# Equiconvergence of some sequences of rational functions

## Full text

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Internat. J. Math. & Math. Scl.

VOL. 15 NO. 2 (1992) 221-228

EQUICONVERGENCEOFSOME SEQUENCESOF RATIONAL FUNCTIONS

221

M.A. BOKHARI

Departmentof Mathematical Sciences

### K.F.U.P.M.

Dhahran,Saudi Arabia

DepartmentofMathematics

UniversityofAlberta

ABSTRACT. The phenomenon of equiconvergencewasfirstobserved by Walshfortwosequences ofpolynomialinterpolants toaclassoffunctions. HereweobtainaalogesofYuanren’sresultsfor

Walsh equlconvergence usingrational functionsasinSaff and Sharma. Weextend this toHermite interpolation andanearlierresult of

### [I]

isimproved and corrected.

KEY WORDS

### AND

PHRASES. Rational functions, polynomials, sequences, interpolation, equiconvergence.

1980 AMSSUBJECTCLASSIFICATION CODE. 41A05,41A20.

1. INTRODUCTION.

Walsh equiconvergence theoremisconcernedwiththe class

offunctions

### f(z)

whichare

analyticinthe disk

butnotanalyticin

where p

1. If

### A,,

the theoremofWalsh asserts that the difference between the

### Lagrange

interpolauton the n rootsof

unityandtheTaylor polynomial ofdegree n-1 about the origin tends tozero as n oo inthe

disk

### D.

Hereweshall be interestedinthe recent extensionby Saff and Sharma

### [2]

of Walsh’s theoremto rational functions withagiven denominator z" r" where r is areal number

### >

1. Laterthepresent authorsextendedtheseresultsby replacing

### Lagrange

interpolationwithHermite interpolation. The object of this note istwo-fold: Wefirst replace interpolation in thezerosof zm+"+l 1 by interpolationin the zerosof zm+"+l -a’+"+l where

p and m is an

integer

### _

-1. We alsoobtain the analogueofLou

recentextension

### [3]

of the Walsh equiconvergence theorem using rational functions in the spirit of Sail and Sharma

Lou

### [3]

wasthefirst toobserve that thesumof the

### t

"help polynomials" discussed in

### [4]

haveanatural

interpretationasthe

### Lagrange

interpolantinthe nth roots of unity ofTaylorpolynomial ofdegree

### n-

1. Secondlyweapply thesamepoint ofview to Hermite interpolationandtherebyimprove

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222 M.A. BOKHARI AND A. SHARMA

2. LAGRANGEINTERPOLATION.

Forafixed integer m

### _>

-1, apositive number

### >

1 andanumber a with

p, let

### R,+,.,(z)

denote the rational function of theform

"=

### z-

-whichinterpolates

### f(z)

inthe zerosof z"+"+]-

Let

### r,+m.,(z)

denoteanother rational functiongiven by

p.+()

:=

### z-

-whichinterpolates

inthezerosof

that if

1 and

G

then

rain

### I’

is attainedwhen P(z)interpolates

in thezeros of

### -").

Thus the

rational functionin

### (2.2)

ischaracterized by the property

### (2.3).

Itis easy toseethat

d

### "_

where F isthecircle

### Itl

p- e forsomesmall e

0. Itfollowsasin

that

for

=:

### .

If KCD, iscompact,then

lim sup

where

sup

if p

or, then for all

w,haye 0,

lira

for m -1

for m O,1,2,...

where

a,z. From

and

### (2.6),

we see that Theorem 2.1 in

holds also for

/0

Since

### r.+m,,,(z)

hasarepresentation similar to

### (2.4),

it iseasy toseethat

where

### (2.7)

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EQUICONVERGENCE OF SEQUENCES OF RATIONAL FUNCTIONS 223 THEOREM 1. Let p

1 and let m

### >_

-1 Deafixed integer. If

E

and if

and o

1, then

0

for

when a

and for

if a

### =-).

For a 1, thistheoremgivesaresult ofSalfand Sharma

### [2].

Aslightlymoregeneraltheorem canbeprovedifwe set

t-

### -fltn),

where is areal number

p,

and where

denotes the

### Lag;range

interpolant of degree

n

to

:=

atthezerosof

Let

### Lm+n(z,a,f)

denote theLagrangeinterpolant

to

### f

at thezerosof zm+"+l a

Weshallnowprove

THEOREM 2. Let

E

(p

and let m

### _>

-1 be afixed integer. If

:=

cz,

cz,

then

lhn

=0,,

n-oo Z ty

for

o,, if a

al :=

and for

o if a

al. For Z and

### a-l,

weget Theorem1. PROOF. Weknow that

where

:=

### [tt(m+"+li-:t(m+’+l)-

-z Itiseasy toseethat

--z,

### J

L.+.(z,a,K(.,\$))

1 tt(m+n+l) --at(re+n+1) tin+n+1

m+n+l otm+n+l

### :

Z Z

Thisgivesus anintegralrepresentation for

sothat

1

### Kl(z,t)

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22 H.A. BOKHARI AND A. SHAIO

where

lm+n+

ot(m+n+l)

[ta(|t(m/l)

(-From

and

### (2.13),

wecaneasily obtain

3.

Let

E

and let u

1.

### For

fixedintegers m, r, s

-1, 1

r

set N=n+m+l. Let

where

### CoN-(z,.f)

isapolynomialofdegree

### _<

aN-1 whichinterpolates

inthe oof

wh

Foy

<.,

:----

:----

where

interpolates

inthezerosof

n-

If weset

thenweshallprove

3. f

d

th (,),

N,,.tz,

### f)

0 inthefogsituation:

For

p :=

p wh

:=

wh X

p,-,+t l/(#--r)

For

}

when 1

#

### <

p.

Thenvergisfo d gmetdc

sutof D.

MA.

q 1 d 0, Thmm

Thr2.1in

when r.

For s

r, c

d

### (c)

inTh3t thestatistdTh2.1 in

### [1].

Wesh proveatlymorn

t

H

Thr3 wh t 1.

the

we

the

idtity

where

### 7i,r(z’)

i,apolynomialin z ofdegree

given by

i---- 1,2,

### (3.4)

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EQUICONVERGENCE OF SEQUENCES OF RATIONAL FUNCTIONS 225

THEOREM 4. If

6

(p

and if a

### (ll,ll < )

are any complex

numbers, then foranyinteger

### _

1, thereexistpolynomials P,N-Ij(z,

### f)

ofdegree sN 1 in z(j 1

depending onlyon

### -a")

andits power seriessuch that

t-I

[N,r,o(z’

PN-I,j(z,

### r]

for z6D where A(’’)

isgivenby

### (3.2)

andtheregion D is given below:

If

then D

If p

then

D

max p

If 1 <a<p, then

D

### #

}-P

The convergenceisuniformandgeometricin everycompact subset of D. PROOF. It iseasy toseethat

N’’(z’

r"

dt where

:=

Rewriting

where

t"

From

### (3.3),

it followsimmediatelythat

In

### [6]

it wasshownby K.G. Ivanovand Sharmathat

### "- max(l,

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226 H.A. BOKHARI AD 6. S[[A[[A

where

### )

arepolynomialsin z ofdegree

### sN

1 for each j andaregivenby

where

tN t-z

and

Fom

and

### (3.13)

itis easy toseethat

### M(t,z)

isapolynomialin z ofdegree aN 1. Forany positiveinteger

## _

1, we seefrom

### (3.11)

that

Inorder to estimatetheexpressionontheright above,we see onusing

that for

and

t), wehave

tN t--z

and

### I=1"" I=1

"(’+)

itl(+.)-Hencewith

R

p and

### >

p, wehve

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EQUICONVERGENCE OF SEQUENCES OF RATIONAL FUNCTIONS 227

<C

R(’+t)-R(,+t)(m+:)

x

Fromthe aboveweseethat

R(

when

if o

01 :=

If /7

u

pl, then

holds if

I/(,-)

If u

p, then

holdsif

and

?s :=

d

=.

### (3.7)

It is ey to s that

d

the

D inthe ces

### (c)

of

Theorem4.

Tscompletestheproofof theTheory.

Itmay benotthat p >p d ps>p, but p>p oyif

d p2

a if

/

r, both

d

### (3.17)

give the resultthat

convergence holfor

### #

a. When t 1, weget thertof Thr 3.

MA. The "helpfctions"

### P,_xj(z,f)

inThrem4 sm to be&fferent

those obtned

inThrum 2.2.

### #

r, Thrum4pdthe correction to Th

rein2.2inourlier paper

### [1].

It wod be interesting to terpret the polynomi P,_j(z,y) (j 1,...,-

havingmeinteo]ry

hbn donein

d

wet

:’-

n-yn)r

andif

### Tn-if

denotestheTaylor polynomialof

ofdegree

### _<

n-1, weshallshow that

t-1

### PsN-l,J(Z’f)

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228 R.A. BOKHARI AND A. SHARMA

where H.N-l(z,a,g) denotesthe Hermiteinterpolant ofdegree 8N- 1 to g on the zerosof

N

### ,N)..

Recallingthatif gt(z):= z

where 0

j

n 1, then

wecanverify that

Similarly,

1

tjN

ffil

t"-"

Z

t Z 1

-j

### om

the abovewe c get

ily.

### (3.18)

isnot extly d the se kind the

rtsin

d

Howevitpd

### interetation

forthe "help function"

in tes d iteration of

### orators.

It wod be interting to s if the regis of

### eq&convergen

cbeeded byappHtion d sbility metho& h bn done in

byR.

for thecd

1.

M.A.and

### SHARMA, A.

Equiconvergence ofcertain sequencesofrational

inter-polants

Methods of

Analysis

### .in

Approximation Theory I\$NM 76

Micchelli,Pai

Limaye)

281-292.

2.

E.B. and

A.

### On

equiconveextceof ce.xtainsequences of

### rationM

int-.

p01ants,Lecture NotesinMath.

### Grves,

Morals, Salf andVarga)Vol. 1105,

Proc. ofRational

### Approx.

and Interpolation., Springer-Verlag, 1983.

3.

### LOU, Y.R.

Extensionsofatheorem of

### .I.L.

Walshonoverconvergence,

Theory and

itsApplications

19-32.

4. CAVARETTA

A.and

### VARGA, R.S.

Interpolationinthe roots ofunity: anextensionofatheorem ofI. L. Walsh,

155-191.

5.

### WALSH,

.I.L. Interpolationandapp&tion by

in

### the

complex plane,

A.M.S.Colloq. Publications,

### XX,

Providence,lI., 5t ed.,1969.

6.

K.G.and

### SHARMA,

A. Morequantitative resultsonWalshequiconvergenceII. Her-mite Interpolation and t-approximation, Approx. Theory

ItsApplications

47-61.

7.

### M.R.,

:IAK]MOVSKI, A. and

### SHARMA, A.

Equiconverence ofsomecomplex interpolatory polynomials,NumerischeMath.57

635-649.

8.

### BRUCK,

R.Generaliztions of WIsh’sequiconverence theorembytheapplication of

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