Intrinsic functions on complex semisimple algebras

LINEAR

ALGEBRA

Intrinsic

,4ND

ITS

APPLICATIONS

Functions on Complex

429

Semisimple

Algebras

J. I’. LONG College of Steubenville

Steubenville,

Ohio

AND

C. G. CULLEN

University

of Pittsbuvgh

Pittsburgh,

Pennsylvmia

Communicated

by Marvin

Marcus

1. INTRODUCTION

Let

X be a finite

dimensional

associative

algebra

over the complex

field V: and let (5 be the group of all automorphisms phisms

of ‘u which 1.1.

DEFINITION

said to be intrinsic (i) (ii)

pointwise

A function

F(QZ)

VZE a,

=&Y(Z), of intrinsic [6].

a domain

functions

VQE6. on 2l was introduced

In this paper we complete

this case it is known

and motivated

the characterization

for the case in which ‘u is semisimple

of such

over %?’(zero radical).

In

that

where each ‘Xi is a simple algebra over 59 [l 1. Moreover, the only simple algebras

over V are the total matrix

%n = (~2 x n matrices Linear Copyright

3 of ‘3 into ‘u, is

on %I if WE@,

by Rinehart

invariant.

F, mapping

ZED*QZED,

The study functions

leave 9

and antiautomor-

0

1970

up to isomorphism,

algebras

over %7} [l 1.

Algebra and Its Applications

by American

Elsevier

Publishing

3( 1970), 429-440 Company,

Inc.

1. P.

430 Because

of these

structure

theorems

LONG

AND

we restrict

C. G. CULLIW

our attention

here

to the algebra

All our results carry over directly the techniques The

described

elements

In general polynomial

of the algebra

semisimple

The spectrum (not necessarily

‘u are thus block

for any ti x n matrix

diagonal

B we shall denote

of B will be denoted A(B) that n(B)

A = A, @A,

0e.e

using

matrices:

the minimum

polynomial

B) = fi (x - &) = xn - al[B]xn-l t=l

distinct);

algebras

[3].

of B by WZ~(X) and the characteristic

c,(x) = det(x1 -

For

to arbitrary

by Hall

by

+ G~[B]~+~

and will always contain

n points

is,

= {L

22,. . >A,}.

@A,E%

we know that

and that

A(A) = ; A(AJ. i=l To help in describing later

we introduce

some of the complicated now the notion

situations

of the tagged

which will occur

spectrum

of A.

DEFINITION1.2. For A = A, @ A, @ . . * @ A, E ‘u, the tugged spectrum of A is T/l(A) The Linear

elements

of T A(A)

= ((i, L)]~LEA(AJ}~

are called

Algebra and Its Applications

the tugged eigenvahes

3(1970),

429-440

of A.

FUNCTIONS

ON

SEMISIMPLE

The set T A(A) distinct,

always contains

vz =

and it should be clear that,

Q-‘AQ

then

= B),

in %

~~=i

tagged

We also note that,

n, points,

not necessarily

if A and B are similar in ‘u(3Q E ‘u 3

= T/l(B).

T/l(A)

and A has no repeated

431

ALGEBRAS

Moreover,

eigenvalues,

then

if i”J(x) is any complex

if T/l(A)

= TA(B)

A and B are

similar

polynomial,

ThP(A)) = WP@))I(k 4 ETW)I. 2. THE

INDUCED

FUNCTIONS

If F is an intrinsic known

that

function

on a simple

F is a polyfunction,

A whose coefficients In the

case

depend

under

block

consideration

each block

of A.

THEOREM

* -- @A,

be

A i, F(A),

E Ki.

algebra,

then it is

polynomial

in

[7].

‘u = F?4?,1 @ -. . @ %Tnl, it no

is always a polyfunction.

IS . a polynomial

What

in the corresponding

we have

Let F be an intrinsic function defined on a domain 33

2.1.

of the semisimple

here,

function

of F(A)

Specifically

complex

is a complex

on A as well as on F

longer follows that an intrinsic is true is that

i.e., F(A)

complex

algebra VI = V,, @ * * * @ V, .

arbitrary

in

Then

D

and

there exist

let

F(A)

unique

= F(A\,

Let A = A, @ 0.

-* @F(A),;

covnjdex polynomials

pz,4(x),

i = 1, 2,. . . , t satisfying

F(A), = Pi.4&), (ii)

degPiA(4 G degm,Jx).

Proof. intrinsic commutes similar

with Ai.

of the theorem for complex

In general,

2.1 involves

commutes

The detailed

to the proof given

conclusion theorem

The proof of Theorem to show that F(A)i

using the fact that

with every element

proof will be omitted

F is

of %,,i which

since it is very

in [7] for the case where ‘u is simple.

The

then follows from the classical double centralizer

matrices

the coefficients

all of A, not only on Ai.

[4]. of the polynomials

P_,(X)

will depend

One can see this by considering

on

the intrinsic

function F(A, defined

on the algebra

@A,)

= tr(A,)A,

@ (det A)A,”

‘9X= %’ @ V,. Linear

Algebra

and Its

Afiplications

3(1970), 429-440

432

J. P. LONG

Rinehart’s

[7] characterization

algebra G?Fn involved A.

of intrinsic

an induced function

In the case under consideration

not only on the eigenvalues those eigenvalues

come.

of A and of F(A) to T A(P(A)). it follows

AXD

C. G.

functions

CULLBN

on the simple

on the spectrum

of the argument

here the eigenvalues

of F(A)

depend

of A but also on the blocks of A from which

It is then natural

and to consider

From Theorem

to compare

the induced

the tagged spectra

functions

2.1 and the remarks

from T/I(A)

following Definition

1.2

that

TW+(A)) = ((h$/A4))(6 4 ETA(A)}, where

the polynomials

Consider

pzll(x) are those

now the induced

described

in Theorem

2.1.

mapping

(6 1) + (iJ AA(l)) from TA(A) inner

to T A(F(A)).

automorphisms

This

induced

of ‘u since,

= p-1

nature

Ai, Bi, P, E 5??Ei

of F,

@ ***

OP,-l)(p,,(A,)O...Opt,(A,))(P,0...0P,) 0.

the uniqueness

a. OPtA(J’clAtPt) part

If the tagged

eigenvalues

of the choice

the induced

of the element

with those

If A has repeated tagged

eigenvalues.

derogatory

tagged eigenvalues,

such that

TA(C)

mapping

= TA(A),

Algebra

and Its

in U

(2.1)

3(1970),

is independent

eigenvalues.

of the element of F contains

we define the induced

Applicntiom

piA (x) =

any element

* * @ Ct (me,(x) = cc-(x),

(6 4 + (6 p,(4). Linear

that

then it need no longer be true

if the domain

C = C, 0.

then

tagged

(2.1) is independent

However,

element

2.1 it follows

is similar in ‘u to A. Thus, for the case of

eigenvalues,

that the induced mapping

0 * . . Oh(&).

(2.1) for A and B are identical.

of A are distinct,

with the same tagged eigenvalues tagged

= h(4)

of Theorem

piB(?c) and hence that the induced mappings

distinct

under

= P-lF(A)P

= &(P,-lA,Pl) From

is invariant

P E 2l, then

B=B1@...@B,,

P = PI 0 * . * @ P,,

F(B) = F(P-lAP)

mapping

if B = P-lA P,

A=A,@...@A,,

and, from the intrinsic

(2.1)

4X-440

with those

a block noni = 1, 2,. . . , t)

mapping

to be (2.1)’

FUNCTIONS

Since spectra

ON

SElMISIMPLE

any two-block

nonderogatory

must be similar

the choice

in 8, so, if the domain

complex

numbers

@ A, E 53 with

for which

Trl(C)

there

T&4)

= T/l(A).

exists

mapping

(2.1),

function

induced

be described in terms of the coefficients A,.

of A ==A,

@***@A,

As observed

= Xni -

c1 [.4&V-l

is the characteristic The

coefficients

+ f . . + (-

polynomial of each

l)“~-la,,_,[Ai]x

the eigenvalue

polynomials

determines

The elements Now let derogatory

of Ai with the inte$er

of A, with the point ~(4~).

=

{(o(A,),

of V,(%?) = V,,(U) 2,;

of the set CP(A)

o(&),

1,;.

x

set of Y = Z~PZ,points V,(U)

. . ; G,),

and

Its

of

i, we now tag each

x

. ** x

Vnt(%):

&)piE-+Q}.

will be called the characteristic

Algebra

a unique Instead

Using this device, we can

P E V,(%?) be a characteristic point A E ‘D. Any other block-nonderogatory Linear

l)%,,[AJ

of Ai.

of these

with A = A, @ * . * @ A, the following

CW)

of

polynomial

+ (-

tagging

distinct)

polynomials

AJ

simply

(not necessarily

may

. * . c,(x),

o(AJ = (a~[A,l, a2[Ai1,. . ., o,~_~[A,~) E VIR_lW.

associate

mapping

into

point

of the eigenvalues

A =

of the Ai, and on the

that the induced 1, the characteristic

CA(X) = Ci(X)&)

-

ni

depends only on the tagged

of the characteristic

in Section

can be factored

ci(x) = cAi(x) = det(x1

C:=i

by F, sends Lii to p,,@,,).

polynomials

This suggests

F.

set of m =

a block-nonderogatory

this mapping

of A, the characteristic

the blocks

seen to be dense

= {(i, 3Lij)llZij~il},

Since A is block-nonderogatory, given intrinsic

C.

of ‘u are easily

= 1,. . . , t, j = 1,. . , 9-zi} is any

then the eigenvalue

of

YDof F is open in 3, there will always be a block

A, @*..

eigenvalues

of ‘u with equal tagged

(2.1)’ will be independent

element

elements

C E 2, such that

If II = {lijli

elements

nonderogatory

nonderogatory

nonderogatory

433

in 3, the mapping

of the block

The block

ALGEBRAS

of

points

some matrix

Applications

of A.

block-non-

B E ‘11with

3(1970),

429-440

434

J. I’. LONG

P as one of its characteristic property

(i) of Definition

the induced functions Equivalently

there

independent

3

is a mapping

Although

f from

il,;. . . ;Wt),&)

are the polynomials

of the choice

the foregoing

THEOREM

An intrinsic

2.2.

of the semisimple

# F(B),

defined

by

. .~P,,(&)),

of Theorem

2.1.

This mapping matrix

is

A in %.

with

F, defined on an open domain

function

algebra ‘u = Wn, @ - * * @ %7*, induces

a domain of V,,,(q) = V,JU)

a function

x **a x ;,,(%?)into V,(%?).

f

The

which is a characteristic point

A E D,. If A = A, @ * * . @ A, E D has P =

of some block nonderogatory

. . ; o(A,), A,) as a characteristic point, then.

1,;.

f(P) = (P,‘4@l)~P,,(&2)~.

= (f,(P)>.

. .,P,,(&))

i = 1, 2,. . , , t are the polynomials

where pi*(x), value f(P)

to V,(e)

-(P,,(u~.

function f is defined at any point P E V,(U) @(A,),

in ‘L[ to A and, by

of A and B must agree.

V,(e)

discussion

C. G. CULLEN

in general F(A)

of the block-nonderogatory

We summarize

mapping

must be similar

1.1, B E 3.

(2.1) on the tagged spectra

f: P = @(A,), where the PiA

points

AND

is independent

of the choice

. .>fi(P)),

of Theorem

of the block

The

2.1.

nonderogatory

matrix

AED. Theorem

2.2 could

component

Clearly

functions

fl, fz,. . ., ft.

component

functions

fi will be involved

3. DISTINCT

TAGGED

function

In Theorem

where

the

in terms

of the t

function

f and its

in what

follows.

@A,

be an element

of the domain tagged

D of the

eigenvalues.

2.1 we saw that

polynomials function

of the ith block of F(A). depends

LinearAlgebra

and

Its

CD**- OPtA(

$iA (x) are unique

2.2 we know that

of the induced P E CP(A)

completely the induced

F, and assume that A has distinct

F(A) =#,,(A,) Theorem

Both

EIGENVALUES

Now let A = A, @a.* intrinsic

be stated

and

deg pi, (x) < ni.

the value of the ith component

f at a characteristic Moreover,

Apfdications

3(1970),

fi

point of A is an eigenvalue

once A is chosen,

only on the component

From

function

the value of fi at

of P in position

429-440

n, + n2 +

FUNCTIONS

ON

. . - + n,.

Thus,

becomes by

SEblISI&Z’LE

once A is fixed,

a function

fiA and

each of the component

of one complex

the value

For each il E A(AJ

variable.

of this function fi.4(Z) = fi(.

of degree ni -

435

ALGEBRAS

; cT(Ai), 2;.

1 or less which interpolates

this polynomial

is unique and can be written formula

[5],

$iA (x) is a polynomial

fiA at

the function

Since the ni points of A@,)

interpolation

fi

.).

fiA(A)= piA(A),i.e.,

we know that

functions

this function

at z by

points of A(AJ. Lagrange

We denote

i.e.,

are assumed explicitly

each of the

to be distinct,

using the classical

if A(A i) = {Ail, 4%. . , &J,

then

Now piA(

2 E ‘u, as given

value of the primary summarize

3.1.

THEOREhl

f,,(Z)

Let F be an intrinsic

functions

@ction

fiA(z) be

If A has distinct tagged eigenvalues, =fl.d(Al)

1.

SOME

TOPOLOGICAL

If the intrinsic

function

of Theorem

eigenvalues

of A are distinct. some

The algebra define

the norm

We

defined on a domain ‘D

. ., ft

of Theorem

the function,

2.2.

be the

Consider

of z only,

.).

O-..

Of,,(A,),

with stem function

f,,(z).

CONSIDER.4TIONS

conclusions to introduce

[5].

the

then

Of2n(As)

where fix9(z) is the $wimary function

f,,(z)

. . @ gnt and let fl,

ftA(z) = fi(. . . ;a(AJ,z,.

F(A)

is precisely

with

of the indarced function

@ A, E a, and let

A = A, @..*

above

with stem function

of this section

algebra 2I = %Yn,0.

of the semisim$le roqbonent

function

the discussion

by the formula

F is sufficiently

3.1 hold without

topological

well behaved

the assumption

To investigate

at A E 3, the that the tagged

this matter

it is necessary

considerations.

21 can be turned

into

a metric

topological

space

if we

of A E X by Linear

Algebva

and

Its Applications

8(1970),

429-440

436

J. P. LONG

Concepts

of neighborhood,

well defined

limit,

continuity

AND

C. G. CULLEN

of functions,

in ‘% and the usual elementary

techniques

etc.,

are thus

of analysis

are

applicable. As remarked in 9.

earlier, the block-nonderogatory

More explicitly,

any neighborhood same tagged

of A there

eigenvalues

small E’S to selected similar

if A E D has repeated exists

as A.

elements tagged

of D are dense

eigenvalues,

a block-nonderogatory

To see this it is necessary

off-diagonal

positions

of a Jordan

then in

B with the only to add

canonical

matrix

in ‘u to A.

The proofs given by Rinehart t = 1) can be modified proved

in detail

the following

facts,

which

are not

here:

1. If D is open in ‘u, then 2. If limZi

[7] for the case ‘u = Fn (i.e., the case

to establish

= 2,

and

CP(D)

is open in V,(U).

P, E CP(Za),

then

Vi3P,

E CP(Z,)

such

that

lim Pi = Pa. 3. If CP(%)

is dense at Pa, then 3 is dense at any 2, E D such that

Pa E CP(Z,). 4. If P is continuous tions of Theorem are continuous

at A E rD and D is open, then the induced

2.2 are defined at any point

on the open set CP(D)

func-

of I’,(%)

and

PO E CP(A).

5. Let (alo,. . . , c(_,, no) be a point of V,(U) and let 0,” in the equation + ... O = 0 be determined so that 31Ois a root. xn _ g1op-1 + (lpJ, There

exists

a deleted

disk,

that

(-

I)TJ~ = 0, with on determined

5. REPEATED

the

open disk, 0 < I;1 -

in

TAGGED

equation

THEOREM

5.1.

component

process

Let F be an intrinsic

to establish

Algebra

our main

theorem.

function defined ow a domain D

function

of Theorem

@ A, E 3) and let f,.4(z) be the junction,

and

+

roots.

algebra ‘u = %?‘*,@ . - * @ Vzt and let fI,. . ., ft be the

fiA(z) = f((. . . ; o(AJ, z;. . .). Linear

for any 3,

1)“~%$_,x

so that 1 is a root, has distinct

functio?ts of the induced

A=A,@--0

li”j < 6, such that,

(I~~X”-~ + . . * + (-

EIGENVALUES

We can now use a limiting

of the semisimple

?G”-

Its

Applications

3(1970), 429-440

2.2.

Consider

of z only, satisfying

FUNCTIONS

OX

SEMI SIRII’LE

437

ALGEBRAS

Then

where fiA(Z) is the primary

(i) (ii) analytic

function

with stem functiorz fiA(z) if either

A has distinct tagged eigenualues,

OY

A is an interior point of a, F is co&nuous

at A, and f,A(z) is

in a z neighborhood of each repeated eigenvalue

of Ai.

Proof. Case (i) is just Theorem 3.1. Case (ii). Since A is an interior point of D, each characteristic point of A is an interior point of CP(D), the domain of the induced functions fi. Now let {$s} b e a set of open balls in V,(g), each containing exactly one characteristic point of A, namely P,. These neighborhoods can be taken sufficiently small that, for any point of the restricted type

J’,’ = @(A,), z 1; o(&)> 2,;. . . ; o(A,), zt)

(5.1)

within Cs, except possibly P,q, any A’ E 3 with P,’ E CP(A’) has distinct tagged eigenvalues (Remark 5 of Section 4). Let JV be the subset of matrices from D whose characteristic points are all within U & and of the special type (5.1). Since the set CP(M) is dense at each P, E CP(A), JV is dense at A (Remark 3 of Section 4). Since F is continuous at A, F(W) = F(A).

lim WC,+‘-, IV-A

Since each WE JV has distinct given by (Theorem 3.1) F(W)

= f&W,)

tagged

0.

eigenvalues,

each F(W)

is

* . @f,,,(W,),

where I&W,) is the value at Wi of the primary function with stem function f,,&). Hence the firV(Wi) are computed from the Lagrange polynomials which interpolate the fLrcrat the n, distinct eigenvalues of W,. Remark 4 of Section 4 guarantees the continuity of each fi at z E A(AJ. Walsh [S] proved that, when the functions f,il(z) are continuous at each I EA(AJ and are analytic in a neighborhood of the repeated A, each f,&z) approaches a unique limiting polynomial hiA as the interpolation points approach the points of A(AJ through distinct values. The hiA are the Lagrange-Hermite interpolation polynomials: L&ear

Algebra

and

Its

Applications

3(1970),

429-440

438

J,

I’.

LONG

AND

C. G. CULLEK

where

and the tlk are the di distinct cities

elements

of _4(ili) with respective

multipli-

sk.

As W in N

approaches

the eigenvalues F(4)

=

A, the distinct

of A through

lim F(W) =

distinct

lim [ficV(W1) 0. IV+

W+A

of W approach

Hence

* . @f&W,)]

A

Wibv(W~) - hIA

= lim

eigenvalues

values.

- 4~Wt))l

0. * . 0 (fdwt)

W+A

lim PIA

t Now, (/Jz)

-

+

W+A

for

each

hiA(

coefficients,

i,

-+ 0.

(fiw(W,)

hzA(Wi)) 4 0 since

-

Moreover,

so h, (WJ

at ‘qi, of the primary

hA(W

3. . +

each hiA

polynomials with fixed

is precisely

the value,

Now hiA

---f hi_4(A J. function

the

is a polynomial

with stem function

fiil(z) [S].

Therefore

as asserted. In the case in which ‘11has isomorphic ponent

functions

Xi z ‘uj under determined

fi have the

automorphism

tion to ‘uj is I’-l,

other

mapping

I’:

simple components,

noteworthy

properties.

‘LIi ---f ‘uj and let 9

of % whose restriction

and whose restriction

the com-

Suppose be the

that

uniquely

to ‘ui is r, whose restric-

to Xk, k # i, i, is the identity.

Then

~(A)=9(A,0...0Ai0...0A,0...0A,) =A,O...OAjO...OAi+...OA,. Since F is intrinsic,

we must

have F(QA)

(5.2) = QF(A).

nF(A)=f,,(A,)O...Ofj,,(Aj)O...OflA(ili)O...OftA(At) and Limar

Algebra

and Its Applicatiotis

8(1970).

429-440

Now

FIiNCTIONS

ON

SEMISIMPLE

so we must

have

439

ALGEBRAS

fk‘4 (Ak) = fm (Ak) fjA (AJ = fin‘4(AJ fiA@i) = fjn‘4bu fiA(Ai),

Since conditions

etc.,

(5.3)

rE (U

are computed

are equivalent

for each

point

fyi@ 1 3-)= fl;Lc~))1

6.

(al

*a* +

+

?$)-ARY

In this section from a suitable Let

f

point

f

from

sj, in A(AJ,

si, in A(AJ.

an intrinsic

from a domain

r of V,(g)

of some intrinsic

function

characteristic

fl, f2,. . . , ft; i.e., f be the induced

F, it is necessary

if P E T is a characteristic

point

point

is made

then,

s,--1

si-I

to V,(U).

In order that

CP(A)CT,

,...,

V,(9)

be denoted

once the choice

,...,

sj-1

Li, of multiplicity

(f,(P), . , . , f,(P)).

every

)...)

we show how to construct

functions

the

(5.3)’

FUNCTIONS

function

be a function

component

functions,

sk, in A(A,),

Ai, of multiplicity

s=O,l,Z

for each

s=O,l,Z

kfj,

s=O,l,Z

I;;;,(&),

(5.3)

as values of primary

Ak, of multiplicity

for each point /$(A,, =

kfj.

to the conditions

k#i,

fL.4 (At),

=

I

kfi,

to V,(U)

on A

and let the

for each P E r, f(P) function

(Theorem

= 3.2)

that

of some A E %, then

of A is in r,

i.e.,

of A = Ai 0.

for each

function

i = 1,2 ,...,

CP(A)

*. @A,

C r;

(6.1)

such that

t,

fiA(4 = f,(Q,), 1,;. . ; b(Ai)>2;. . . ; opt), A,) is independent each

fiA(z) is

of the choice analytic

of iij E A(A j), j # i;

at each nonsimple Linear

Algebra

and Its

point

(6.2)

1 EA(AJ;

Applications

3(1970),

(6.3) 429-440

440

J. I’.

AND C. G. CULLEN

LONG

if qni = 2& = Xj = kYnj and &? is the automorphism then

the conditions

DEFINITION

Let

6.1.

to V,(U) satisfying A = A, 0-e.

(5.3)’

hold.

f be

(6.4)

a function

the conditions

(6.1),

@ A, E ‘u such that F(A)

(5.3),

=f,,4(A1)

from

(6.2),

a domain

r

of V,(U)

(6.3), and (6.4).

CP(A)

C r,

o***

OftA(

where fi, (AJ is the value at A i of the primary

For each

define (6.5)

function

with stem function

The function F defined by (6.5) will be called the (nl + n2 + - * . +

fiA(z). n,)-a7y

function

with

stem

It follows easily intrinsic

[S]

intrinsic

that

an

function

fl =

Aa

en,

f.

(n, + . * * + n,)-ary

on ‘u.

THEOREM 6.1. algebra

function

from (6.4) and the fact that

0.

We state

(n,

+

n2

functions

are

on ‘u is always

an

this as . ** +

+

primary

function

. * @ %Tntis an

n,)-ary

on the semisimple

function

function

on some

continuous

intrinsic

intrinsic

domain

D

of ‘u.

Theorem

5.1 says

that

are (~2~+ - * - + n,)-ary characterize

appropriately

functions.

the intrinsic

Thus Theorems

function

functions

5.1 and 6.1 essentially

on the semisimple

algebra

‘2I = FR, @

* . - @IV,‘. REFERENCES 1 A. A. Albert,

Structure

2 C. G. Cullen Monthly

and

74( Jan.

3 C. A. Hall,

of

C. A.

Amer.

algebras,

Hall,

Functions

A4ath.

on

Sot.

Colloq.

semi-simple

Pub.

algebras,

24(1961).

Amer.

Math.

1967).

Intrinsic

functions

on semi-simple

Canad.

algebras,

J. Math.

19(1967),

590-59s. Lectures

4 N. Jacobson,

in Abstract

Algebra,

Vol.

2, Van

Nostrand,

Princeton,

N. J.,

1953. 5 1~. F. Rinehart, Monthly 6 R. F.

The equivalence

62(1955), Rinehart,

Math.

J.

7 R. I;. Rinehart, 8 J. I,. Walsh, domain,

Elements

27(1960),

Intrinsic Math.

Received October, Linear

Algebra

and

of a matric

function,

Amer.

Math.

of

a theory

of intrinsic

functions

on

algebras,

Duke

1-19. functions

Interpolation

Amer.

of definitions

395-413.

on matrices,

and approximation Sot.

Colloq.

Pub.

Duke

20(1935),

1969 Its Applications

3(1970),

Mat/~. J. 28(1961),

by rational

429-440

Chap.

functions III.

291L300.

in the complex