Overconvergence

of

some

simultaneous

Hermite-Pad

e

interpolants

Report

Marcel

G.

de

Bruin

A.

Sharma

Technische

Universiteit

Delft

Delft

University

of

Technology

Faculteit

der

Technische

Wiskunde

en

Informatica

Faculty

of

Technical

Mathematics

and

Informatics

ISSN

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Delft

The

Netherlands

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Baltzer

Journals

September

Overconvergence

of

some

simultaneous

Hermite-Pad

e

interpolants

Marcel

G.

de

and

A.

of

Pure

Mathematics

Delft

University

of

Technology

P.O.

Box

GA

Delft

The

Netherlands

E-mail

m.g.debruin@twi.tudelft.nl

Department

of

Mathematical

Sciences

University

of

Alberta

Edmonton

Alberta

Canada

E-mail

asharma@vega.math.ualberta.ca

In

a

previous

paper

the

authors

gave

an

upper

bound

on

the

limsup

of

the

n-th

root

of

the

difference

between

two

rational

Hermite-Pad

e

interpolants

to

a

d-tuple

of

meromorphic

functions

with

a

given

fixed

number

of

poles

-LRB-

disjoint

pole

sets

for

different

functions

-RRB-

This

paper

addresses

the

problem

of

the

number

of

points

for

which

the

upper

bound

is

not

achieved

the

cases

jzj

ae

and

jzj

ae

where

ae

is

the

radius

of

convergence

for

the

functions

with

their

poles

removed

have

to

be

distinguished

Subject

classification

AMS

-LRB-

MOS

-RRB-

Introduction

The

history

of

the

problem

goes

back

to

a

result

of

J.

L.

Walsh

on

equiconvergence

of

polynomial

interpolants

for

analytic

functions

on

the

disk

jzj

ae

ae

-LRB-

but

not

analytic

on

the

closure

-RRB-

This

result

was

extended

by

E.

B.

Saff

A.

Sharma

and

R.

S.

Varga

-LSB-

-RSB-

to

equiconvergence

of

Hermite-Pad

e

interpolants

of

meromorphic

functions

having

a

given

number

of

poles

in

the

disk

M.

P.

Stojanova

-LSB-

-RSB-

extended

the

results

further

introducing

an

extra

parameter

in

the

interpolant

for

the

new

interpolants

reduce

to

Pad?e

approximants

for

the

rational

functions

do

not

lead

to

known

approximants

In

-LSB-

-RSB-

the

results

of

-LSB-

-RSB-

were

generalized

to

the

situation

of

simultaneous

Hermite-Pad

e

interpolation

to

a

d-tuple

of

meromorphic

functions

-LRB-

having

given

disjoint

pole

sets

-RRB-

again

introducing

a

parameter

For

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

d

the

results

of

-LSB-

-RSB-

are

recovered

for

d

the

results

are

quite

analogous

to

those

in

-LSB-

-RSB-

Parallel

to

this

development

V.

Totik

-LSB-

-RSB-

gave

so-called

qualitative

results

on

polynomial

interpolation

-LRB-

the

number

of

points

for

which

a

certain

upper

bound

is

not

achieved

-RRB-

This

was

then

extended

to

the

rational

interpolation

case

in

-LSB-

-RSB-

The

purpose

of

this

paper

is

to

generalize

the

results

in

-LSB-

-RSB-

to

simultaneous

rational

Hermite-Pad

e

interpolation

of

d-tuples

of

meromorphic

functions

The

Interpolation

Problem

and

Its

Solution

Let

d

d

be

natural

numbers

and

let

Fd

be

d

functions

meromorphic

in

the

disc

Dae

fz

C

jzj

aeg

ae

given

by

Fi

-LRB-

z

-RRB-

fi

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

fi

-LRB-

z

-RRB-

X

k

ai

k

zk

lim

sup

k

jai

k

ae

-LRB-

-RRB-

where

Bi

-LRB-

z

-RRB-

iY

j

-LRB-

z

zi

j

-RRB-

i

j

i

X

j

i

j

i

-LRB-

-RRB-

Here

it

is

assumed

that

the

poles

of

the

functions

form

disjoint

sets

-LRB-

i.e.

zi

j

zk

when

i

k

and

all

j

-RRB-

and

all

lie

in

Dae

Let

be

an

integer

and

put

n

oe

where

oe

d.

Let

now

C

n

satisfy

j

j

ae

then

the

Lagrange-interpolant

to

the

Taylor

sections

n

X

k

ai

k

zk

on

the

zeros

of

zn

n

will

be

denoted

by

efi

-LRB-

z

-RRB-

n

X

j

A

-LRB-

-RRB-

i

j

zj

-LRB-

explicit

formulae

for

the

A

-LRB-

-RRB-

i

j

are

easily

derived

cf.

-LSB-

-RSB-

-RRB-

-LRB-

When

efi

-LRB-

z

-RRB-

is

the

full

Taylor

series

of

efi

-LRB-

z

-RRB-

-RRB-

The

interpolation

problem

can

then

be

stated

as

find

d

rational

functions

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

with

deg

U

i

-LRB-

z

-RRB-

n

i

-LRB-

i

d

-RRB-

deg

B

-LRB-

z

-RRB-

n

that

interpolate

the

d

rationals

efi

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

on

the

zeros

of

zn

n

and

B

-LRB-

z

-RRB-

is

monic

B

-LRB-

z

-RRB-

n

P

k

k

zk

n

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

It

has

been

proved

in

-LSB-

-RSB-

that

this

problem

has

a

unique

solution

for

n

suffi

ciently

large

a

solution

that

moreover

satisfies

A.

lim

n

k

k

with

n

X

k

kzk

dY

i

Bi

-LRB-

z

-RRB-

-LRB-

-RRB-

B.

lim

sup

n

max

-LRB-

z

-RRB-

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

n

j

jae

oae

o

ae

j

jae

o

ae

-LRB-

-RRB-

for

each

compact

subset

H

jzj

o

o

that

omits

the

singularities

of

the

functions

Fi

-LRB-

z

-RRB-

i

d

We

are

now

in

a

position

to

reformulate

the

result

from

-LRB-

-RRB-

in

a

form

suitable

for

further

study

of

the

upper

bound

Define

for

z

not

one

of

the

singularities

of

the

function

Fi

S

-LRB-

z

Fi

-RRB-

lim

sup

n

j

i

-LRB-

z

-RRB-

n

-LRB-

-RRB-

with

i

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

-LRB-

z

-RRB-

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

-LRB-

-RRB-

Furthermore

define

K

-LRB-

z

-RRB-

j

jae

jzjae

jzj

ae

j

jae

jzj

ae

-LRB-

-RRB-

Then

-LRB-

-RRB-

takes

the

form

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

-LRB-

for

z

with

Bi

-LRB-

z

-RRB-

-RRB-

-LRB-

-RRB-

The

question

now

arises

whether

the

upper

bound

in

-LRB-

-RRB-

is

attained

for

any

z

and

if

so

whether

this

is

a

natural

phenomenon

or

not

For

polynomial

interpolation

the

problem

has

completely

been

solved

-LRB-

cf.

-LSB-

-RSB-

-RRB-

Just

as

in

-LSB-

-RSB-

the

results

to

be

formulated

in

the

next

section

show

a

gap

in

the

behavior

for

z

outside

the

disk

jzj

ae

Main

Results

Let

us

call

a

point

z

jzj

ae

exceptional

for

the

function

Fi

when

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

-LRB-

-RRB-

Then

we

have

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Theorem

For

each

i

dg

there

are

at

most

exceptional

points

for

Fi

in

jzj

ae

The

total

number

of

exceptional

points

for

the

d-tuple

-LRB-

Fd

-RRB-

in

the

open

disk

Dae

is

at

most

d

-LRB-

-RRB-

Theorem

For

any

set

of

d

-LRB-

-RRB-

points

j

in

jzj

ae

and

any

subdivision

into

d

sets

of

points

say

i

j

-LRB-

j

i

d

-RRB-

there

exists

a

d-tuple

of

meromorphic

functions

of

the

type

-LRB-

-RRB-

-LRB-

-RRB-

such

that

for

each

i

dg

the

points

i

j

-LRB-

j

-RRB-

are

exceptional

for

Fi

Remark

This

covers

the

case

jzj

ae

the

construction

in

the

proof

of

Theorem

could

be

adapted

to

cover

the

case

of

multiple

poles

zi

j

as

well

but

for

sake

of

simplicity

an

example

with

simple

poles

will

be

exhibited

The

proof

of

Theorems

and

will

be

given

in

Section

Section

will

be

devoted

to

some

technical

lemmas

Theorem

For

each

i

dg

there

are

at

most

oe

exceptional

points

for

Fi

in

jzj

ae

The

total

number

of

exceptional

points

for

the

d-tuple

-LRB-

Fd

-RRB-

outside

the

closed

disk

jzj

ae

is

at

most

d

-LRB-

oe

-RRB-

Theorem

For

any

set

of

d

-LRB-

-LRB-

oe

-RRB-

-RRB-

points

j

in

jzj

ae

and

any

subdivision

into

d

sets

of

-LRB-

oe

-RRB-

points

say

i

j

-LRB-

j

-LRB-

oe

-RRB-

i

d

-RRB-

there

exists

a

d-tuple

of

meromorphic

functions

of

the

type

-LRB-

-RRB-

-LRB-

-RRB-

such

that

for

each

i

dg

the

points

i

j

-LRB-

j

-LRB-

oe

-RRB-

-RRB-

are

exceptional

for

Fi

Remark

The

gap

between

Theorems

and

is

obvious

whether

this

is

a

matter

of

the

method

of

proof

or

a

question

of

intrinsic

behavior

of

rational

interpolation

is

a

matter

of

further

research

After

treating

some

technical

lemmas

in

Section

the

proofs

will

be

given

in

Section

Technical

Lemmas

The

lemmas

given

have

to

be

used

in

the

proofs

of

Theorems

and

and

in

those

of

the

other

theorems

Theorems

and

however

do

need

some

extra

technical

lemmas

to

be

treated

in

Section

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Lemma

For

each

z

satisfying

jzj

ae

B

-LRB-

z

-RRB-

dY

i

Bi

-LRB-

z

-RRB-

the

following

statements

are

equivalent

-LRB-

i

-RRB-

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

-LRB-

ii

-RRB-

i

-LRB-

z

-RRB-

z

i

-LRB-

z

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

for

a

q

q

Proof

A.

-LRB-

i

-RRB-

implies

-LRB-

ii

-RRB-

As

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

we

have

S

-LRB-

z

Fi

-RRB-

qK

-LRB-

z

-RRB-

for

a

certain

q

with

q

Thus

i

-LRB-

z

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

and

because

j

z

j

can

be

estimated

by

a

constant

also

z

i

-LRB-

z

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

the

result

is

then

obvious

B.

-LRB-

ii

-RRB-

implies

-LRB-

i

-RRB-

This

takes

more

time

we

have

to

distinguish

two

cases

The

first

step

uses

a

telescoping

sum

i

-LRB-

z

-RRB-

z

s

i

s

-LRB-

z

-RRB-

s

X

k

-LRB-

z

k

i

k

-LRB-

z

-RRB-

z

-LRB-

k

-RRB-

i

k

-LRB-

z

-RRB-

-RRB-

which

implies

i

-LRB-

z

-RRB-

z

s

i

s

-LRB-

z

-RRB-

s

X

k

z

k

i

k

-LRB-

z

-RRB-

z

i

k

-LRB-

z

-RRB-

-LRB-

-RRB-

Put

-LRB-

ii

-RRB-

in

the

form

j

i

-LRB-

z

-RRB-

-LRB-

z

-RRB-

i

-LRB-

z

-RRB-

j

c

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

then

we

find

from

-LRB-

-RRB-

i

-LRB-

z

-RRB-

z

s

i

s

-LRB-

z

-RRB-

c

s

X

k

z

k

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

k

-LRB-

-RRB-

Apply

-LRB-

-RRB-

with

s

first

i

-LRB-

z

-RRB-

z

i

-LRB-

z

-RRB-

c

X

k

z

k

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

k

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Now

we

have

to

distinguish

between

jzj

ae

and

jzj

ae

for

jzj

ae

we

find

an

upper

bound

c

X

k

ae

j

j

k

q

j

j

ae

k

c

q

j

j

ae

X

k

qk

q

j

j

ae

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-LRB-

-RRB-

with

q

For

jzj

ae

we

find

an

upper

bound

in

which

the

sum

does

contribute

c

X

k

jzj

j

j

k

q

jzj

ae

j

j

ae

k

c

q

jzj

ae

j

j

ae

X

k

q

jzj

ae

k

c

q

jzj

ae

j

j

ae

X

k

jzj

ae

k

-LRB-

-RRB-

c

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-LRB-

jzjae

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

jzjae

-LRB-

q

-LRB-

K

-LRB-

z

-RRB-

-RRB-

jzj

ae

-LRB-

-RRB-

-LRB-

-RRB-

c

jzjae

A

jzj

jzjj

j

ae

ae

jzj

ae

jzj

jzjj

j

ae

-LRB-

-RRB-

The

next

form

we

need

uses

s

for

jzj

ae

we

get

-LRB-

as

before

-RRB-

the

upper

bound

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-LRB-

-RRB-

For

jzj

ae

we

can

use

the

bound

derived

in

-LRB-

-RRB-

there

the

upper

limit

in

the

sum

was

replaced

by

to

cover

the

case

s

jzj

jzjj

j

ae

-LRB-

-RRB-

The

proof

of

the

lemma

will

now

be

finished

by

deriving

a

contradiction

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

First

put

jzj

ae

and

assume

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

Then

we

first

read

-LRB-

-RRB-

-LRB-

-RRB-

as

z

-LSB-

-RSB-

i

-LRB-

z

-RRB-

i

-RSB-

-LRB-

z

-RRB-

O

h

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-LRB-

-RRB-

-LRB-

here

-LSB-

r

-RSB-

denotes

the

integral

part

of

r

-RRB-

Then

z

S

-LRB-

z

Fi

-RRB-

lim

sup

z

-LSB-

-RSB-

i

-LRB-

z

-RRB-

max

ae

lim

sup

i

-RSB-

-LRB-

z

-RRB-

lim

sup

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

oe

maxfK

-LRB-

z

-RRB-

qK

-LRB-

z

-RRB-

g

K

-LRB-

z

-RRB-

S

-LRB-

z

Fi

-RRB-

This

implies

S

-LRB-

z

Fi

-RRB-

z

ae

jzj

j

j

ae

ae

jzj

K

-LRB-

z

-RRB-

K

-LRB-

z

-RRB-

a

contradiction

with

the

assumption

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

Thus

for

jzj

ae

B

-LRB-

z

-RRB-

-LRB-

ii

-RRB-

=-RRB-

-LRB-

i

-RRB-

For

jzj

ae

the

bounds

-LRB-

-RRB-

-LRB-

-RRB-

can

be

written

as

z

-LSB-

-RSB-

i

-LRB-

z

-RRB-

i

-RSB-

-LRB-

z

-RRB-

O

jzjj

j

ae

-LRB-

-RRB-

and

now

the

assumption

S

-LRB-

z

Fi

-RRB-

K

-LRB-

z

-RRB-

jzjj

j

ae

leads

to

-LRB-

K

-LRB-

z

-RRB-

-RRB-

lim

sup

i

-RSB-

-LRB-

z

-RRB-

max

ae

lim

sup

z

-LSB-

-RSB-

i

-LRB-

z

-RRB-

lim

sup

jzjj

j

ae

oe

max

-LRB-

z

jzjj

j

ae

jzj

j

ae

-RRB-

z

jzjj

j

ae

z

K

-LRB-

z

-RRB-

This

implies

K

-LRB-

z

-RRB-

z

z

ae

jzjj

j

K

-LRB-

z

-RRB-

ae

jzj

K

-LRB-

z

-RRB-

K

-LRB-

z

-RRB-

again

a

contradiction

Thus

for

jzj

ae

B

-LRB-

z

-RRB-

-LRB-

ii

-RRB-

=-RRB-

-LRB-

i

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Lemma

For

i

-LRB-

z

-RRB-

we

have

the

following

representation

i

-LRB-

z

-RRB-

X

s

R

-LRB-

i

-RRB-

s

j

j

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-LRB-

-RRB-

with

X

s

R

-LRB-

i

-RRB-

s

B

-LRB-

z

-RRB-

oe

X

k

oeX

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

zs

z

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

o

m

-LRB-

-RRB-

where

the

coefficients

rs

-LRB-

s

oe

-RRB-

are

given

by

rs

s

-LRB-

s

oe

-RRB-

-LRB-

-RRB-

Before

embarking

upon

the

proof

of

this

lemma

we

need

to

state

some

facts

about

the

coefficients

s

es

vs

ai

j

A

i

j

to

be

found

in

-LSB-

-RSB-

B

-LRB-

z

-RRB-

dY

i

Bi

-LRB-

z

-RRB-

oe

X

s

eszs

j

s

esj

-LRB-

s

oe

-RRB-

-LRB-

-RRB-

lim

s

es

-LRB-

s

oe

-RRB-

j

s

j

for

all

s

-LRB-

-RRB-

rs

s

jrsj

j

j

ae

-LRB-

oe

-RRB-

-LRB-

s

oe

-RRB-

-LRB-

-RRB-

jai

jj

-LRB-

ae

-RRB-

j

-LRB-

j

i

d

-RRB-

-LRB-

-RRB-

A

i

s

X

r

ai

r

-LRB-

oe

-RRB-

s

r

-LRB-

oe

-RRB-

-LRB-

s

-RRB-

A

i

s

oe

A

i

oe

s

-LRB-

s

-RRB-

-LRB-

-RRB-

s

A

i

s

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

e

s

with

je

sj

-LRB-

ae

-RRB-

s

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

oe

for

s

-LRB-

-RRB-

s

A

i

s

ai

-LRB-

oe

-RRB-

oe

s

oe

-LRB-

oe

-RRB-

s

with

j

sj

j

j

ae

oe

je

s

oe

j

for

s

-LRB-

-RRB-

A

i

s

ai

s

ffi

s

with

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

jffi

sj

-LRB-

ae

-RRB-

s

j

j

ae

oe

s

-LRB-

-RRB-

jA`i

sj

-LRB-

ae

-RRB-

s

s

-LRB-

ae

-RRB-

s

j

j

ae

oe

s

-LRB-

-RRB-

oe

X

k

i

k

p

i

s?k

oe

X

k

kA`i

s?k

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

with

i

k

for

k

k

i

p

i

s

for

s

s

oe

i

-RRB-

Proof

of

Lemma

From

the

definition

-LRB-

-RRB-

of

the

quantities

i

-LRB-

z

-RRB-

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

-LRB-

z

-RRB-

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

we

see

that

a

good

form

for

the

products

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

is

needed

With

U

i

-LRB-

z

-RRB-

oe

i

X

s

p

i

szs

-LRB-

i

d

-RRB-

we

find

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

i

X

k

i

kzk

oe

i

X

s

p

i

szs

oeX

s

oe

X

k

i

kp

i

s?k

zs

oeX

s

oe

X

k

kA`i

s?k

zs

-LRB-

-RRB-

where

the

equations

and

conventions

of

-LRB-

-RRB-

have

been

used

Consider

the

dif

ference

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

X

s

A

i

szs

insert

-LRB-

-RRB-

and

the

explicit

form

of

B

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

X

s

A

i

szs

oeX

s

oe

X

k

kA`i

s?k

zs

oe

X

k

kzk

X

s

A

i

szs

oeX

s

oe

X

k

kA`i

s?k

zs

oeX

s

oe

X

k

s?k

kA`i

s?k

zs

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

The

first

term

on

the

right

hand

side

is

the

full

double

sum

while

the

second

term

consists

of

a

double

sum

with

the

extra

condition

s

k

Subtracting

the

two

double

sums

putting

o

s

k

we

are

left

over

with

two

other

double

sums

one

with

k

oe

and

with

o

oe

k

for

fixed

k

the

other

with

k

oe

and

k

o

The

conclusion

is

-LRB-

renaming

o

s

-RRB-

Bi

-LRB-

z

-RRB-

U

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

X

s

A

i

szs

oe

X

k

oe?k

X

s

k

kA`i

szs

k

oe

X

k

X

s

k

kA`i

szs

k

-LRB-

-RRB-

Insert

-LRB-

-RRB-

with

resp

into

-LRB-

-RRB-

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

ae

-LRB-

z

-RRB-

X

s

szs

oe

X

k

oe?k

X

s

szs

k

oe

X

k

X

s

k

szs

koe

-LRB-

z

-RRB-

ae

B

-LRB-

z

-RRB-

X

s

A

i

szs

oe

X

k

oe?k

X

s

kA`i

szs

k

oe

X

k

X

s

k

kA`i

szs

koe

with

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

X

s

-LRB-

s

A

i

s

-RRB-

zs

B

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

szs

k

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

kA`i

szs

k

B

-LRB-

z

-RRB-

oe

X

k

X

s

k

szs

k

-LRB-

z

-RRB-

oe

X

k

X

s

k

kA`i

szs

k

-LRB-

-RRB-

To

be

able

to

compare

terms

where

two

of

the

three

suffices

are

the

same

but

different

from

the

third

-LRB-

in

and

-RRB-

we

have

to

throw

in

some

connecting

terms

remains

as

is

but

both

and

are

split

into

three

expressions

each

fB

-LRB-

z

-RRB-

-LRB-

z

-RRB-

g

oe

X

k

oe?k

X

s

szs

k

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

-LRB-

s

A

i

s

-RRB-

zs

k

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

-LRB-

k

-RRB-

A

i

szs

k

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

and

fB

-LRB-

z

-RRB-

-LRB-

z

-RRB-

g

oe

X

k

X

s

k

szs

k

-LRB-

z

-RRB-

oe

X

k

X

s

k

-LRB-

s

A

i

s

-RRB-

zs

k

-LRB-

z

-RRB-

oe

X

k

X

s

k

-LRB-

k

-RRB-

A

i

szs

k

-LRB-

-RRB-

It

is

now

necessary

to

weed

out

certain

terms

-LRB-

those

that

can

be

estimated

with

the

order

of

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

as

indicated

in

-LRB-

-RRB-

-RRB-

to

find

the

representation

-LRB-

-RRB-

-LRB-

-RRB-

In

the

procedure

the

estimates

-LRB-

-LCB-

h

-RRB-

are

used

many

times

the

six

pages

of

calculations

will

not

be

given

in

full

here

B

-LRB-

z

-RRB-

X

s

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Replace

in

the

expression

s

A

i

s

by

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

this

introduces

an

error

that

can

be

bounded

by

jB

-LRB-

z

-RRB-

j

-LRB-

z

-RRB-

j

X

s

-LRB-

ae

-RRB-

s

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

oe

jzjs

according

to

-LRB-

-RRB-

Also

jB

-LRB-

z

-RRB-

j

-LRB-

z

-RRB-

j

can

be

bounded

by

a

constant

leading

to

an

upper

bound

for

the

error

of

the

form

M

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

oe

X

s

jzj

ae

s

The

factor

j

j

ae

oe

is

used

as

q

-LRB-

apart

from

a

constant

-RRB-

and

there

are

now

three

cases

-LRB-

i

-RRB-

jzj

ae

take

small

enough

to

get

P

jzj

ae

s

jzj

ae

s

C

-LRB-

ii

-RRB-

jzj

ae

later

on

we

are

going

to

remove

-LRB-

i.e.

-RRB-

so

the

worst

case

here

is

P

as

q

j

j

ae

we

can

introduce

a

slightly

larger

to

get

rid

of

the

factor

-LRB-

q

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

M

-LRB-

-RRB-

-RRB-

-LRB-

iii

-RRB-

jzj

ae

P

jzj

ae

s

-LRB-

jzj

ae

-RRB-

-LRB-

jzj

ae

-RRB-

c

jzj

ae

just

the

factor

that

turns

the

upper

bound

into

K

-LRB-

z

-RRB-

for

jzj

ae

As

a

second

step

in

the

derivation

of

-LRB-

-RRB-

we

want

to

replace

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

by

B

-LRB-

z

-RRB-

From

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

-LRB-

B

-LRB-

z

-RRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

with

j?j

oe

X

k

jzjs

-LRB-

the

sum

contains

oe

terms

a

finite

fixed

number

-RRB-

we

find

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

where

j

-LRB-

z

-RRB-

-LRB-

-RRB-

c

which

shows

that

the

total

error

because

of

writing

as

on

the

right

hand

side

of

-LRB-

-RRB-

can

be

bounded

by

-LRB-

use

-LRB-

-RRB-

too

-RRB-

c

X

s

-LRB-

ae

-RRB-

-LRB-

oe

-RRB-

s

j

j

-LRB-

oe

-RRB-

jzjs

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

c

j

j

ae

-LRB-

oe

-RRB-

q

X

s

jzj

ae

s

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

treating

P

jzj

ae

s

in

the

same

manner

as

on

the

previous

page

As

the

steps

in

estimating

the

other

quantities

-LRB-

-RRB-

are

globally

the

same

as

those

for

only

the

method

to

reach

the

result

will

be

indicated

oe

X

m

rmzm

oe

X

k

oe?k

X

s

ekai

szs

k

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Method

B

-LRB-

z

-RRB-

-LRB-

z

-RRB-

Poe

m

rmzm

replace

by

ek

s

by

ai

s

use

the

bounds

-LRB-

c

d

h

-RRB-

keeping

in

mind

that

oe

is

finite

and

fixed

B

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

k

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Method

Replace

-LRB-

z

-RRB-

by

B

-LRB-

z

-RRB-

and

use

-LRB-

b

f

-RRB-

replace

then

by

ek

and

use

-LRB-

d

f

-RRB-

finally

replace

s

A

i

s

by

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

and

use

-LRB-

-RRB-

B

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

rkai

szs

k

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Method

Replace

-LRB-

z

-RRB-

by

B

-LRB-

z

-RRB-

and

use

-LRB-

c

i

-RRB-

then

replace

A

i

s

by

ai

s

use

-LRB-

h

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Method

Straightforward

application

of

-LRB-

c

i

-RRB-

j

j

ae

oe

acts

as

qoe

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Method

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

and

is

therefore

bounded

apply

-LRB-

g

h

-RRB-

and

use

j

j

ae

oe

for

qoe

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Method

-LRB-

z

-RRB-

bounded

-LRB-

i

-RRB-

j

j

ae

oe

for

qoe

Insert

-LRB-

-LCB-

g

-RRB-

into

i

-LRB-

z

-RRB-

as

given

by

-LRB-

-RRB-

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

X

s

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

oe

X

m

rmzm

oe

X

k

oe?k

X

s

ek

ai

szs

k

B

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

k

B

-LRB-

z

-RRB-

oe

X

k

oe?k

X

s

rkai

szs

k

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Combination

of

the

first

and

third

term

on

the

right

gives

B

-LRB-

z

-RRB-

ae

B

-LRB-

z

-RRB-

X

s

ai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

oe

X

k

oe?k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

koe

B

-LRB-

z

-RRB-

ae

oe

X

k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

k

oe

X

k

oe?k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

koe

as

the

inner

sum

from

Poe

k

Poe?k

s

is

empty

i.e.

zero

for

k

oe

The

expression

between

curly

braces

can

be

written

as

oe

X

k

oe?k

X

s

ekai

-LRB-

oe

-RRB-

s

-LRB-

oe

-RRB-

zs

k

oe

X

k

oeX

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

zs

which

is

just

the

first

term

on

the

right

in

-LRB-

-RRB-

The

second

and

fourth

term

in

-LRB-

-RRB-

lead

with

B

-LRB-

z

-RRB-

Poe

m

emzm

to

oe

X

m

oe

X

k

oe?k

X

s

rmekai

szs

k

m

oe

X

m

oe

X

k

oe?k

X

s

o

emrkai

szs

k

m

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

oe

X

m

oe

X

k

oe?k

X

s

-LRB-

rmek

rk

em

-RRB-

ai

szs

k

m

oe

X

k

oe?k

X

o

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

o

m

-LRB-

o

s

-RRB-

which

is

the

second

term

on

the

right

in

-LRB-

-RRB-

if

we

just

change

the

order

of

summation

oe

X

k

oe?k

X

o

oe

X

o

oe

o

X

k

This

proves

-LRB-

-RRB-

and

the

error

made

in

-LRB-

-RRB-

is

O

-LSB-

-LRB-

qK

-LRB-

z

-RRB-

-RRB-

-RSB-

showing

-LRB-

-RRB-

too

Proof

of

the

Theorems

for

jzj

ae

We

can

now

give

the

proofs

for

the

theorems

about

the

exceptional

points

inside

the

disk

Proof

of

Theorem

Assume

that

there

is

at

least

one

function

Fi

say

for

which

there

are

exceptional

points

in

jzj

ae

S

-LRB-

Fj

-RRB-

K

-LRB-

>=-RRB-

-LRB-

-RRB-

From

Lemma

we

then

find

j

-LRB-

>=-RRB-

j

-LRB-

>=-RRB-

O

-LSB-

-LRB-

qK

-LRB-

>=-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Now

apply

Lemma

the

error

made

by

replacing

j

-LRB-

>=-RRB-

and

j

-LRB-

>=-RRB-

by

the

corresponding

formulae

-LRB-

-RRB-

is

of

the

same

order

as

the

error

on

the

right

in

-LRB-

-RRB-

Because

the

rm

do

depend

on

the

value

of

we

have

to

distinguish

between

rm

's

for

j

-LRB-

>=-RRB-

and

those

for

j

-LRB-

>=-RRB-

the

last

ones

will

be

denoted

by

brm

The

left

side

of

-LRB-

-RRB-

then

consists

of

two

different

terms

a

double

sum

-LRB-

-RRB-

and

a

triple

sum

-LRB-

-RRB-

B

-LRB-

>=-RRB-

oe

X

k

oeX

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

B

-LRB-

>=-RRB-

oe

X

k

oe

X

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

-LRB-

-RRB-

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

k

o

m

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

brm

ek

brk

em

-RRB-

ai

o

k

o

m

-LRB-

-RRB-

Note

that

the

change

of

into

does

not

affect

the

value

of

oe

-LRB-

for

oe

d

-RRB-

oe

d

for

-LRB-

-RRB-

d

-RRB-

-LRB-

-RRB-

d

First

it

will

be

shown

that

can

be

estimated

away

into

the

same

order

term

as

in

-LRB-

-RRB-

using

-LRB-

d

-RRB-

and

j

j

ae

=-RRB-

j

j

qae

for

a

q

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

k

o

m

-LRB-

qae

-RRB-

oe

X

o

oe

o

X

k

oe

X

m

max

k

oe

j

ekj

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

o

-LRB-

qae

-RRB-

k

o

m

max

k

oe

j

ekj

j

j

ae

-LRB-

oe

-RRB-

qae

ae

oe

X

o

oe

o

X

k

oe

X

m

qae

ae

o

-LRB-

qae

-RRB-

k

m

-LRB-

-RRB-

Given

a

q

q

we

can

find

another

for

small

enough

such

that

qae

ae

furthermore

the

three

sums

all

contain

a

finite

number

of

terms

the

bound

is

of

the

form

f

M

j

j

ae

-LRB-

oe

-RRB-

q

i.e.

of

the

order

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

because

j!jj

ae

The

other

triple

sum

gives

the

same

bound

-LRB-

an

extra

factor

j

j

-LRB-

ae

-RRB-

does

not

matter

-RRB-

showing

that

-LRB-

-RRB-

can

be

written

with

-LRB-

-RRB-

as

B

-LRB-

>=-RRB-

ae

oe

X

k

oeX

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

oe

X

k

oe

X

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

oe

O

-LSB-

-LRB-

qK

-LRB-

>=-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Rewrite

the

second

term

between

the

curly

brackets

oe

X

k

oe

X

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

oe

X

k

oe

X

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

showing

that

there

is

some

cancellation

in

-LRB-

-RRB-

B

-LRB-

>=-RRB-

ae

oe

X

k

k

X

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

oe

X

k

oe

X

s

oe

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

oe

O

-LSB-

-LRB-

qK

-LRB-

>=-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Now

use

jB

-LRB-

-RRB-

j

is

bounded

j

j

qae

for

a

q

with

q

and

-LRB-

-RRB-

to

get

oe

X

k

oe

X

s

oe

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

s

max

k

oe

j

j

j

ae

-LRB-

oe

-RRB-

oe

X

k

oe

X

s

oe

qae

ae

s

-LRB-

ae

-RRB-

k

This

can

be

treated

as

before

qae

ae

-LRB-

small

enough

-RRB-

finite

number

of

terms

in

the

sums

thus

this

term

is

of

the

order

-LRB-

qK

-LRB-

>=-RRB-

-RRB-

Using

B

-LRB-

>=-RRB-

and

putting

o

s

k

we

find

the

important

formula

oe

X

k

X

o

ekai

-LRB-

oe

-RRB-

o

o

k

O

qK

-LRB-

>=-RRB-

j

j

-LRB-

-RRB-

Rewriting

the

sums

another

factor

Poe

k

ek!k

B

-LRB-

>=-RRB-

can

be

divided

out

and

we

get

X

o

ai

-LRB-

oe

-RRB-

o

o

O

qK

-LRB-

>=-RRB-

j

j

which

because

K

-LRB-

>=-RRB-

j

jae

leads

to

X

o

ai

-LRB-

oe

-RRB-

o

o

O

q

ae

O

q

ae

oe

-LRB-

-RRB-

This

set

of

equations

for

the

unknowns

ai

k

-LRB-

oe

-RRB-

k

-LRB-

oe

-RRB-

has

a

non-vanishing

determinant

-LRB-

Vandermonde

-RRB-

and

application

of

Cramer

's

rule

shows

ai

-LRB-

oe

-RRB-

j

O

q

ae

oe

j

As

f

-LRB-

oe

-RRB-

j

j

N

j

N

this

implies

lim

sup

k

jai

k

lim

sup

oe

jai

-LRB-

oe

-RRB-

j

j

-LRB-

oe

-RRB-

j

ae

ae

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

a

contradiction

with

-LRB-

-RRB-

Now

we

know

that

the

number

of

exceptional

points

is

at

most

it

will

be

shown

that

there

are

indeed

d-tuples

of

meromorphic

functions

for

which

each

of

the

functions

possesses

that

maximal

number

of

exceptional

points

Proof

of

Theorem

Let

now

d

-LRB-

-RRB-

points

be

given

in

Dae

n

and

consider

the

subdivision

into

d

sets

of

points

each

i

j

-LRB-

j

i

d

-RRB-

There

will

be

constructed

a

d-tuple

-LRB-

F

F

d

-RRB-

of

the

form

-LRB-

-RRB-

and

-LRB-

-RRB-

with

the

properties

F

i

analytic

at

i

j

-LRB-

j

-RRB-

S

-LRB-

i

j

F

i

-RRB-

K

-LRB-

i

j

-RRB-

-LRB-

j

-RRB-

for

all

values

of

i

d.

Define

for

each

i

dg

the

coefficients

fai

ngn

by

ai

m

ae

mai

with

ai

-LRB-

m

:-RRB-

X

s

ai

m

s!si

j

-LRB-

j

m

:-RRB-

-LRB-

-RRB-

For

m

fixed

we

can

write

BBBB

i

i

i

i

i

i

CCCCA

BBBB

ai

m

ai

m

ai

m

CCCCA

ai

m

BBBB

CCCCA

ae

mai

BBBB

CCCCA

-LRB-

-RRB-

The

determinant

of

the

system

is

i

i

i

-LRB-

i

i

i

-RRB-

and

application

of

Cramer

's

rule

leads

to

a

very

nice

formula

The

numerator

for

the

unknown

ai

m

j

-LRB-

j

-RRB-

is

a

Vandermonde

determinant

with

the

column

containing

ji

missing

and

instead

thereof

a

column

containing

i

appears

-LRB-

and

the

order

of

the

columns

is

interchanged

-RRB-

If

we

denote

this

Vandermonde

by

Vj

-LRB-

i

i

-RRB-

we

get

ai

m

j

ae

m

Vj

-LRB-

i

i

-RRB-

i

i

V

-LRB-

i

i

-RRB-

ai

-LRB-

j

-RRB-

-LRB-

-RRB-

Very

important

here

is

that

all

determinants

in

-LRB-

-RRB-

are

independent

of

m

and

we

can

conclude

ai

n

ae

ai

n

-LRB-

n

-RRB-

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

For

an

arbitrary

m

we

can

write

m

k

s

for

a

certain

k

and

s

this

immediately

implies

lim

sup

m

m?smod

jai

m

lim

sup

k

ae

k

k

s

jai

-LRB-

k

s

-RRB-

ae

-LRB-

-RRB-

However

for

the

sequence

m

k

k

lim

k

jai

k

k

lim

k

ae

k

ae

-LRB-

-RRB-

We

conclude

lim

supm

jai

m

and

the

functions

f?i

-LRB-

z

-RRB-

X

m

ai

mzm

-LRB-

i

d

-RRB-

F

i

-LRB-

z

-RRB-

f?i

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

-LRB-

i

d

-RRB-

-LRB-

-RRB-

with

Bi

-LRB-

z

-RRB-

Q?ij

-LRB-

z

zi

j

-RRB-

zi

zi

-LRB-

-RRB-

-LRB-

-RRB-

satisfy

the

condition

-LRB-

-RRB-

stated

at

the

beginning

of

this

proof

if

we

choose

ai

in

such

a

manner

that

f?i

-LRB-

zi

j

-RRB-

-LRB-

j

i

-RRB-

Looking

back

at

-LRB-

-RRB-

-LRB-

-RRB-

and

the

way

-LRB-

-RRB-

followed

from

-LRB-

-RRB-

we

have

i

-LRB-

i

j

-RRB-

i

j

i

-LRB-

i

j

-RRB-

B

-LRB-

i

j

-RRB-

-LRB-

oe

-RRB-

X

o

ai

-LRB-

oe

-RRB-

o

oi

j

O

-LSB-

-LRB-

qK

-LRB-

i

j

-RRB-

-RRB-

-RSB-

-LRB-

j

i

d

-RRB-

-LRB-

-RRB-

Because

the

second

line

of

the

definition

of

the

ai

m

in

-LRB-

-RRB-

implies

that

the

term

with

B

-LRB-

i

j

-RRB-

vanishes

we

conclude

i

-LRB-

i

j

-RRB-

i

j

i

-LRB-

i

j

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

i

j

-RRB-

-RRB-

-RSB-

and

thus

-LRB-

Lemma

-RRB-

S

-LRB-

i

j

Fi

-RRB-

K

-LRB-

i

j

-RRB-

thus

i

j

is

exceptional

More

Technical

Lemmas

The

case

jzj

ae

is

more

intricate

and

we

need

further

results

on

the

coefficients

R

-LRB-

i

-RRB-

s

from

-LRB-

-RRB-

Lemma

The

quantities

R

-LRB-

i

-RRB-

s

from

Lemma

satisfy

R

-LRB-

i

-RRB-

s

O

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

s

s

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

R

-LRB-

i

-RRB-

s

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

-LRB-

oe

-RRB-

s

oe

-LRB-

-RRB-

Proof

For

-LRB-

-RRB-

we

have

to

calculate

the

coefficient

of

zs

in

-LRB-

-RRB-

to

this

end

it

is

necessary

to

interchange

the

summation

in

the

part

with

B

-LRB-

z

-RRB-

oe

X

k

oeX

s

k

oeX

s

min

-LRB-

s

oe

-RRB-

X

k

This

part

gives

a

contribution

to

R

-LRB-

i

-RRB-

s

for

all

values

of

s

B

-LRB-

z

-RRB-

oeX

s

min

-LRB-

s

oe

-RRB-

X

k

ekai

-LRB-

oe

-RRB-

s?kzs

-LRB-

oe

-RRB-

oe

X

m

emzm

oeX

o

min

-LRB-

o

oe

-RRB-

X

k

ekai

-LRB-

oe

-RRB-

o?k

zo

-LRB-

oe

-RRB-

oeX

s

ae

X

m

o

s

m

oe

o

oe

em

min

-LRB-

o

oe

-RRB-

X

k

ekai

-LRB-

oe

-RRB-

o?k

oe

zs

-LRB-

oe

-RRB-

There

are

now

three

cases

s

oe

oe

s

-LRB-

oe

-RRB-

-LRB-

oe

-RRB-

s

that

will

influence

the

value

of

min

-LRB-

o

oe

-RRB-

in

the

innermost

sum

-LRB-

i

-RRB-

s

oe

sX

m

s?m

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

max

k

oe

j

ekj

sX

m

s?m

X

k

j

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

-LRB-

oe

-RRB-

s?m?k

c

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

s

as

we

are

considering

a

finite

number

of

terms

in

the

summation

only

-LRB-

ii

-RRB-

oe

s

-LRB-

oe

-RRB-

oe

X

m

min

-LRB-

s?m

oe

-RRB-

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

c

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

s

as

we

still

have

an

upper

bound

of

-LRB-

oe

-RRB-

resp

-LRB-

oe

-RRB-

on

the

number

of

terms

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

-LRB-

iii

-RRB-

-LRB-

oe

-RRB-

s

Now

the

double

sum

is

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

leading

to

the

same

bound

of

the

order

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

s

as

before

There

could

however

be

another

contribution

to

the

coefficient

of

zs

from

the

triple

sum

z

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

o

m

-LRB-

-RRB-

It

will

be

shown

that

actually

the

lowest

power

of

z

following

from

-LRB-

-RRB-

is

z

j

with

j

oe

i.e.

zoe

Then

it

would

be

clear

that

only

in

a

finite

number

of

cases

-LRB-

from

zoe

to

zoe

oe

-RRB-

the

triple

sum

gives

a

contribution

to

the

estimate

X

o

k

m

s

o

oe

m

oe

k

oe

o

-LRB-

rmek

rk

em

-RRB-

ai

o

max

k

oe

j

ekj

oe

X

o

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

o

-LRB-

oe

-RRB-

-LRB-

just

estimating

the

sums

over

m

and

k

by

the

number

of

terms

-RRB-

Now

this

can

again

be

bounded

from

above

by

M

j

j

ae

-LRB-

oe

-RRB-

-LRB-

ae

-RRB-

s

as

the

difference

between

s

and

o

is

a

finite

number

the

extra

terms

are

taken

care

of

by

the

constant

M

There

still

remains

one

step

to

prove

the

validity

of

-LRB-

-RRB-

Make

a

change

of

summation

index

in

-LRB-

-RRB-

by

shifting

o

to

o

z

oe

X

o

oe

o

X

k

oe

X

m

-LRB-

rmek

rk

em

-RRB-

ai

o

zk

o

m

-LRB-

oe

-RRB-

X

j

c

jz

j

-LRB-

-RRB-

with

c

j

X

o

k

m

j

o

k

m

o

oe

m

oe

k

oe

o

-LRB-

rmek

rk

em

-RRB-

ai

o

-LRB-

-RRB-

We

now

restrict

ourselves

to

j

oe

From

o

k

we

see

that

m

j?o

j

oe

and

also

from

m

o

k

j?o

oe

o

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

This

shows

that

the

extra

conditions

on

k

m

can

be

omitted

and

the

coefficient

c

j

satisfies

c

j

j

X

o

j?o

X

k

-LRB-

rj?o

kek

rk

ej?o

k

-RRB-

ai

o

j

oe

-LRB-

-RRB-

The

inner

sum

leads

to

a

surprising

result

j?o

X

k

rj?o

kek

rk

ej?o

k

j?o

X

k

rj?o

kek

j?o

X

k

rk

ej?o

k

j?o

X

ej?o

j?o

X

k

rk

ej?o

k

for

o

j

oe

this

shows

that

the

triple

sum

-LRB-

-RRB-

only

contributes

to

the

powers

zs

with

s

oe

The

final

part

of

the

proof

is

concerned

with

-LRB-

-RRB-

because

s

oe

we

only

have

to

look

at

the

B

-LRB-

z

-RRB-

part

as

was

done

in

the

first

part

of

the

proof

The

coefficient

of

zs

with

s

oe

appeared

to

be

exactly

as

stated

in

-LRB-

-RRB-

looking

at

case

-LRB-

iii

-RRB-

-LRB-

oe

-RRB-

s

oe

-LRB-

oe

-RRB-

The

intersection

of

the

s-values

-LSB-

oe

-RSB-

and

-LSB-

-LRB-

oe

-RRB-

-RSB-

is

just

-LSB-

-LRB-

oe

-RRB-

oe

-RSB-

and

there

both

indices

k

m

run

to

the

maximal

value

of

oe

Lemma

Let

-LRB-

Fd

-RRB-

be

meromorphic

on

jzj

ae

as

in

-LRB-

-RRB-

with

simple

poles

and

let

oe

If

jzj

ae

then

we

have

for

oe

sufficiently

large

z

ae

i

-LRB-

z

-RRB-

z

i

e

-LRB-

z

-RRB-

oe

B

-LRB-

z

-RRB-

X

s

-LRB-

oe

-RRB-

oe

X

k

s

ekai

;-LRB-

-RRB-

szs

k

B

-LRB-

z

-RRB-

-LRB-

oe

-RRB-

X

s

ai

;-LRB-

-RRB-

szs

B

-LRB-

z

-RRB-

X

s

-LRB-

oe

-RRB-

s

X

k

ekai

;-LRB-

-RRB-

szs

k

Cq

j

j

ae

ae

-LRB-

-RRB-

where

e

-LRB-

oe

-RRB-

e

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Proof

Recall

the

defining

equations

for

the

k

as

given

in

-LSB-

-RSB-

formula

-LRB-

-RRB-

with

all

multiplicities

oe

X

k

k

oeX

s

A

i

s?kzsi

j

-LRB-

j

i

i

d

-RRB-

-LRB-

-RRB-

where

the

zi

j

-LRB-

j

i

-RRB-

are

the

simple

poles

of

Fi

in

jzj

ae

We

are

going

to

consider

the

difference

of

-LRB-

-RRB-

with

and

with

that

will

lead

to

a

system

of

equations

for

the

rm

m

showing

the

growth

order

of

rm

as

a

function

of

improving

upon

-LRB-

-RRB-

oe

X

k

k

oeX

s

A

i

s?kzsi

j

oe

X

k

oeX

s

s?kzsi

j

oe

X

k

kzki

j

oe?k

X

s

k

A

i

szsi

j

oe

X

k

zki

j

oe?k

X

s

k

szsi

j

oe

X

k

-LRB-

k

-RRB-

zki

j

oe?k

X

s

k

A

i

szsi

j

oe

X

k

zki

j

oe?k

X

s

k

-LRB-

A

i

j

s

-RRB-

zsi

j

-LRB-

-RRB-

The

method

is

now

as

follows

ffl

the

first

sum

will

be

written

as

Poe

k

rkzki

jfi

-LRB-

zi

j

-RRB-

with

error

j

j

ae

-LRB-

oe

-RRB-

qoe

for

a

certain

q

q

ffl

the

second

sum

will

be

estimated

by

the

same

error

form

We

are

then

in

the

situation

of

the

following

system

of

equations

oe

X

k

rkzki

jfi

-LRB-

zi

j

-RRB-

j

j

ae

-LRB-

oe

-RRB-

qoe

-LRB-

j

i

i

d

-RRB-

-LRB-

-RRB-

and

application

of

Cramer

's

rule

shows

rk

O

q

j

j

ae

oe

-LRB-

-RRB-

The

treatment

of

the

first

sum

starts

with

an

application

of

-LRB-

-RRB-

A

i

s

ai

s

ffi

s

jffi

sj

-LRB-

ae

-RRB-

s

-LRB-

s

-RRB-

and

the

definition

A

i

s

oe

A

i

oe

s

s

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Indeed

-LRB-

using

oe

-RRB-

the

first

sum

can

be

written

oe

X

k

-LRB-

k

-RRB-

zki

j

oe?k

X

s

k

A

i

szsi

j

oe

X

k

rkzki

j

ae

X

s

k

oe

A

i

oe

szsi

j

oe?k

X

s

A

i

szsi

j

oe

oe

X

k

rkzki

j

ae

X

s

k

oe

-LRB-

ai

oe

s

ffi

oe

s

-RRB-

zsi

j

oe?k

X

s

-LRB-

ai

s

ffi

s

-RRB-

zsi

j

oe

-LRB-

-RRB-

The

sum

over

the

negative

values

of

s

X

s

k

oe

-LRB-

ai

oe

s

ffi

oe

s

-RRB-

zsi

j

X

s

k

j

joe

ae

-LRB-

ae

-RRB-

oe

s

-LRB-

ae

-RRB-

oe

s

oe

jzsi

jj

j

joe

ae

-LRB-

ae

-RRB-

oe

-LRB-

ae

-RRB-

oe

oe

X

s

k

jzi

jj

ae

s

j

j

ae

oe

because

P

s

k

jzi

jj

ae

s

is

bounded

by

a

constant

The

ffi

s-part

in

the

sum

over

the

positive

values

of

s

oe?k

X

s

ffi

szsi

j

oe?k

X

s

jzi

jj

ae

s

This

turns

-LRB-

-RRB-

into

oe

X

k

rkzki

j

ae

oe?k

X

s

ai

szsi

j

oe

and

thus

the

first

sum

is

oe

X

k

rkzki

j

X

s

ai

szsi

j

oe

X

k

rkzki

j

ae

X

s

oe?k

ai

szsi

j

oe

-LRB-

-RRB-

Now

use

-LRB-

-RRB-

and

jzi

jj

ae

X

s

oe?k

ai

szsi

j

X

s

oe?k

jzi

jj

ae

s

jzi

jj

ae

oe?k

jzi

jj

ae

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

with

max

q

jzi

jj

ae

and

the

-LRB-

finite

number

of

-RRB-

extra

factors

jzi

jj

ae

k

have

been

put

into

the

constant

Using

the

known

estimate

-LRB-

-RRB-

for

rk

-LRB-

-RRB-

can

be

written

as

oe

X

k

rkzki

jfi

-LRB-

zi

j

-RRB-

j

j

ae

oe

-LRB-

-RRB-

The

second

sum

in

-LRB-

-RRB-

is

estimated

directly

using

-LRB-

f

g

-RRB-

oe

X

k

zki

j

oe?k

X

s

k

-LRB-

A

i

s

s

-RRB-

zsi

j

oe

X

k

jjkae

X

s

k

-LRB-

ae

-RRB-

oe

s

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

-LRB-

oe

-RRB-

jzi

jjs

oe?k

X

s

-LRB-

ae

-RRB-

s

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

oe

jzi

jjsoe

j

j

ae

oe

-LRB-

-RRB-

with

j

j

ae

Combination

of

-LRB-

-RRB-

and

-LRB-

-RRB-

leads

to

-LRB-

-RRB-

Now

comes

the

reason

for

this

better

estimate

for

rm

because

of

the

bound

-LRB-

-RRB-

it

is

possible

to

estimate

away

the

triple

sum

part

in

the

representation

-LRB-

-RRB-

of

i

-LRB-

z

-RRB-

To

this

end

we

use

the

form

of

the

triple

sum

as

given

in

-LRB-

b

-RRB-

-LRB-

oe

-RRB-

X

j

oe

X

o

k

m

j

o

k

m

o

oe

m

oe

k

oe

o

-LRB-

rmek

rk

em

-RRB-

ai

o

z

j

-LRB-

oe

-RRB-

-LRB-

oe

-RRB-

X

j

-LRB-

oe

-RRB-

-LRB-

oe

-RRB-

cqoe

j

j

ae

oe

max

k

oe

j

ekj

-LRB-

ae

-RRB-

o

jzj

j

where

the

sum

over

the

triples

-LRB-

o

k

m

-RRB-

has

been

replaced

by

an

upper

bound

on

the

number

of

terms

One

easily

sees

-LRB-

oe

is

finite

-RRB-

that

the

triple

sum

can

be

bounded

by

j

j

ae

oe

jzj

ae

oe

for

jzj

ae

Therefore

-LRB-

-RRB-

now

leads

to

i

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

oe

X

k

oeX

s

k

ekai

-LRB-

oe

-RRB-

s?k

-LRB-

oe

-RRB-

zs

j

j

ae

oe

jzj

ae

oe

jzj

ae

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

The

derivation

of

-LRB-

-RRB-

uses

-LRB-

-RRB-

for

different

values

of

e

-LRB-

oe

-RRB-

d

=-RRB-

eoe

-LRB-

oe

-RRB-

d

eoe

e

oe

e

d

=-RRB-

oe

-LRB-

oe

-RRB-

d

oe

oe

The

form

i

-LRB-

z

-RRB-

z

i

e

-LRB-

z

-RRB-

consists

of

two

parts

when

we

apply

-LRB-

-RRB-

a

part

containing

B

-LRB-

z

-RRB-

as

a

factor

say

PB

and

the

error

term

say

E

Then

PB

B

-LRB-

z

-RRB-

ae

oe

X

k

X

s

k

ekai

s?k

oe

X

k

X

s

k

ekai

s?k

oe

B

-LRB-

z

-RRB-

ae

oe

X

k

X

s

k

ekai

s?k

oe

X

k

X

s

k

ekai

s?k

B

-LRB-

z

-RRB-

oe

X

k

ek

ae

k

X

s

k

ai

s?k

X

s

ai

s?k

-LRB-

-RRB-

E

j

j

ae

-LRB-

-RRB-

jzj

ae

j

j

ae

jzj

ae

-LRB-

-RRB-

The

first

sum

in

-LRB-

-RRB-

can

be

estimated

by

jB

-LRB-

z

-RRB-

j

oe

X

k

j

ekj

k

X

s

k

-LRB-

ae

-RRB-

s?k

j

j

jzjs

which

leads

to

another

term

as

those

two

already

written

down

in

-LRB-

-RRB-

Here

we

find

j

j

ae

first

without

a

trace

of

q

But

as

we

are

now

considering

the

case

jzj

ae

this

can

be

written

j

j

ae

jzj

ae

ae

jzj

and

ae

jzj

plays

the

role

of

q.

Thus

i

-LRB-

z

-RRB-

z

i

e

-LRB-

z

-RRB-

B

-LRB-

z

-RRB-

oe

X

k

X

s

ekai

s?k

cq

j

j

ae

jzj

ae

jzj

ae

-LRB-

-RRB-

To

turn

-LRB-

-RRB-

into

-LRB-

-RRB-

apply

the

following

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

ffl

shift

the

summation

index

s

to

s

ffl

divide

through

by

z

ffl

change

the

order

of

summation

in

Poe

k

P

s

ekai

;-LRB-

-RRB-

s?k

The

final

step

runs

as

follows

oe

X

k

X

s

ekai

;-LRB-

-RRB-

s?k

oe

X

k

k

X

s

k

ekai

;-LRB-

-RRB-

s

k

X

s

-LRB-

oe

-RRB-

oe

X

k

s

-LRB-

oe

-RRB-

X

s

oe

X

k

X

s

-LRB-

oe

-RRB-

s

X

k

-LRB-

ekai

;-LRB-

-RRB-

s

k

-RRB-

as

can

be

seen

by

splitting

up

the

set

of

values

-LRB-

k

s

-RRB-

Lemma

Let

all

the

zeros

of

the

monic

polynomial

Q

-LRB-

z

-RRB-

Psk

kzk

-LRB-

s

-RRB-

of

degree

s

belong

to

the

disk

jzj

ae

Furthermore

let

the

quantities

An

satisfy

lim

sup

n

n

ae

-LRB-

-RRB-

sX

k

kAs?k

O

q

ae

s

for

a

q

with

q

-LRB-

-RRB-

Then

we

have

lim

sup

n

n

ae

Proof

cf.

M.

P.

Stojanova

-LSB-

-RSB-

Remark

-LRB-

-RRB-

implies

that

Anzn

has

radius

of

convergence

ae

and

-LRB-

-RRB-

shows

that

Q

-LRB-

z

-RRB-

Anzn

has

radius

of

convergence

ae

the

result

is

clear

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Proof

of

Theorems

for

jzj

ae

Proof

of

Theorem

Assume

that

there

is

at

least

one

function

Fi

say

with

oe

exceptional

points

j

in

jzj

ae

S

-LRB-

j

Fi

-RRB-

K

-LRB-

j

-RRB-

j

oe

-LRB-

-RRB-

According

to

Lemma

this

implies

i

-LRB-

j

-RRB-

j

i

-LRB-

j

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

j

oe

-LRB-

-RRB-

The

first

step

in

the

proof

is

to

show

that

can

be

written

in

a

different

form

using

Lemma

X

s

R

-LRB-

i

-RRB-

s

sj

X

s

R

-LRB-

i

-RRB-

s

R

-LRB-

i

-RRB-

s

sj

X

s

R

-LRB-

i

-RRB-

s

sj

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

j

oe

-LRB-

-RRB-

Just

use

-LRB-

-RRB-

and

concentrate

on

the

sum-part

the

error

is

just

as

it

should

be

this

leads

to

-LRB-

-RRB-

at

once

oe

oe

X

s

R

-LRB-

i

-RRB-

s

sj

j

oe

oe

X

s

R

-LRB-

i

-RRB-

s

sj

oe

oe

X

s

R

-LRB-

i

-RRB-

s

sj

oe

oe

X

s

R

-LRB-

i

-RRB-

s

sj

The

first

sum

in

-LRB-

-RRB-

and

part

of

the

second

-LRB-

Poes

-RRB-

will

be

bounded

by

the

same

error

term

as

present

in

-LRB-

-RRB-

showing

X

s

oe

R

-LRB-

i

-RRB-

s

R

-LRB-

i

-RRB-

s

sj

X

s

R

-LRB-

i

-RRB-

s

sj

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

because

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

itself

in

view

of

-LRB-

-RRB-

From

Lemma

-LRB-

-RRB-

X

s

R

-LRB-

i

-RRB-

s

sj

X

s

c

j

j

ae

-LRB-

oe

-RRB-

j!jj

ae

s

j

j

ae

-LRB-

oe

-RRB-

j

j

ae

-LRB-

oe

-RRB-

j!j

j

ae

ae

j!jj

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

using

q

-LRB-

ae

-RRB-

mini

j

j!i

jj

To

estimate

Poes

-LRB-

R

-LRB-

i

-RRB-

s

R

-LRB-

i

-RRB-

s

-RRB-

sj

we

have

to

split

the

sum

in

two

parts

-LRB-

oe

-RRB-

X

s

oeX

s

-LRB-

oe

-RRB-

For

the

first

sum

the

same

estimate

as

for

-LRB-

-RRB-

works

-LRB-

oe

-RRB-

X

s

R

-LRB-

i

-RRB-

s

R

-LRB-

i

-RRB-

s

sj

-LRB-

oe

-RRB-

X

s

c

j

j

ae

-LRB-

oe

-RRB-

j!jj

ae

s

j

j

j

j

ae

-LRB-

oe

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

The

other

part

of

this

sum

uses

formula

-LRB-

-RRB-

from

Lemma

as

s

-LRB-

oe

-RRB-

this

formula

holds

for

both

R

-LRB-

i

-RRB-

s

and

R

-LRB-

i

-RRB-

s

Keeping

in

mind

that

the

change

does

not

affect

oe

we

get

R

-LRB-

i

-RRB-

s

R

-LRB-

i

-RRB-

s

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s

m?k

-LRB-

oe

-RRB-

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

oe

X

m

oe

X

k

em

ekai

-LRB-

oe

-RRB-

s?m?k

-LRB-

oe

-RRB-

-LRB-

-LRB-

oe

-RRB-

s

oe

-RRB-

-LRB-

-RRB-

Combining

-LRB-

-RRB-

and

-LRB-

b

c

-RRB-

we

find

-LRB-

-RRB-

Now

put

o

s

oe

in

-LRB-

-RRB-

oe

X

o

R

-LRB-

i

-RRB-

o

oe

R

-LRB-

i

-RRB-

o

oe

o

oe

j

oe

X

o

oe

R

-LRB-

i

-RRB-

o

oe

o

oe

j

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

As

oe

-LRB-

d

-RRB-

is

only

a

finite

number

more

than

we

can

write

oe

as

exponent

in

the

order

term

and

divide

through

by

oe

j

-LRB-

-RRB-

oe

X

o

R

-LRB-

i

-RRB-

o

oe

R

-LRB-

i

-RRB-

o

oe

oj

oe

X

o

oe

R

-LRB-

i

-RRB-

o

oe

oj

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

O

qoe

j

j

ae

oe

-LRB-

ae

-RRB-

oe

j

oe

-LRB-

-RRB-

This

is

a

system

of

oe

equations

in

oe

unknowns

having

determinant

-LRB-

Vandermonde

in

j

-RRB-

different

from

zero

apply

Cramer

's

rule

R

-LRB-

i

-RRB-

m

R

-LRB-

i

-RRB-

m

O

qoe

j

j

ae

oe

-LRB-

ae

-RRB-

oe

oe

m

oe

oe

-LRB-

-RRB-

R

-LRB-

i

-RRB-

m

O

qoe

j

j

ae

oe

-LRB-

ae

-RRB-

oe

oe

oe

m

-LRB-

-RRB-

Manipulating

with

-LRB-

b

-RRB-

it

is

possible

to

give

an

order

estimate

for

R

-LRB-

i

-RRB-

m

that

is

valid

for

a

range

of

consecutive

m-values

extending

downward

as

low

as

the

value

oe

Put

instead

of

in

-LRB-

-RRB-

oe

is

replaced

by

oe

and

oe

does

not

change

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

O

qj

j

-LRB-

ae

-RRB-

oe

oe

oe

m

oe

oe

Incorporate

the

factor

in

the

order

term

shift

m

to

m

and

we

find

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

maxfoe

oe

oe

g

m

oe

oe

-LRB-

-RRB-

-LRB-

replacing

the

lower

bound

oe

oe

by

maxfoe

oe

oe

g

is

convenient

for

the

sequel

-RRB-

Insert

-LRB-

-RRB-

in

the

original

-LRB-

-RRB-

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

maxfoe

oe

oe

g

m

oe

oe

after

which

the

factor

is

again

incorporated

in

the

order

term

and

m

is

shifted

to

m

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

maxfoe

oe

oe

g

m

oe

oe

Finally

we

replace

by

-LRB-

thus

oe

by

oe

oe

does

not

change

-RRB-

and

put

the

exponent

oe

back

to

oe

in

the

order

term

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

maxfoe

oe

oe

m

oe

oe

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Comparing

-LRB-

-RRB-

and

-LRB-

-RRB-

we

see

that

the

steps

in

the

proof

have

extended

the

validity

range

of

m

values

downwards

by

a

value

of

without

changing

the

form

of

the

order

term

only

the

constant

that

comes

with

it

If

this

method

is

repeated

finitely

many

times

-LRB-

not

depending

on

-RRB-

we

keep

the

same

order

of

magnitude

The

aim

is

to

do

this

in

such

a

way

that

there

is

no

gap

and

oe

A

-LRB-

-RRB-

oe

oe

oe

A

-LRB-

-RRB-

oe

The

condition

on

A

is

A

-LRB-

oe

-RRB-

and

therefore

we

find

in

-LRB-

oe

-RRB-

steps

more

R

-LRB-

i

-RRB-

m

O

qj

j

-LRB-

ae

-RRB-

oe

oe

m

oe

oe

-LRB-

-RRB-

There

is

however

in

-LRB-

-RRB-

an

explicit

form

for

the

R

-LRB-

i

-RRB-

s

combining

that

form

with

-LRB-

-RRB-

consequtive

s

values

from

-LRB-

-RRB-

leads

to

oe

X

m

em

oe

X

k

ekai

-LRB-

oe

-RRB-

s?m?ka

-LRB-

oe

-RRB-

O

q

ae

oe

j

j

ae

oe

oe

s

oe

Divide

out

-LRB-

oe

-RRB-

-LRB-

the

extra

is

absorbed

in

the

constant

-RRB-

oe

X

m

em

oe

X

k

ekai

-LRB-

oe

-RRB-

s?m?k

O

q

ae

oe

-LRB-

ae

-RRB-

oe

oe

s

oe

As

-LRB-

oe

-RRB-

s

runs

from

-LRB-

oe

-RRB-

oe

-LRB-

-RRB-

oe

to

-LRB-

oe

-RRB-

oe

-LRB-

-RRB-

oe

we

can

let

-LRB-

oe

-RRB-

s

run

through

all

integers

by

letting

run

through

all

integers

We

conclude

oe

X

m

em

oe

X

k

ekai

n?m?k

O

qoe

-LRB-

ae

-RRB-

-LRB-

-RRB-

oe

O

ae

n

-LRB-

-RRB-

Now

is

still

a

number

in

the

open

interval

-LRB-

-RRB-

application

of

Lemma

twice

shows

lim

sup

n

jai

n

ae

a

contradiction

with

-LRB-

-RRB-

-LRB-

-RRB-

Proof

of

Theorem

Let

for

each

i

a

set

of

-LRB-

oe

-RRB-

distinct

points

i

j

in

jzj

ae

be

given

Because

the

functions

Fi

will

be

constructed

separately

the

index

i

on

the

points

will

be

dropped

for

notational

convenience

From

now

on

i

is

fixed

Define

the

sequence

fai

by

ai

;-LRB-

-RRB-

n

s

Aae

-LRB-

-RRB-

n?s

-LRB-

oe

-RRB-

s

n

-LRB-

-RRB-

ai

;-LRB-

-RRB-

n

s

s

-LRB-

oe

-RRB-

n

-LRB-

oe

-RRB-

-LRB-

-RRB-

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Here

A

is

a

constant

that

will

be

chosen

in

a

special

way

later

on

Now

the

equations

-LRB-

-RRB-

-LRB-

-RRB-

define

the

ai

m

for

m

's

with

m

-LRB-

oe

-RRB-

-LRB-

-RRB-

completely

and

thereafter

only

for

those

m

's

that

satisfy

-LRB-

oe

-RRB-

m

mod

-LRB-

-RRB-

There

are

two

sets

of

coefficients

to

define

yet

-LRB-

i

-RRB-

the

finite

set

fai

kg

k

-LRB-

ii

-RRB-

the

infinite

set

fai

;-LRB-

-RRB-

n

sg

s

-LRB-

oe

-RRB-

and

n

-LRB-

oe

-RRB-

The

second

set

is

defined

inductively

using

-LRB-

oe

-RRB-

X

s

sj

ai

;-LRB-

-RRB-

n

s

B

-LRB-

j

-RRB-

ae

X

s

-LRB-

oe

-RRB-

oe

X

k

s

ekai

;-LRB-

-RRB-

n

s!s

k

j

X

s

-LRB-

oe

-RRB-

s

X

k

ekai

;-LRB-

-RRB-

n

s!k

s

j

oe

-LRB-

-RRB-

where

s

-LRB-

oe

-RRB-

n

-LRB-

oe

-RRB-

j

-LRB-

oe

-RRB-

Indeed

the

left

hand

side

contains

the

unknowns

ai

;-LRB-

-RRB-

n

s

with

s

-LRB-

oe

-RRB-

while

the

indices

on

the

right

hand

side

are

of

the

forms

-LRB-

-RRB-

n

s

with

-LRB-

oe

-RRB-

s

defined

by

-LRB-

-RRB-

s

-LRB-

oe

-RRB-

i.e.

n

-LRB-

s

-RRB-

with

s

-LRB-

oe

-RRB-

-LRB-

-RRB-

with

n

-LRB-

oe

-RRB-

The

crux

of

the

proof

is

now

to

show

that

A

can

be

chosen

in

such

a

manner

that

lim

sup

n

jai

nj

ae

-LRB-

-RRB-

From

-LRB-

-RRB-

it

is

clear

that

-LRB-

-RRB-

holds

for

the

subsequence

-LRB-

even

with

limit

instead

of

lim

sup

-RRB-

ai

;-LRB-

-RRB-

m

s

with

-LRB-

oe

-RRB-

s

fixed

and

m

Apply

ing

Cramer

's

rule

on

the

equation

-LRB-

-RRB-

again

the

Vandermonde

determinant

on

-LRB-

oe

-RRB-

plays

a

role

treating

the

ai

's

on

the

right

as

quantities

to

be

traced

ai

;-LRB-

-RRB-

n

m

X

s

-LRB-

oe

-RRB-

cm

s

ai

;-LRB-

-RRB-

n

s

X

s

-LRB-

oe

-RRB-

ecm

sai

;-LRB-

-RRB-

n

s

-LRB-

-RRB-

where

the

constants

cm

s

ecm

s

only

depend

on

the

j

and

ek

m

-LRB-

oe

-RRB-

Introduce

the

explicit

forms

of

-LRB-

b

-RRB-

into

the

right

hand

side

of

this

equation

to

find

ai

;-LRB-

-RRB-

n

m

X

s

-LRB-

oe

-RRB-

cm

sAae

-LRB-

-RRB-

n?s

X

s

-LRB-

oe

-RRB-

ecm

sAae

-LRB-

-RRB-

n?s

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

-LRB-

the

second

sum

is

-LRB-

-RRB-

the

first

sum

originally

contains

ai

;-LRB-

-RRB-

n

s

with

-LRB-

oe

-RRB-

s

i.e.

ai

;-LRB-

-RRB-

-LRB-

n

-RRB-

-LRB-

s

-RRB-

with

-LRB-

oe

-RRB-

s

they

give

rise

to

Aae

-LRB-

-RRB-

-LRB-

n

-RRB-

-LRB-

s

-RRB-

Aae

-LRB-

-RRB-

n?s

too

-RRB-

Pulling

out

the

common

factor

Aae

-LRB-

-RRB-

n?m

we

can

write

ai

;-LRB-

-RRB-

n

m

dmAae

-LRB-

-RRB-

n?m

m

-LRB-

oe

-RRB-

n

-LRB-

oe

-RRB-

-LRB-

-RRB-

with

dm

a

constant

that

can

be

given

explicitly

by

dm

X

s

-LRB-

oe

-RRB-

cm

saem?s

X

s

-LRB-

oe

-RRB-

ecm

saem?s

-LRB-

-RRB-

As

m

runs

through

finitely

many

values

only

in

-LRB-

-RRB-

we

see

that

lim

sup

n

jai

;-LRB-

-RRB-

n

f

-LRB-

-RRB-

n

mg

A

ae

m

-LRB-

oe

-RRB-

-LRB-

-RRB-

and

from

-LRB-

-RRB-

it

is

clear

that

lim

sup

n

jai

;-LRB-

-RRB-

n

f

-LRB-

-RRB-

n

mg

A

ae

-LRB-

oe

-RRB-

m

-LRB-

-RRB-

While

-LRB-

b

-RRB-

cover

all

natural

numbers

we

find

-LRB-

-RRB-

taking

A

Finally

the

-LRB-

-RRB-

values

ai

ai

ai

are

chosen

in

such

a

manner

that

f?i

-LRB-

z

-RRB-

X

m

ai

mzm

-LRB-

-RRB-

satisfies

f?i

-LRB-

zi

j

-RRB-

j

i

-LRB-

-RRB-

Because

oe

d

this

is

always

possible

Now

put

F

i

-LRB-

z

-RRB-

f?i

-LRB-

z

-RRB-

Bi

-LRB-

z

-RRB-

-LRB-

-RRB-

then

-LRB-

F

F

d

-RRB-

are

functions

satisfying

the

conditions

-LRB-

-RRB-

-LRB-

-RRB-

and

moreover

the

j

for

F

i

lead

with

Lemma

to

i

-LRB-

j

F

i

-RRB-

j

i

e

-LRB-

j

F

i

-RRB-

B

-LRB-

j

-RRB-

B

-LRB-

j

-RRB-

X

s

-LRB-

oe

-RRB-

oe

X

k

s

ekai

;-LRB-

-RRB-

s!s

k

j

-LRB-

oe

-RRB-

X

s

ai

;-LRB-

-RRB-

s!sj

B

-LRB-

j

-RRB-

X

s

-LRB-

oe

-RRB-

s

X

k

ekai

;-LRB-

-RRB-

s!k

s

j

oe

cq

j

j

ae

j!jj

ae

M.

G.

de

Bruin

and

A.

Sharma

Quantitative

Interpolation

Now

the

quantity

between

curly

brackets

is

identically

zero

because

of

-LRB-

-RRB-

and

the

order

term

is

cq

-LRB-

j

-RRB-

Then

writing

e

we

see

that

i

-LRB-

j

F

i

-RRB-

j

i

-LRB-

j

F

i

-RRB-

O

-LSB-

-LRB-

qK

-LRB-

j

-RRB-

-RRB-

-RSB-

which

implies

S

-LRB-

j

F

i

-RRB-

K

-LRB-

j

-RRB-

by

Lemma

j

is

exceptional

for

F

i

j

-LRB-

oe

-RRB-

References

-LSB-

-RSB-

E.

B.

Saff

A.

Sharma

and

R.

S.

Varga

An

extension

to

rational

functions

of

a

theorem

of

J.

L.

Walsh

on

the

difference

of

interpolating

polynomials

R.A.I.R.O.

Anal

Num?er.

No.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

P.

Stojanova

Equiconvergence

in

rational

approximation

of

functions

Constr

Approx.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

G.

de

Bruin

and

A.

Sharma

Equiconvergence

of

some

simultaneous

Hermite-Pad

e

in

terpolants

Mod?el

math?em

et

Anal

Num?er.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

V.

Totik

Quantitative

results

in

overconvergence

of

complex

interpolating

polynomials

J.

Approx

Th.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

P.

Stojanova

Overconvergence

in

rational

approximation

of

meromorphic

functions

Mathematic

Balkanica

-LRB-

New

Series

-RRB-

-LRB-

-RRB-

-LCB-