On some constants in simultaneous approximation

ON SOME

CONSTANTS IN

SIMULTANEOUS

APPROXIMATION

K.

BAI.ZS

Budapest University of Economics Budapest 5, Pf. 489, II-1828 llungary

T. KILGORE

Division ofMathematics

Auburn University,

Auburn,

Alabama 36849

(Mailing

address of

authors)

(Received March 3, 1993 and in revised form February 18, 1994)

ABSTRACT.Pointwise estimates fortheerrorwhich is feasible in simultaneousapproximation ofa function andits derivativesbyanalgebraic polynomial wereoriginallypursued from theoretical

motivations,which did notimmediately require theestimationof theconstantsinsuch results. llowever,recent numericalexperimentation withtraditional techniques of approximation suchas

Lagrange

interpolation,slightly modified byadditionalinterpolation ofderivatives at -1-1, showsthat

rapid convergence ofanapproximating polynomialtoafunction and ofsome derivatives tothe

derivativesofthefunction isofteneasy to achieve. The newtechniquesaretheoretically basedupon olderresultsaboutfeasibility,contained in work ofTrigub,Gopengauz.Telyakovskii,andothers,giving

newrelevancetothe investigation of constantsinthese older results. Webegin thisinvestigation here. Helpfulinobtainingestimatesforsomeof theconstants isanewidentity for thederivativeofa

trigonometric polynomial,basedon awell known identity ofM.Pdesz. Oneofourresultsis a newproof ofatheorem of

Gopengauz

which reduces theproblemof estimating theconstanttheretothe question of estimating the constantin asimpler theorem of Trigub usedin the proof.

AMSSubject Classification: Primary41A28, 42A05,Secondary65

KEYWORDS

AND PIIRASES.

Simultaneous approximation, theorem of

Gopengauz

1992AMS

CLASSIFICATION

CODES.

PRIMARY

41,SECONDARY65.

1.

INTRODUCTION

Thereareseveral resultswhichgive pointwiseestimatesof theerrorin simultaneousapproximation

ofafunction

f

E

Cq[-1,

1]

andofits q derivatives by analgebraic polynomial and its corresponding derivatives,in termsof the modulus of continuity of

f(q),

whichwerecallis definedby

w(f(’);

)=

suPl,-,l<lf(’)(:r,)

THEOREM 1.

(see

Trigub

[I])

Let

f

E

C’[

-1,

1]

Then

.for

eachn

>

2q, there existsapoly,tom,al P,

of

degreeat most n such that

for

k O,...,q and

for

-1

<_

z

<_

+

f(q).

v/1

x’-’

withM independent

of

n and

f.

,)

+

(.)

A theorem of

Gopengauz

[2],

based upon thetheorem of Trigub, shows that there is thepossibility

ofexactapproximationattimendpoints:t:l"

THEOREM 2. Let

f

ECq[-1,

1].

Then

for

each n

>

4q

+

5, there eztstsapolynom,al

at mostn such that

for

k O ,q and

for-1

<

x

<

(1.)

withIt" independent

of

n and

f.

Such resultsasthesewereoriginallypursuedforthe sake of completing the theory of algebraic

polyno-mialal)proximation. Theoreticalinterestin similarresultshas been widespreadandsustained, leadigto aliteraturetooextensivetociteorl)araphrasehere. Insucha_{theoretical context,}however,the question ofobtainingavaluefor theconstants is noturgent. Infactaclose examination ofthe originalproofsof theseresultsand ofthoseinsuch related workasthatof Telyakovskii

[3]

shows that theproofsare

ex-tremelyuneconomicalconcerningconstants,and thequestion of estimatingauyoftlerelevantconstants

(obviously

difficult)

hasbeen littleaddressedin thesubsequent work.

Morerecently, results suchasthe theorem of

Gopengauz

havebeenusedasessential tools inshowing

that

Lagrange

interpolation

(and

otherlinear projections,

too)

canbe modified by interpolationofsome derivatives at+1withgoodeffect: the derivatives of the functionbeing approximated aresimultaneously approximated by thederivatives of the approximatingpolynomial, atarate which approaches whatis

theoretically feasible. Evidence obtainedfrom computerexperimentation indicatesthat polynomial

inter-polation e.g. onnodesgenerated by ChebyshevorJacobipolynomials,when,no(liftedbythe introduction

of interpolation of derivativesat+1,cangivenumerical results for convergencetomanystandard"bad"

functions

(the

infamous

Runge

function and some

others)

and their derivatives which a’re very good

indeed. For example,Tasche

[4]

an(l llaszenski andTasche

[5]

havecombined these methodswith

cotn-putation by afast algorithm and havedeveloped amethod of approximation which seems comparable

in itspractical efficiencytocubic spline approximation and seems togive superior approximation fora

comparablenumber ofdatapoints if the functionisseveraltimesdifferentiable.

Thus,there is the potentialofapplying new methods of simultaneous approximation in nunerical

mathematics,but estimatesoftheerror incurred in approximation require reasonableestinatesofthe constants in such basic theoremsonsimultaneousapproximationasthat of

Gopcngauz.

l[ere,webegin

toaddressthisproblemwith the following result:

THEOREM 3. For a

function

f

C(q)[-1, 1]

letP,, be apoly,,o,nial satisfy,ng

(1.1)

for

someC

(which

may ormay notdependuponn orf, as we

choose)

andalso satisfying

Then

(1.2)

follows

witha constant h"

<_ max(,le’C,

7C

+

7).

Inparticvlar, the relationbetween h" and

Cisabsolute andindependent

of

all other quantities involved.

Fromour theorem,the theoremofGopengauz will alsofollow,once it isshown how to construct

poly-nomials satisfying

(1.1)

and

(1.3).

We willcompletetheproofof

(1.2)

by givinga newconstruction for suchpolynomials. Thus,

(1.2)

will be deriveddirectly fromTrigub’s

(1.1),

bypassingasecondconclusion

of Trigub’stheorem

(not

stated here indetail)givingapointwise estimate for

I;?+’)(z)l.

Thereare two reasonsforthis innovation. Oneis that the second conclusionof Trigub,whileinterestingin itself,isnot needed here. The secondisthat for both thefirstand the second conclusion ofTrigub’stheoremtohold, the value of the constant M in the theorem must be large enough toaccommodate both conclusions, requiring theuserof Trigub’s theoremtobeginhisworkwithalargervalueofM. Another factrelevant

hereisthat the original proof of

Gopengauz

andalsothederivationofapointwiseestimatefor

p,?+l

in

Trigub’stheorem depend upon aninequalityofBrudnyi givingsuch pointwiseestimates of derivatives.

To date,noestimateof theconstant inBrudnyi’s inequality exists,either. Our proofwillpermit another improvementwhichcanhave importanceinapplications: theminimal value ofn forwhichtheestimate isvalid canbelowered from4q

+

5to2q

+

1.

2.

A LEMMA

The following result will beusefulinprovingtheTheorem:

LEMMA

1. Let

T,(O)

beatrigonometric polynomial

of deree

at mostn. Thenitsderivative can bewrittenas

(sin)

(-1)

’+

T’.(O)=

---mYT.(O+t,)knsin

(-sn-

-)-,

(2.1)

inwhichm:= 2n andin which

2j-

2j-t

:=

2m r= 4n 7r.

(2.2)

PROOF: Settingm:=2n,weinvokethe formula ofM.Riesz

[6],

whichgives foranarbitrary

trigono-metricpolynomial

,

ofdegreeatmost m

"

(-1)+

(z)

---m

,(z.=

+

tl)(sin

1/2tl),

where

t

(-)" (-t)" _{Replacementin this identity of}

(z)

_{by & sin mz gives for z 0 the}

2m 4n

useful fctthat

1=

=,

(sin’$t,)"

(2.3)

Havingnoticed this, we now sume that

O.(z)

is anarbitrary trigonometric polynomiM ofdegreeat mostn and choose

Forz 0wethus obtain

Finally, since

,

wasarbitrary, we may set

,(z)

T,,(z

+

0),

obtaining

(2.1),

and the proof of the

Lemmaiscomplete, l-I

3.

PROOFS

Webegin with the proof of Theorem 3, and thenwewillshow howthisresultcan beusedintheproof of Theorem2.

PROOF: TheproofofourTheorem

(Theorem 3)

divides itselfnaturally, into twocases: that k q

(includingthe possibility that k q 0 is oneof the cases, and that k

<

qisthe other. Case 1

(k

q):

Weprovethat

If’)(+l)

P.’(+I)I

0 and

If’(x)- P.c’(x)I

<

Cw

f(’);

+

-implies

vzi’-:)

If(’)(x)- ef’)(x)l

<

C’

to

f’);

n

(3.1)

in whichC depends only uponC. The specialcasethat q 0isalso covered here.

Clearly,

(3.1)

holdswith anyC’

>

6Cif

i

x

>

orif x -!-1. Therefore wewillassume that

0

< /1

x

<

,

and thereisnoloss of generalityinassuming furtherthat x lies near1. Wethen may

maketheestimates

IfC’)(z)

P’)(z)l-<

If(’)()

ff’)(1)l +

IP’)(x)-and

ql x

2)

3.3

Iff’)(x)- f(’)(1)l

_<

to(f(’);

x) _<

to

f(’);

n

A

moreprecise pointwiseestimatefor

IP’)(z)-

P’)(1)I

isneeded which is zerowhen I. Toobtain

suchanestimate,we noticefirstthatfor arbitrary 6

[-I,I]

IP’)(t)-

P’)(1)I

<

IP’)(t)

f(’)(t)l

+

Iff’)(t)

whence

v/1

)

+

+

to(f(’);

t)

(3.4)

IP’)(t)-

P,’)(1)I

<

Cw

f(’);

,

’i

Wenowdefine :=arccosx and cos for0

<

a

<

.

Thenwehave0

<

sin #

<

sin

,

and furthermore forn

_>

wehave

.

Nowwedefine

T,(#):=

P’)(cos#)- P’)(1),

and we notethat

Using the inequality(3.,11inthe identity(2.1)and noting(2.3),wecanestablish a

Imintwise

estinate for

IT;’,(O)l

onthe interval

[0,0].

1=1

2

lll’X

(

[Pq)(cOS

(0

+ ’J

))

-(,!

Sill

)’2

Pq)(1),)

(sin

?1221

+nax

f(;

sin 0

b

(sin

)

n sin

A(O)+

(0).

Nowweusethefact that

w(f(); A)

_< (1

+

A)w(f(’);

6)

toestimate

A(O)

andthen B($). Weobtain

(

) (f,);

sinO.’

(sin".-t)

A(O)

<

nCmax

+lsi’*(O+t’)l+

sinO

nsinO*

n / nsin

u

_<

c

%x

+

Ii,, o

t,i,,I

+o.los

i,,

t,

+

,,

_i.

;

,,

.

_i,,

Therefore

/o

’

A(O)dO <

2nC

rnax

(

O

+

(1

csO*)[cstlsin

O

+

[2sin

-t’

cos

lt’

+

O,

)

.

(f(q);

sin0).

(sin

)

n n sin

andweeasilyobtain

Z’A(O)dO2nC(

2

)

(

sinO)

(

sin

0.)

+

+

-n

+

w

its);

n

<

7Cw

frO;

__n

(3.6)

We will now similarly estimate

B(O)

and then its integral. It is helpful to notice that always if 0

<

0

<

0,then

0

t

cos

>]cosl

forall j, whichwewilluseinestimatingoneoftheterms.

B(O)

<_

2nm]tx

1+

sinO"

w n n <2nmax l+2n

mn

O.

n n

Nowwemay estimate

fo.

(

(0

+

sin

0.) sin

)

B(O)dO

<

2nmax

O.+2n

(1-cos0,)+

(1-cs0*)lsin[2

sin

0

(f(q);

Fromthisweobtain

fo

"

(

sin

0)

(f();

sin

0)

Theestimatefor

[P’)(x)-

Pq)(1)l

maynowbe obtained by combining

(3.6)and (3.7):

s,,,o.),,

Now,combiningtheesti,,,ates

(3.2)

and

(3.3)

and

(3.8),

wehaveshow,, that

[f(q)(x)-

P’)(x)[

(7C

+

7)

f(q);

n

provided that 0

<

1

x

g,

andclearly this estimateisdominantontherestof the interval

[-1, !]

aswell. Wehaveestablishe(l

(3.1)

withaconstantC’

<

7C

+

7. Thiscompletes the argument forCase1.

Case2: Itremainstoshowforq that if the polynonials

E,

satis@ (1.1)

and

(1.3)

withconstant

Cthen for k 0,... ,q- we can obtain

w

f();

n

(3.9)

in whichC" 4

min{2q-,e}C.

Clearly,

(3.9)

holds with constant

C

2q-C

if

1

x

,

and weneed toconsiderthose values ofx such that0

<

1

x

<

Withoutany loss ofgeneralitywe can assume that x liesnear We then have for k q

(q:

.,),-,

(1-

.,),-,

(-

#)II(’)(.)-forsomeintermediate point zlyingbetween and I. Wemayfurtherestimate

]l(’)(z)-P’)(z)].

<C

f(,); 41-x

+

in which wehavealsoused thefact that

1

z

1

x

.

Therefore

I(’()

P()I

<

’-C

I(’;

+

Toshowthat

2-C

canbereplaced(independentlyof

q)

by

eC,

werepeat theargument, noting that

if0<l-<

((-/9

then

(1-:)-On the otherhand,if

v/1

z

>

((_),),_-4-r

weobtain

+

<

+

((q_

k)!),__

n

-FromStirling’s formula,onemay notethat

(q-

k)! >

e-*(q-

k)

q-:,

whence

((q-

k)!),---r

Combining thepreviousestimates,wehaveshown

Finally, it isclear that (3.11)i.,l,lies(3.9)if

v/1

-

_>

_.t.,,

(), theotl,erhand,if

v/I

x

_<

1/4,

tl,en wemayuse thewell know. fact tlat

b-w(fq);) 2-w(f(q)’)

when 0<

<

and themeanvalue thcoren ofLagrangctoestablish

z)-

n

from which

II((x)-

Pff(x)l

<

2mi,,{2

,-

e}C(l-x),--.n.

+--71 7Z 7

andwehave established

(a.9).

Thisargument completes the proof ofCase2.

Theconstant Kin Theorem canbe definedas

max{C’,C"},

andTheorem

a

is proven.

Next,

weshow howourTheoren

a

canbeused toestablishthe Theorem ofGopengauz (Theore2): POON In view ofour result

(Theorem a),

it is only necessary toconstruct polynomials 1, whid

satisfy

(1.1)

and alsosatisfy

(1.a).

Our method will be, beginning with polynonials Q,, whid, satisfy

(1.1),

toconstructpolynomials

,,...,

,,

in reeursivefashion, and foreach j 0,...,q thepolynomial

,

will beseenstilltosatisfy

(1.1)

(withanewconstant

M

dependi,gonj) and tosatisfy

FinMly, he polynomials

P.

.,

reseen tosisfy

(1.1)

with newconstn

M

replacing the original

consn

M

nd thesmeime tosatisfy

(1.3).

Wedefine

.,0

by

Thenboth conclusionssurely follow for j 0, and wehave

M

Before constructing

.,...,

,,

wedenote bymthegreatesteveninteger such that mq

<

n, and

welet

T(x):= eos(m

areeosx),

theChebyshev polynomial ofdegreem. The propertiesof

T

whkhwe

need herearethat

T(I)=

;IT,]

;and

T;(1)= m;T;’(-1)

-m Now,assumi,,g that

agivon j

{1,...,

q}

the polynoial

0,-

hasalready been defined,welet

Q,.,(z)"

Q,._(z)

+

{(-1)’(f)(

1)-o

(’).,,_,(1))

+

(fo)(1)

,.,_

(-),

j!m.+

(1

Q.,_,(:)

+

,,().

Itisthenseeneasily that

II,,..,

_<

M.,_,(n-l-+)q-a(f(’+)

_.

).

Therefore,weobtain fork 0,...,7 (usingtheMarkov inequalityto estiltatethe deriwttives of

h’,,a)

II II

in which

Fromtheconstructionof

Q-o

thestate,neat

(3.12)

will alsofollow. Onlythecasek jneeds special

ROl hasexactlyoneterm initsexpansion whicltisnotzero at+1 attention wenotethatthederivative_{_.,o}

nanely

{(

1)’(fO)(1)

....

,_,(-1))

+(fO)(1)-O

O)

,(1))

}.(-1)’

,1

Ill21

Thevalue of thisterm at is

(-7;’,,()),.

(fO)(1)

O0)

(m)

fO)(1)-

(,)

rl)

’

,1

anditsvalueat -1 is

O0

(-1)’(m’)’

10)(_I

00)_,(_i),

(-1)’(/0)(-

1)

""’i-’(-1))

Thus it follows that

f0)(4-1)

-,,oO0)(+1)

0. Aspreviously stated,we nowlet

I

:=

Q,..

An estimate

for theconstant

M

maynowbeobtNned by noting thatthefollowing relation followsfi’om

(3.13)

M,M

1+

(3.1,1)

=0

Atthispoint, theproof of the theorem of

Gopengauz

is _{completed by applying the theoren of}this article

(Theorem

3),

using the value of

C

:=

M.

Theorem3then provides an estimatefor theconstant

inthe theorem of

Gopengauz.

0

REFERENCES

1. It.

TRIGUB,

Approximation offunctions by polynomials with integralcoefficients (in Russian),

lsv. Akad. Nauk.

SSSR,

Ser. Matem.26

(1962),

261-280.

2. I.

GOPENGAUZ,

AtheoremofA. F.Timanontheapproximationoffunctionsby polynomialson afinitesegment,Mat. Zametki

(1967),

163-172.

3. S.

TELYAKOVSKII,

Twotheoremson the approximation of functions by algebraic polynomials, Mat. Sb. 70

(112)

(1966),

252-265.

4. M.

TASCHE,

Fast algorithms fordiscrete Chebyshev-Vandermonde transforms andapplications, Proceedingsof the NNt’OAdvanced WorkshoponAlgorithmsfor Approximations, Oxford,1992.

5. G.

BASZENSKI

and M.

TASCIIE,

Fast algorithms for simultaneous polynomial approxination, in "Multivariate approximation andwavelets," K.Jetterand F.

Utreras,

editor,kVorld Scientific,

1992.

6. M.

RIESZ,

Eine trigonometrische Interpolationsformel und einige Ungleichungen fiir Polynome,