?d??x?t?d??t??m?????????N?
where itisknown???thattheChristo?el?DarbouxkernelK
m
canbeexpressedas
K
m
?d??x?t???
m??
X
k ??
p
k
?d??x?p
k
?d??t?
?
?
n??
?d??
?
n
?d??
p
m
?d??x?p
m??
?d??t??p
m
?d??t?p
m??
?d??x?
x?t
??????
Itiswellknown???thatforeachn?????????thepolynomialp
n
?d??hasndistinct
zeros? all inthe smallest interv alS???containingthesupportofd??Wedenotethese
zerosbyx
k?n
?d??? withtheordering
x
n?n
?d???x
n???n
?d???????x
??n
?d???
The Cotesnumbersarede?nedby
?
k?n
?d????
?
K
n
?d??x
k?n
?d???x
k?n
?d???
?
??
?k???????n?n?????????N?
Oneofthemostimportantpropertiesoftheseisthe?Gaussian?quadratureformula?
?

Page 4

aStieltjesintegral?Thus?let?
n
bethemeasure thatassociates themass?
k?n
?d??with
x
k?n
?d???k???????n?Then?????canbewrittenin theform
Z
P?t?d?
n
?t??
Z
P?t?d??t??P??
?n??
?n???????N? ?????
Inthe remainderof thissection?we assume that?isamassdistribution? i?e??it isa
positive Borelmeasure?all ofwhosemomentsare ?nite?and? hasin?nitelymanypoin ts
ofincrease? Thus?theorthonormalpolynomialsp
k
?d??arede?nedforallnon?negative
integersk?
Wenowproceed to de?netheoperatorswhichwillbe thediscreteanaloguesofthe
delaV alle?ePoussinmeans?Foranyintegerm???andBorelmeasurablefunction
f de?nedon thesmallest in terval containingthe support of??wede?nethediscrete
Fouriercoe?cientsoffby
a
k?m
?d??f???
m
X
j??
?
j?m
?d??f?x
j?m
?d???p
k
?d??x
j?m
?d???
?
Z
f?t?p
k
?d??t?d?
m
?t??k?????????
The operatorsanalogous tothe dela Valle?ePoussin operators are nowde?nedby
?
n?m
?d??f???
n
X
k ??
a
k?m
?d??f?p
k
?d??
?
?n??
X
k?n??
?
??
k
n
?
a
k?m
?d??f?p
k
?d???n?m?????????
The followingTheorem???listssome basicprop ertiesof theoperators?
n?m
?In the
sequel?iffisafunctionde?nedon aBorelsetA?IR?wewrite
kfk
??A
??sup
t?A
jf?t?j?
Duringtheproofof Theorem????wewillpointoutthat
?
n?m
?d??f??
?
n
?n
X
k ??
s
k
?d?
m
?f??
Motivatedby??????wede?nethesublinearoperator
?
?
n?m
?d??f???
?
n
?n
X
k?n??
js
k
?d?
m
?f?j??????
?

Page 5

Iff?Z
m
?IR?then?
n?m
?f???
?n??
?LetG?Z
m
???????x?S????I?S???bean
intervalcontainingx? andJbeaninterval suchthatf?x
k?m
?d?????ifx
k?m
?d???J?
Thenthefollowingestimateholds
j?
n?m
?d??f?x?j??
?
n?m
?d??f?x??kfGk
??Z
m
nJ
q
K
?n
?d??x?x?
?????
?
?
?
?
?
?
?
?
v
u
u
u
t
Z
InJ
?
G
?
?t?
d?
m
?t??
??
?n
?d??
n
v
u
u
u
t
Z
S???n?I?J?
?
G
?
?t??x?t?
?
d?
m
?t?
?
?
?
?
?
?
?
?
where
?
?n
?d????max
??j??n
?
j??
??
?d??
?
j
?d??
?
Proof? Let?p
k
?d?????
??
k
?d??p
k
?d???Fromtheorthogonalityrelations?weseethatfor
somepolynomial P??
k??
?k?m?
Z
?p
k
?d?
m
??p
k
?d??d???
k
?d??
??
Z
?
k
?d???p
k
?d?
m
?p
k
?d??d?
??
k
?d??
??
Z
fp
k
?d???Pgp
k
?d??d???
k
?d??
??
?
Similary ?
Z
?p
k
?d?
m
??p
k
?d??d?
m
??
k
?d?
m
?
??
?
Inview of thefactthatm??n? thequadratureform ula????? impliesthat
?
k
?d?
m
???
k
?d???k??????n???
Usingthequadrature formulaagain?weobtain fork????????n??
Z
??p
k
?d?
m
???p
k
?d???
?
d??
Z
?p
k
?d?
m
?
?
d??
Z
?p
k
?d??
?
d?
??
Z
?p
k
?d?
m
??p
k
?d??d?
?
Z
?p
k
?d?
m
?
?
d?
m
??
??
k
?d?????
Wehave thus sho wnthat
?

Page 6

A simplecomputationthenleads to
?
n?m
?d??f??
?
n
?n
X
k?n??
s
k
?d?
m
?f??
Theoperator?
n?m
isthusadiscretizationofthe delaValle?ePoussin?typeoperatorfor
theorthonormalp olynomialexpansions? The?rstinequalityin????? isno wclear?Since
s
k
?d?
m
?P??PforeveryP??
n
?andk?n????????n?weobtain that?
n?m
?d??P??P
forallP??
n
?Also? it isclear thatforanyfunction fde?nedonZ
m
we have?
n?m
?d??f??
?
?n??
?
Theestimate ?????isobtainedusinganargumentsimilartotheonein?????????LetI
beaneighbourhood ofx?Thenwede?ne
A
x
f?t???
?
f?t??ift?I?
?? otherwise?
and
B
x
f?t? ??
?
?
?
?
?
??ift?I?
f?t??A
x
f?t?
x?t
? otherwise?
toobtain thesplitting
f?t??A
x
f?t???x?t?B
x
f?t??
Forn?k??nwehave
js
k
?d?
m
?A
x
f?x?j
?
?
?
?
?
?
?
?
?
Z
InJ
f?t?K
k
?d??x?t? d?
m
?t?
?
?
?
?
?
?
?
?
?
?
Z
jK
k
?d??x?t?j
?
d?
m
?t?
?
?
?
?
Z
I
jf?t?j
?
d?
m
?t?
?
?
?
?K
k
?d??x?x?kf?Gk
?
??Z
m
nJ
Z
InJ
?
G
?
?t?
d?
m
?t??
Hence?
?
?
n?m
?d??A
x
f?x??
q
K
?n
?d?
m
? x?x?
v
u
u
u
t
Z
InJ
?
G
?
?t?
d?
m
?t?kf?Gk
??Z
m
nJ
? ?????
?

Page 7

?
?
k
?d??x?t
?x?t?B
x
f?t?d?
m
?t?
?
?
k??
?d??
?
k
?d??
?p
k
?d??x?a
k??
?d?
m
?B
x
f??p
k??
?d??x?a
k
?d?
m
?B
x
f???
Since
js
k
?d?
m
?f?A
x
f?x?j??
?n
?d???jp
k
?d??x?a
k??
?d?
m
?B
x
f?j?jp
k??
?d??x?a
k
?d?
m
?B
x
f?j??
w
Since
e getusingBessel?sinequalitythat
?
?
n?m
?d??f?A
x
f?x?
?
?
n
?n
X
k?n??
js
k
?d?
m
?f?A
x
f?x?j
?
??
?n
?d??
n
q
K
?n
?d??x?x?
v
u
u
t
?n
X
k?n
ja
k
?d?
m
?B
x
f?j
?
?
??
?n
?d??
n
q
K
?n
?d?? x?x?
s
Z
jB
x
f?t?j
?
d?
m
?t?
?
??
?n
?d??
n
q
K
?n
?d??x?x?
v
u
u
u
t
Z
Z
m
n?I?J?
?
G
?
?t??x?t?
?
d?
m
?t?kf?Gk
??Z
m
n?I?J?
?
?????
?
?
n?m
?d??f?x???
?
n?m
?d??A
x
f?x???
?
n?m
?d??f?A
x
f?x??
thesecondestimatein ?????isprov edinview of????? and ???????
?GeneralizedJacobiweights
A gener alized Jac obi weightis afunctionoftheform
w?x? ??
?
?
?
?
?
?
Y
k??
?x??
k
?
?
k
?x????? ???
?? otherwise?
?????
where??? isaninteger??????
?
??????
?
?? ??and?
k
???fork?????????The
classof generalizedJacobiw eigh ts willdenotedby GJ? orthonormalp olynomialswith
?

Page 8

m
k??
w
mm
Ifwisthe Legendreweight??
k
???k?????????thenitiseasytoseethat
w
m
?x???
m
?x???
p
??x
?
?
?
m
?
In thesequel?weadoptthefollowingconventionregardingconstan ts? The letters
c? c
?
????willdenotepositiveconstantsdependingonlyontheweightfunctionandother
?xedparametersoftheproblem? buttheirvaluemaybedi?erentin di?erentoccurences?
evenwithin thesameformula?ThenotationA?Bdenotes thefactthat cA ?B?c
?
A?
Theorem???Letw? GJ?d??x???w?x?dx? and
G?x? ?????x
?
?
???
q
w?x?
j?m
????????
IfL???n????n?m?Ln ar eintegers?andfis afunction de?nedonZ
m
?then
jj?
n?m
?d??f ??
?
p
m
p
w
m
jj
????????
?jj?
?
n?m
?d??f ??
?
p
m
p
m
jj
????????
?cjjfGjj
??Z
m
? ?????
Pr oof?We recall from????a few factsab out the GJp olynomials?W riting
x
j?m
?d?? ?? cos?
j?m
??x
j?m
?j??????? m?
we have
???
j?m
????
j?m
??
j???m
?
?
m
?j??????? m? ?????
where?
??m
?? ???
m???m
????F urther??
n
?d??? ??
K
?n
?d?? x?x??
n
w
n
?x?
?x????? ??? ?????
and
?
j?m
?
?
m
w?x
j?m
?
q
??x
?
?j??????? m??????
Theorem???nowimpliesthatforx????????andanyintervalIcontainingx?
?
?
n?m
?d??f?x??cjjfGjj
??Z
m
s
n
w
n
?x?
?
?
s
Z
I
G
??
?t?d?
m
?t??
?
n
s
Z
??????nI
?G?t??x?t??
??
d?
m
?t?
?
?
Using??????weobtainforx???????
q
w
n
?x??
?
n?m
?d??f?x??cjjfGjj
??Z
m
?
q
S
?
?
?
m
q
S
?
?
??????
?

Page 9

S
?
??
x
j?m
???????nI
?x?x
j?m
?
?
??????
WewillestimateS
?
andS
?
forx???thecasewhenx??issimilar?Intheremainder
oftheproofx??cos?isa?xednumber? with????????
Case?????????
p
m?
WewriteI
?
?? ???????
p
m??and
I??fx?cos????I
?
g??????
In viewof?????andthefactthat?????
?
m
S
?
?
?
m
X
?
j?m
?I
?
sin
?
?
j?m
?
c
m
X
?
j?m
?I
?
?
?
j?m
?c
Z
????
p
m
?
t
?
dt?c???
?
p
m
?
???
??????
Since??
?
p
m
?
?
p
m
?wededucethat
S
?
?c???
?
p
m
?
???
?c?sin??
?
p
m
?
???
?c?
???
p
m
?x????????
It iseasytocheckthatsin ???if???????????and hence? that
S
?
?c
X
?
j?m
??????nI
?
sin
?
?
j?m
??
?
j?m
??
?
?
?
???????
If?
j?m
?????????then?
?
j?m
??
?
?c?Using ?????? andthefactthat?????we obtain
X
?
j?m
????????
sin
?
?
j?m
??
?
j?m
??
?
?
?
?c
X
?
j?m
????????
sin
?
????
j?m
?? cm
Z
????
?
t
?
dt? cm?
Since??? and??
?
p
m
???
p
m?
m?m
?
m
????????
?cm
?
?
??
?
p
m
?
???
?
Hence?
X
?
j?m
????????
sin
?
?
j?m
??
?
j?m
??
?
?
?
? cm
?
?
??
?
p
m
?
???
? ??????
?

Page 10

?
j?m
????????nI
?
??
j?m
???m
of
?
j?m
????????nI
?
? cm
Z
?
??
?
p
m
t
???
dt?cm
?
??
?
p
m
?
???
?cm
?
?
??
?
p
m
?
???
?
Along with??????and??????? thisgives
S
?
? cm
?
?
??
?
p
m
?
???
? cm
?
?
?
???
p
m
?x?? ??????
The estimates????????????? ??????yield thatwhen??????
p
m?wehave
?
??
p
m
?x?
q
w
n
?x??
?
n?m
?d??f?x??cjjfGjj
??Z
m
? ??????
Case?????
p
m???????
Inthis case?we takeI
?
????????m????????m????andI??fcos????I
?
g? The
followingestimateswillbeusedin theremainder thisproofoften?sometimeswithout
anexplicitreference?
?
?
?
??
?
p
m
?
?
?
?
?
??
?
m?
?
???
?
m?
?????
?
m?
???
?
p
m
???????
Also?in viewof??????thenumb er of?
j?m
?s inI
?
isatmostc???c
?
??
?
p
m
?
??
? Hence?
S
?
?c
X
?
j?m
?I
?
?
?
j?m
?c
?
??
?
p
m
?
??
?c?
???
p
m
?x????????
As inCase?? wededuce easilythat
X
?
j?m
?????????
sin
?
?
j?m
??
?
j?m
??
?
?
?
? cm?
If?????then
?
m
?m
????????
?c?
???
?
and if??????then it isclearthat??m? c?
???
?Thus? ineithercase?m?
m
?
?
??
?
p
m
?
???
? andwe get
X
?
j?m
?????????
sin
?
?
j?m
??
?
j?m
??
?
?
?
?cm
?
?
???
p
m
?x?? ??????
??

Page 11

S
??j
??
X
?
j?m
?I
??j
?
?
j?m
??
?
j?m
??
?
?
?
?j?????????
Then
X
?
?
j?m
?????????nI
?
sin
?
?
j?m
??
?
j?m
??
?
?
?
?c
?
X
j ??
S
??j
? ??????
WeestimateS
???
andS
???
?theestimates forS
???
andS
???
aresimilar? Using??????
S
???
?
c
?
?
?
??
?
m?
?
?
X
?
j?m
??????m??
?
??
j?m
???
?
? c?
???
m
Z
?????m??
?t?
n
??
??
dt? cm
?
?
???
? cm
?
?
??
?
p
m
?
???
? ??????
If??I
???
?then????????Using????? andthefactthat????we get
S
???
?c
X
?
j?m
?I
???
?
???
j?m
?cm
Z
?????m??
t
???
dt
?cm
?
??
?
m?
?
???
? cm
?
?
??
?
p
m
?
???
???????
F rom?????????????? ???????andsimilar estimates forS
???
andS
???
?we obtain
X
?
j?m
?????????nI
?
sin?
j?m
??
?
j?m
??
?
?
?
?cm
?
?
??
?
p
m
?
???
?cm
?
?
???
p
m
?x??
Alongwith???????this yields
S
?
? cm
?
?
???
p
m
?x?? ??????
F rom ?????????????and???????weconclude that
?
??
p
m
?x?
q
w?x??
?
n?m
?d??f?x??cjjfGjj
??Z
m
? ??????
The estimates?????????????? and analogous estimatesfor x????? ??yield???????
Weendthissectionbyobservingthe?continuous analogue?ofTheroem ????Itis
probablynotnew? butweareunable tolocateaprecisereference?Theproofofthe
following theoremissimilar to thatof Theorem????butsimpler?
Theorem???Letw?GJ?d??x???w?x?dx?andG?L?m?nbeasinTheorem ????Let
f????????IRbeameasurablefunctionsuchthatfGisessentiallyboundedon???????
Thenforx????????
?
?
p
m
?x?
q
w
m
?x?
?
n
?n
X
k?n??
js
k
?d??f?x?j?cjjfGjj
????????
? ??????
??

Page 12

Thenberissolutiontheequation
c
?
andc
?
suchthat
??c
?
?
xQ
??
?x?
Q
?
?x?
?c
?
?????x????????
Themostcommonlydiscussedexamplesincludeexp??jxj
?
??????Intheremainder
ofthissection?wwilldenotea?xedFreud?typeweightfunction?fp
k
gwilldenotethe
sequenceofpolynomialsorthonormalonIRwithrespecttothemeasurew
?
?x?dx?From
allnotations?wewillomitthementionofthismeasure?thusx
k?n
willbethek?thzeroof
p
n
?etc?
Associatedwiththeweightfunctionwaretwosetsofnumb ers?The Freud?numb erq
x
istheleastpositivesolution oftheequation
q
x
Q
?
?q
e
x
?
weight
?x?x???
uma
x
theof
x?
?
?
Z
?
?
a
x
tQ
?
?a
x
t?
p
??t
?
dt?
Itisnotdi?culttoseethata
x
?q
x
?q
?x
?x???Oneofthemost importantproperties
ofa
x
isthefollowing?Foreveryin tegern?? andP??
n
??cf? ???????????
max
x?IR
jP?x?w?x?j?max
jxj?a
n
jP?x?w?x?j?
and? if??p???Nistheleast in tegernot exceedingn???p?then
Z
IR
jP?x?w?x?j
p
dx??
Z
jxj?a
N
jP?x?w?x?j
p
dx??????
Ourmaintheorem inthissectionisthe follo wing?
Theorem ???L etwbeaFreud?typfunction?andQ?? log???w?satisfy the
followingLipschitzcondition?
jQ
?
?a
y
cos???Q
?
?a
y
cos??j?c
y
a
y
j???j
?
?y???
wherecand?arepositiveconstants independentofy? Iff?Z
m
?IR? and L?????then
fore achintegern??and?????n?m?Ln?
k?
n?m
?f?wk
??IR
?k?
?
n?m
?f?wk
??IR
?ckwfk
??Z
m
? ?????
whereZ
m
?fx
k ?m
g
m
k??
?andc isapositivec onstant depending only onw?L?and??
??

Page 13

K
n
?x?x??c
n
q
n
w
??
?x???????
?b?Wehave
?
n
?q
n
??????
?c?F or any???andjxj??????a
n
?
K
n
?x?x??c???
n
q
n
w
??
?x???????
?d?Forany???andx
k ?n
?x
k???n
?x
k???n
in???????a
n
? ?????a
n
??
x
k???n
?x
k ?n
?x
k?n
?x
k???n
?
q
n
n
?
?e? Ifjx?tj?cq
n
?n?jxj?jtj?c
?
q
n
?thenw?x??w?t??
?f?L etb?IRand ??p???Then
c
Z
c
?
q
n
?
w
??p
?x????x
?
?
b
dx ?
n
X
j??
?
j?n
w
?p
?x
j?n
????x
?
j?n
?
b
?????
?c
?
Z
c
?
q
n
?
w
??p
?x????x
?
?
b
dx?
Proof?Parts?a?and?b?areprovedin????Using?????? it isnotdi?cult toverify
?cf????? thatQ
?
?Aq
n
??n?q
n
for anyA? ??P art?e? isthena simple applicationof
themeanvaluetheorem?P art?f?w asprov edbyKnopmac herandLubinsky???under
sligh tlydi?erent conditions on theweightfunction?As stated?theresultispro v edin????
?Theorem?????? usingtheirideas?Parts ?c?and?d?were provedby Lubinskyand Sa?
????under somewhatstrongerconditionsontheweight function?Weareunable to?nd
a pro of ofthesefacts underthe statedconditions?Therefore?wepro videabriefpro of?
utilizingtheideasin ?????????? Itisproved in?????Lemma???????b?? that thereexistsa
sequence of real?nonnegativepolynomialsR
?
k
??
k
?suc hthat
w?a
k
x?R
?
k
?x?
?
?c?ifx?IR?
?c
?
???x
?
?n
????
?? ifjxj???
Writingx??a
n
y?a
k
?weobtain
w?a
n
y?R
?
k
?a
n
y?a
k
?
?
? c?ifx?IR?
?c
?
????a
n
y?a
k
?
?
?n
????
?? ifjyj?a
k
?a
n
?
??

Page 14

Next?werecall????? ?????Lemma ??????that
??
foranyA??andk?n?Ak?
c
?
?
n
k
??
?
?
a
n
a
k
???c
?
?
n
k
??
?
?
Hence?forany?? ??we mayc hoosean???such thatwith kbeing the integerpart
of ?????
??
n?
?????a
n
????
?
?
?a
k
? ???
?
?
?a
n
??????
Now? let???andjxj? ?????a
n
be?xed?and? ?? x?a
n
?Wechoosetheintegerk
soastosatisfy??????Using??????andawell known extremalpropertyoftheChristo?el
functionsK
n
?x?x???cf??????weobtainbymakingobvious substitutions?
K
n
?x?x?
??
?inf
P??
n??
P?x?
??
Z
IR
that
jP?t?w?t?j
?
dt
??inf
P??
n??
P?x?
??
Z
jtj?a
n
jP?t?w?t?j
?
dt
??a
n
inf
P??
n??
P???
Z
juj??
jP?u?w?a
n
u?j
?
du
??a
n
inf
P??
n?k??
?P???R?? ??
??
Z
juj??
jP?u?R?u?w?a
n
u?j
?
du?
Sincejxj??????a
n
? thechoice ofkshows thatj?j????????a
k
?a
n
?Then ?????implies
that
K
n
?x?x?
??
?ca
n
w
?
?a
n
?? inf
P??
n?k??
P???
??
Z
juj??
jP?u?j
?
du
?ca
n
w
?
?x?inf
P??
n?k??
P???
??
Z
juj??
jP?u?j
?
du?
Theestimate?????nowfollo wsfromwell knownestimatesfortheChristo?elfunctionsfor
Legendreweights?cf???????andthefactk?n?This prov espart?c?ofthe lemma?
P art?d? followsbyusing ideasasin ???? ??????????
Wearenow inap ositiontoproveTheorem ????
ProofofTheorem????Inthisproof?letx???a
?n
?a
?n
?bea?xednumber?and?be
an integersuchthatx??x
? ???m
?x
??m
??Theconstants inthispro ofwill generallydepend
up on?andL?Necessarily?there exists???suchthatjxj??????a
m
?We use the
estimate ?????withI??x?q
m
?m?x?q
m
?m??withJ equaltotheemptyinterval?and
G?w?ThesetS???inthiscaseisIR?Inviewof?????and ??????weobtaintheestimate
??

Page 15

?
?
?
q
n
I
w?t?n
IRnI
w?t??x?t?
?
?
?
Then umberofpointsx
j?m
inIis boundedfromabove?indep endently ofnandm?F or
each suchx
j?m
?I???????withm inplaceofnandadi?erentvalue of??sho wsthat
?
j?m
w
??
?x
j?m
??c
q
m
m
?
Hence?
Z
I
w
??
?t?d?
m
?t??c
q
m
m
?c
q
n
n
???????
Let
I
?
???????
?
?
?a
m
????
?
?
?a
m
?nI
and
I
?
??????????
?
?
?a
m
??????
?
?
?a
m
????
Inviewof?????withb??andp???
X
x
j?m
?I
?
w
??
?x
j?m
??
j?m
?x?x
j?m
?
?
?ca
??
m
n
X
j ??
w
??
?x
j?m
??
j?m
?cq
??
m
???????
Ifx
j?m
?I
?
?then
jx?x
j?m
j?jx?x
j???m
j?jx?x
j ???m
j
and
w
??
?x
j?m
??
j?m
?
q
m
m
?x
j?m
?x
j ???m
?
Therefore?taking intoaccount thefact thatm?n?weobtain
X
x
j?m
?I
?
w
??
?x
j?m
??
j?m
?x?x
j?m
?
?
?
X
x
j?m
?I
?
x
j?m
?x
j???m
?x?x
j?m
?
?
??????
?
Z
IRnI
dt
?x?t?
?
?c
m
q
m
?c
n
q
n
?
Substituting theestimates?????????????and??????in to???????we arrive at???????
Weendthis sectionby observingthe following am usinginequality?obtainedby using
Theorem ???withap olynomialofdegreen inplaceoff?andobservingthat?
n?m
?P??P
forallP??
n
?
??