NATIONAL BUREAU OF STANDARDS REPORT

185>6

ON THE SPECTRUM OF

A ONE PARAMETRIC FAMILY OF MATRICES

A. M. Ostrowski

American University

and

University of Basle, Switzerland

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Asfcio, Acting Director

THE NATIONAL BUREAU OF STANDARDS

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NBS PROJECT

1101-10-5100 July 305 1952

NBS REPORT

1856

ON THE SPECTRUM OF

A ONE PARAMETRIC FAMILY OF MATRICES*

Ao M. Ostrowski

American University

and

University of Basle*, Switzerland

PREPRINT

This work was performed on a National Bureau of Standards

contract with the American University.

This report is issued fc APP roved for public release by the

in any form, either in wb Director of the National Institute of

frost the Office of the Di Standards and Technology (NIST) on October 9, 2015

, reprinting, or reproduction

lission in writing is obteined

hington 25, D. C.

INTRODUCTION

By the transformation

n VA V

1

*

(/* - 1» > n)

with the matrix A( o( V

|JL V

into the vector y^

•

) the vector f(xp 000

, x

n

)

* • $ 7n

)o The expressions

is transformed

A (A)

If I/O If I A (A) MinIf l^o

are then respectively called the upper and the lower bounds of the

matrix A® They are given by the square roots of the greatest and

smallest eigenvalues of the non-ne gative matrix AA o

A(A) and A (A) are very important in the discussion of metric

properties of the matrix As at least when the euclidean length is

used* On the other hand,, in practice^ only estimates of A and A

can be used which contain the corresponding quantities for some

special matrices® However5 only very few types of matrices are

known for which A and A can be given explicitly9 and in all of

these cases these values are immediately obtained®

In what follows we discuss A and A for a not trivial case

of a one parametric class of matrices given by

2

where all elements of the principal diagonal are equal to the real

parameter 9 all elements to the right of this diagonal are 0 and

all elements to the left are 1,

As a matter of fact, our discussion covers the whole spectrum

( n) (

n

of the matrix Dtf< . If the eigenvalues of this matrix are

denoted in the order of their magnitude

- ^2

^^ ~ 0 0 0 " An

( ^

)

we write

A/ <*) s ~ ^ )

and find that the v^( <K ) are roots of a polynomial which

is a linear combination of two Chebyshev polynomials and contains

o( as a linear parameter* We have

4 (*, h COS

2

e) S * sin (211+1)9* ,(« -1), Sin, .(2n-lje_ n sin Q

It follows then that for each V3 v^( *<) is a monotonically

increasing function of^ 5 -oo<c<<oo» In particular, for

three values of we obtain the values of

A(*)s A(D^n)

) , A(«) s A(An)

)

explicitly. We have

A(o)

2 sin

A(|) cot 9 /\(1) =

-2 sin 1^2

A(0) - 0 , A(y)

1

tan IT HE

2 COS 2n+l o^T

A(l) =

3

For other values of <*. <, A( *0 ard A(«> are continuous functions

of <K o We have in particular

A ( °<)

A ( *

)

A( <*)

A(*)

A( °0 ~ A

2n

n “ 1 n

2 - 1

T" + IkT + 0

11 ,2 tr n / 1 ~ cot — + 0

2 8<* 2n

e* - " + — tan2 + q ( 2 8* 2n l

n -1 n - 1 n ( 1

<K 4, 4, -»===-=-- 4, Q =»=

2 2i[®< V«* J

( <x —¥ 0 ) 9

(°< 00 ) 3

(

£>< TM> - Co ) 5

( 00 ) 5

( A — 00) e

These results are deduced in the section XI of the paper. Some

"finite" estimates for A( A) can he deduced from the results of the

section X®

The equation satisfied by the v^( *<) is deduced in the sections

IV and V while in the preceding sections I => III the properties of

some determinants and of some special polynomials are developed as

the preparation for the discussion of this equation® The discussion

is pursued further in the sections VI and VII® In the section VIII

we discuss in particular v^( ^ ) and v^( «< ) and deduce from all these

results the corresponding results for in the section XI®

We may finally mention that the matrices play a certain

role in the discussion of the rounding off errors in the operations

with the matrices

h

I« SOKE SPECIAL DETERMINANTS

If u,

for n = 09

d,i)

v are independent variables we will denote by ^n

(u*v)

19 2S °°* the egressions

1

u

- 0 v

u

V

1 •*» 1 1

1 »•» 1 1

U ® 0 ° 1 1

oooooooo

0 0 0

ooo v n

(n > 2)

(1,1°) t0 sl, ^ = 11, 1’-

u 1

V u 5

where the determinant is a determinant of the order n in which

all elements of the principal diagonal are equal to u5 all elements

to the right of the principle diagonal are equal to one,, all elements

of the first diagonal to the left of and parallel to the principal

diagonal are equal to v and all other elements are equal to 0«

If we subtract in the determinant for 'njf the second row from

the first and then develop in the elements of the first row,, the

coefficient of (u - v) is ^ while (1 = u) is multiplied by

v 1 1 •»» 1 1

0 u 1 ° ° « 1 1

0 v u 0 0 0 1 1

oooooooooo

0 0 0

° ° ° v u

9

and this is at once verified also for n = 2„ We obtain therefore

(1,2) 'l|fn(%v) - (u=-v)ilf

n<_1

(u5 v) + v(u = l)'4T

n==2 (u5 v) (n * 2) .

5

We put now

(1,3) f

n

(x,'V'r ) =

U 1 1

000 1 1

V U 1

o e « 1 1

0 v u ° ° ° 1 1

0 0 0 ° ° ° u 1

0 0 0 • » o v x

(n > 2)

(I5 3°) 1

UX •= V

where the determinant is obtained from that in (I5 1) in replacing

the last element in the last row by x0 The last line of the determinant in (l5 3) is the sum of the two lines

0 0 o ° ° ® 0 0 x=u

0 0 0 ° ° ° 0 v u

and it follows

(I*U) fn

(x,u,v) = + (x-u) (u,v) (n 7 2) n rt»l

This is immediately verified also for n e 2,

We put finally

W) F

n

(o{ 9 x9 u9 v)

u 1 1

V u 1

1

1

oo ooooooooo

0

0

0

U 1 o< V U

0 v x

(n > 2) ,

( 1, 5

°) F

1

" x 9 Fg ® ux “ c<v

6

where the determinant is obtained from that in (I* 3) in replacing

all elements of the last column by with the exception of the

last element of this column which remains equal to xe

If <K ^ 0S it follows from (l$ 5) 9 (1*3)

F

n(^ 5 x5 u5 v) - <* fJZ ,u,v)

and therefore by (i5 U)

(1*6) F

n

(*,x,u,v) - c<llrn(u5 Y) + (x - .< u^h^Cu.y) o

This formula remains by continuity also true for o< a 0 and is also

immediately verified for n = 2 C

II. 0n(v)

We define now the polynomials 0 (v ) in putting u - 1 - v in ijr

din) 0n

(v) - ynd-v,v) „

From ( I-„ 1), (I# l°) we obtain

(11,2) 10

- 1 , 0±

- 1 - v , 02 . v

2 - 3v + 1

and from (

I

9 2

)

(IIS 3) 0n

- (1- ») 0n.x

- v

2

.

¥e put now in 0n

(r)

v

h cos^ ©

(II,U)

7

and multiply the resulting expression by l/

1 cos2n

@j we obtain then

an expression

(n,5) U (©) = l/

1 cos

2n 1

n

\h cos

2

9,

From (11,5), (11,2) and (11,3) it follows

(11,6) U » 1 , - k cos

2 © - 1

Sin

sin 9

(11,7) Un(©) - 2 cos 2© U

nral (©) - ( n - 2, 3* °°°)

From (11,6) and (II, 7) follows now

( 11, 8 ) U (©) n sin ©

Indeed, (11,8) is true for n ~ 0, 1 and the right hand expression

in (11,8) verifies obviously the recurrent relation (II,7)» From

(11,5) and (II, 8) we have

(11,9) hn

cos

2n n ^

I4. cos

2

©,

sin ( 2n + 1)Q

sin ©

On the other hand, we have by a well known trigonometric formula,

in putting

y

= 2 cos ©,

sin (2n + I)Q

sin ©

n

2n i H)r Cp

r >>

2n-2p

and therefore in introducing n - y -

jx as the new variable of

summation

- (-if l (-if > 2n(-i)“ h-if

(

n

2

+;) (?

n-V

8

In comparing this with (11,9) we obtain finally

(11,10) (-l)\(v) - v

n

-

(

n

f) v

n-1

+ • • * + C-lf Mv

11"5

- ... (-l)

n

.

III. THE ROOTS OF 0 , <b + v0 _ rr)? ~ ^n-l

The right hand side expression in (II5 8) vanishes for

q = 7T (v = 15

• ••

* n) . 0n the other hand we have by (II5 8)

(III5 1) U (0) + U (©) « 2 sin 2n Q cot © ,

(III, 2) Un

(6) - Un-1 (0) - 2 cos 2n © s

therefore U (©) + U (©) vanishes for the n values n

v ' n-lv '

(III, 3) 2JL

2n

while U (9) = TJ _(9) has the n roots n n-1

(hi,U) 2y-i 7T

(» - 1, 0,0

5 n) 5

( ^ - 1, * * e 5 n) .

The roots of 0 (v) are obtained from those of U

n

(9) by the

transformation (II.U) and give

( 111, 5 ) h cos'

(» = 1, , n) . 2n+l

We obtain from (II. 10)

n

(III, 6) 0n(r1 = (-l)

n

TT v

»-l h 005,2 2^:

9

On the other hand by the transformation (lI,U)and by (11,5)

V@) + * UJ°) “ Un-1 n v,_n(©) n-1

become after multiplication by v'n

*n(v) + ( V )

and

i) ( v ) = v0 (v) n

rn-l

To the roots (111,1) and (III, 2) correspond respectively the (n-1)

roots

(111,7)

h cos^ 3j!~-

2n

(V = 1, 0 0 0 , n - 1)

of 0r + v0 (the n-th root becomes infinite since 0 + v0 n is n ^n-1 v ^n ^n-1

only of degree n-1) and the roots

(III, 8)

, 2 2)>“1

h 008 TH- w

(v - 1, 000

, n)

of 0 - v0 o ^n-1

By (11,10) 0n

+ v0

n ^begins with the term

(-dV + (-i)"-1 datii v"

-1 (-n^V + c-i) n" 2 v

1^1

/ vri00! n-X

(-1) n v

and 0

- v0 with the term 2(-~l)

n

v

n

j we obtain finally n 'n-l

10

(111,9) d + Vv ^0 n *n-l

n»l

(„i) n“in fy | v si h cos

2

-fjn

(111,10) 0n - V

^ . (-1) 2 IT V.

y=l 2 2,-1 „ \ 4 cos —t—— 7T

IV. GENERAL REMARKS ON THE SPECTRUM OF AA"

If A(a|ip is a quadratic matrix of the order n corresponding

to the transformation

n

- £

V S1

(pi) 0 °°

5 n) $

the spectrum of the symmetric matrix

(W,2)

is given by the set of n roots of the corresponding secular equation

(CT»3) Ue - s| - o ,

This spectrum can be interpreted in a different way. If we form

the quadratic form

n

(w,U) s(*,) 'Ey-, „ = > S .. X X., o r

i 7 ft r p

y

the extremal problem on the unit sphere

(I

V

5 5) 5(x^) " Extremum (x^ + + x

^ s 1)

gives for the Lagrangian multiplier X in

11

(W»6) S(x

y

) / n M £

\ )>”1

after elimination of the variables x^again the equation (IV,, 3) as

condition^ although usually only two roots of (IV5 3) correspond to

the solutions of (IV5 3)o

In differentiating (IV^6) we obtain the n conditions

(^7) £ a y A x.^ - 0 (y - 1, •••

, n)

pi r r

in which we would have to introduce the values of y from (IV5 X) 0

J1 If however we introduce instead of x^ and y^ another set of

n variables z^ connected with the x^ by a non-singular linear

transformation^ the elimination of the z vfrom the transformed equation (IV<,7) gives the same equation for A since the equation (IV^3)

is then only multiplied by a non zero constant e

V* THE SECULAR EQUATION FOR

(n A 2)

/ \ / x'K' In order to obtain the secular equation for the ' we

have after what has been said in the section IV to apply the

Lagrangian procedure to the expression

(V,l) (<* 1^)^ + ^ X2^‘ o o + (xl

n

• «» + x

n-1 x yn

/ A I Xy seI

If we put

(V,2) + « ° ° + X o o o 9 n)

12

and use the convention z q ~ 0$ (7^1) becomes

(v,3) £ (°< z y - ^Vi + Vi)2 “ A S V I i>-l

in differentiating this with respect to x^ we obtain the set of

equations

n (v»M Vi) +

, I (**k

- «w* Vl) - Aj,

k-j) +1

(V = 1, 0e ° 9 n) e

Here we write on the right hand side z ^ - z^ ^ for and bring

everything on the left

n *<* z>-*Vi + Vi) +

, i (^r^k-rvi1 - 0

ife $ +jl

(V - 19 , n) *

In reordering this and in using the notations

(v,S) v A C< + <K U eK + 1 A

we obtain finally

(V,6)

rVZ^i + tLZj,

n?*!

k=V +1

z, + o< z - 0 k n

vs , + (u + ~ l) z ~ 0 o n-1 v 1 n

v

O = 1* 009

9 n - 1)

The determinant of this system is now just F («* 5 Xj,us v) given by

(I# 5) 3 for x - u + o( - 1, (I5 6) gives therefore for this determinant

13

the expression

+ ( u+ °< ^^qC^v) *

Further we have from (V,5>) u s 1 => v and our determinant becomes

therefore by (11,1)

(V,7) A

n

(vs <x) b *0n

(v) + (c< - l)v0nc=1(v) o

The spectrum of D^ ,; is therefore obtained in solving

with respect to A the equation

(V,8) A

n

(v, *) 1 o< 0n

(v) + (* -l)v jfl^Cv) - 0 (v ^ A *

2 + *< ) «

VI. THE DISCUSSION OF A AND

n

In the following discussion we will keep n fixed and denote

n-1'

the roots (111,5) of 0n

( v ) and the roots of 0^-, ( v) respectively

^ V xy

(VI, 1) x^ (p^l5 ooo

5 n) | x% ——>

i

U cos

2 -.pAl ^ Ij. cos2 7* ^- 2n+l 2n~l

(V - 1, o-^n- 1)

We verify immediately the inequalities

< JL <

2n + 1 2n 2n - 1 Un 2n +

1

2i>+ 1 „ V + 1

< <=«r“—* < (*= 1, 000

5 n-1)

and obtain therefore

(71,2) Xj) <

1 ^ D < x„ < 1

5 2 V7T ^ h cos

jjj

, 2 2y+l ^ i>+l

h cos <x . (v-l<,°°%n-l)

In particular5 the roots of 0^ ^separate the roots of 0n

.

lU

If we develope now

- v

(v)

into partial fractions

(VIS 3)

^1(Y) S , 0n

(v)

J>-1 v - x

»

where the ’’polynomial term” is 1 as follows immediately from (11^10),

we have

(vi,U) v

$ , (x.) n-lv y

and in putting in (VI, 3) v - 0,

n p

(vi,5) Z r

- 1

1) «1 If

It follows from (VI,U), since both the roots of ^ and of 0

8

separate the positive roots of 0 , that all p^ have the sane sign

and therefore by (VI, 5) are positive o

Consider now the quotient

(VI, 6) %SYs *

)

,v)

(1- ®0 0n

(v)

we have from (V,8) " v

1 5

15

and in using (VI, 3)

(VI»7) VY» <*) - 5 Py ? 0 •

It follows from (VI, 7) that Q (v, <*) is steadily decreasing if v

grows through any interval containing none of the x^* Indeed we

have

d ®n * ) dv p

In particular, if v grows from x^ to x

y+^ Qn

(v5 <K)^ decreases

monotonically from + oo to =» oo and assumes therefore in (x^,,

ary real value exactly once„ We see that for any value of o( / 1

the equation (V,8) has exactly one root in any of the Intervals

< Wi) °

VIIo THE ROOTS OF A (o< j7 )

In what follows we denote the roots of A^(«K ,v) with respect

to v, ordered increasingly by

(VII, 1) Vi(©<) * V

2

(o< ) 4 oo. 0

1

Since decreases with increasing -K / 1, the corresponding

<K “ 1

root of laying in (x^,x ^ ) Is increasing with increasing <?< e

If we take now into account the roots of (III, 9) and (111,10) corresponding to - oo and we obtain the subdivision

of the interval from x^ to corresponding to the subdivision

V

\

1 < 0

16

of the e<-axis from 1 to oo 5 from - go to 05 from 0 to § and. from

| to le In the diagram 1 below5 the roots of A from the interval

n <Xy5 x^

+

^) for =< from one of the partial intervals of the lower

line lie in the corresponding interval on the upper line®

x. v «~

U cos'

4-

pjr

2n x.

1

i 2 2 v + 1

ll COS T“”““ IT in 'v+1

DIAGRAM 1

On the other hand if v goes from - go to x^, we can take for the

values between “OO and I. and obtain therefore the following diagram

1

* (-—— -4-—

—

1——

—1

“OO 0 1 1

DIAGRAM 2

Finally,, if v grows from x^ to co$ we can take for all values

from 1 to oo and obtain the following diagrams

*1

D IAGRAM 3

17

The resulting distribution of the roots from x„ to x between

and v <«) has been already indicated in the Diagram. 1® ¥e

see in particular that A (v5 ®< ) has exactly n real and different

n

roots for any except the case cx TM oo where^ as follows from

(III, 9) and from the diagram 3, the n=th root becomes Infinite, but

the n roots remain different 0

In combining the information contained in the Diagrams 1, 2

and 3 we obtain the result, that each ) s y - 1, o °° 9 n is a

monotonically increasing function of «< if °< runs from - oo to oo«

We have in particular

(VII, 2) -

Vvv(-oo) k“2

5l'T

U cos

2 2 j>-1

V. X

:

4 COS

2

^

1

^ ii cos

4 COb 2n+l

(1 < 2 < n) |

v-lC-oo) s "oo , ^(0) * 0

(VII, 3) < ^(1) *1

I 2 7T 4 cos 2n+l

V, (oo)

k cos' 2n

(vii,U) J

v (-oo)

n

v y

v (|) n

1

, 2 n-1

4 COS dr

2n

v (0) n

J

t 1

x

nr-l“, 2 n-l~ 11 008 SPT T

r~rr^i

4 COS TT y (l)=x ^ vr ( oo ) ~ oo o n

v 1 n , 2 mr 9 n ' y h cos ^

18

On the other hand it follows .from ( II, 10 ) and (V,7)

(VII, 5) 0n

(O) - 1 , AjO, <*).<* n

(VII,6) A

n

(v, «) = (-l)

n n / _ n-1 \ n-I

v -n 1 o( + ~j=» I v + o «

o

Therefore we obtain

(VII, 7) Y

1

( ^ )

o O O Y

n

( o<) -

(VII, 8) v

]L ( * ) + °°° + vn

( o< ) B n f <K + n=l

For ck s 1 we obtain from (VII, 7)

frn»9) »®o x„ s 1

n

and, if we apply this to 0n

^(v).

(VII, 10) *1 ' * x

n-1

Since A ( ) has for ary , including oop n different roots,

we see further, that the development of each of v^(c<) in powers of

<K - <sK only e 0 or in poxjers of -- has integer ez

VIIIo THE DISCUSSION OF v.. ( <* ) AND v ( «*

)

1

/ nv J

It follows from (II, 10), that

<-«

n

(VIII, 1)

n= * - nv

’1 /n+lA n-2 f n+2 i n-3 / \i>-l/ _ , / -, \n

( 3

) v -

( ^ J

v '+<’->» + (-1) (2n-2;v+ (-1) ,

19

and we have therefore for the sum and the product of the roots of

this polynomial as they are given in (III, 9 ),

(VIII, 2)

n-1

1 _ 1 (n+±\ _ n -

1

V “1 U cos

2 ^n ^ 2

2n

(VIII,3)

n-1

V S1 U cos

2 &£ n

If o< runs between 1 and co, it follows from (VII, 2) and (VII, 3),

that

< < —j— - vy (oo ) (* > 1, V * 1, ••• , n - 1)

1, oos^ f

v

y

( «.) = v^fco) + 0(1) («< -*a>s ])-l, n - 1)

and therefore in summing, in virtue of (VIII, 2),

(viii,U)

n=»l n=l 2

Z *„ < Z V*) <

2) si

n=*l 2 n

.. Z V*) - + 0(i)

^=1

( «*> 00 )

On the other hand it follows from (II, 10), (III, 6) and (VI, 1)

(VIII,5)

n

y »1

n +

1

2

and therefore

(VIII.6) ( n+1 \

n-1

V 2 ) - X

n

< Ll V*' <

\ / >>»1

2 n n -I

20

Here we have

X

n “

| 2 nir “, .2 Tf 4 COS 7T-“T 4 Sin yrnS: 2n+l

rn —i ! i cjn 2(2n+iJ

if we introduce this in (VIII, 6) and use (VII, 8) we obtain, since

n~l

v (®< ) - n °< = (?) - E *) 5 > -1

(VIII, 7) <

—

< v ( * ) - n * < —-i

v

n(« ) - n * + + 0 (i)

n ( ^ > 1)

( & oo )

On the other hand it follows from (VII, 3) for > 1

(VIII,

11 cos SwX

<*!<«) <. 2 w 4 COS —

2n

1 , 2 7 H

tan 2n+l

'<v"1(

e<) < j“ tan^

(* >D

v.1

1 i , 2 7f n /I \ / , \ + r “tan gn

+ 0(~) ( »< oo )

We assume now *=< < l c Then we obtain from (VII, U) the inequalities

f

;—rh=rj7f < V*

} *

;—rm=irv

^ < °) > h cos h cos

(VIII, 9) < 7~~~

2 ( n-1 ) Jr n

k cos ^srrr-

<V"(‘K) <r

z^n-i , 4 cos ”rr” 7/

, 2 2n-l ^ V

n^ ^ <

4 cos ”tu““ 7T 1 2 n _4 cos -s-— 77 (i <* <

2n+l

21

For v.-, ( «*, ) we have from (VII,3) at once

1

it cos

2 if <yl

(o<) <

i ; 4 cos 2n+l

(VIII, 10) <

0 < *) < —

2

it cos^

(I < « < D

(o <=<<§)

and It remains to consider v^( <*. ) for negative e*. 0 Here we have by

(VII, 2) and (VII, it)

It COS

2

If

2n

<^(«K) < (y - 2 , »°° ,n)

ttt: +

< V X. rrr °<i

it cos -75TM- ?r

( °< “ 00 ) ,

z v*>- £ r—2—i- + o ( i) -*

> “2 v y-2 it cos^

00 )

n n

< Z v„( «< ) < V X

it cos

2 ~~ IT V«2 V y«2 i> ~1

2n

The sum to the right is obtained from (VII, 8) for ^ s 0 as ( ^),

while the sum to the left is given by (VIII, 2) j we obtain

^ <

)>-2

Z <(|j . I v*

V =2 + 0(i) ( o< 00) ,

and it follows now from (VII, 8)

22

f

0 <

1

n &f <

(2n~X)(n-l)

(VIII, XX)<

IJSfT ( P"1

, . - , . s n®( 4- —7

- ! +. o(i) _ , 1 6 V<

(®< < 0 )

(«* ~ 00 )

We have new from (VII, 7)

V ( e< ) - v

2 (®<) vn

(«x)

here we have by (VII, 2) and (VII,U) for «X > 0:

v («*. ) > x“ 1 y —

X

> 0<> y * 2, • • • , n) ,

and it follows from (VI 1,10)

(VIII, 13) Vn ( «< ) < &< («* >

On the other hand, if we assume now \ > > 0, we have

\(<*) < %(§) 2<* 800

9 n )

and therefore by (VIII,12) and (VII, 7) applied to * |

(VIII, 1U) vn (®0>

°< ^(i)

vc (i) 8 ° (| Ai. v

1

(i)° oov

n(4;

_c<

9 2 cos'

(J > ^ > 0)

23

IX. X

y (*)

Consider now the fundamental roots A^( ^ ) of , ordered according to their magnitude

(IX.1) 4 A (^) .

¥e have in virtue of (V,!?)

(3X.2) Ai^) OS ~ sS + (V « 1, •••

, n)

so that the equality in (IX ,l) is impossible. The product of the

(n) is equal to the square of the determinant of D ' %

(IX. 3) A 1

(°^ )

° • • A n

( 2n

On the other hand it follows by (VII, 8) and (IX, 2)

(IX,U) >^(cs) + ... + A

n

(o<) » + ^2)

and further from (IX ,3)

(IX,£)

n

ok 2n

TT M-O

V-2

We have from (IX, 2) , (VII,2) and (VTI,iO

) "> v

y

(o) = (y > 1, 0)

and therefore in virtue of ( VII,10)

TT \ .(«) -» TT x' - l (« -> o) . »-2 y V-l(IX,6)

We obtain now from (IX,£)

2U

(IX, 7) A^(<=< ) ^ ( 0< —^ 0) .

For ^ n

(°< ) it follows from (IX,U)

(IXS 8) A

n(^) - n <*2 + (|?) - (A^oO + •- + An_x

( A)

and further from (IX,2), (VII,2) and (VII,3)

x

y (*o o< - <=S +

(IX, 9) cos

2 V'TT'

+ 0

(9 = 1 , ece

, n - 1 | oo ) ,

and therefore by (VIII, 2)

n~l

V“1

^ ^ (°0 - (n - 1)®< ' ” (n - l) + —=g-— + 0 “) (°< oo

)

Introducing this in (IX ,8) we obtain finally

(IX, 10) A

n(«) - <* 2 + (n - 1) * + + 0 (oc oo)

If ») - oo we have

(IX, 11)

V ( e< ) n

v ' 4

. COS2 n=l ^ 1 . 2 'nr 4 sin

and therefore

(XX,12) X

n(«) - =*' _ . 1 1 .2 if . „ H u

00 S °

(•fe) ("<->- ®) .

2£

Further, in the case of A («* ), we can obtain from the bounds for n p v ( <*) derived in the section VIII3 immediately further estimates for

great j j 0

X e BOUNDS FOR A^-0 FOR SMALL | •<

| .

We have from the definition of since A (v, *<) begins with

n

(X,l) °< 0n

(v) + (°< - 1) v0

n=>1 (v) s TT (v

y (^) - v) •

2

In replacing here v by = <=k and in using (rx s 2) and (IX 3 3) we

obtain then the identity

°< 0n

( - °< 2

) - (=< - l)

2 ^ 0r>=1 ( CK „ ®< 2^ ^ ^

2n

j

and in dividing this by =<

(X,2) =* 2

) - (<* - l) 2 ^(x - “ 2

) + c* 2n»l

and therefore

On the other hand we have from (X s l) for °< - 1/2 and - 0

n n

26

and in replacing in both formulae v by <=<-©<

(1,3) 0n(^ - c< 2

) - o<(i _ °<)0

n„1

(=< - °< 2

) = 2JT (vi,(|)

2 + _

(x,U)

n

n„1

( - °< 2

) - TP ^(o) + ^< 2 ^ ^

2, . Tb obtain a representation of 0 ( - °< ) in the form of a product we

replace in (III 5 6) v by ( =>< - <=<“") and use (VI 5 l) and (VTI,9) . Then

we have

n n 0

(<=* - -< 2

) - TT ( x + = TT (l +1- (cx 2 - o<)^ , n y«l

v

J>=1 \ Xl» /

„

1 i 2 j)lf i i . 2 j)ff , 1, COS

^

and therefore finally

(x,S) 0n

(-< - ^

2

) » TT (1 - 2^) + [). C<

( 1 - c*. )

. 2 sin

If we have now

(X,6) 0 < < I

,

we get from (X 5 !?)

0 < 0 (°0 < 1

and by (IX 9 2) S since Vy(<*-) monotonically increases with <=< 5

(X,7) Vy(0) + < A y(«=< ) < Vy(|) + - (y > i) o

27

But here (!-**)< l/l^ while by (VII, 2) and (VI,l)

Vy(0) > 1/U (9 > 1) „ Therefore all terms in (X,7) are positive and

(x,8) tr (v (o) 4. *

2 - •<) < TT A

,.(<*) < TT(V (i) + * 2 - *) . P-2 y=2 y v=2' '

We have now from (X,8)

Xx (°<)”- \(«)

T

i

(i)+ fr(v„(D + «

2- =<)

V-l

and therefore in virtue of (IX,3), (X,2) and (X,3)

>l(°<) 2 <=< 2n

v

1(D+ =< 2- 0n

(«< ®< 2

)0n-1

(i -

2* 2n

cx “ ^+d- (A) (1-2 °<)0

n_1

(o< - of)

We obtain therefore fin ally9 as ^,3/°*” ^ Positive and < 1?

2

(X,9) A

1(«^)>

(1- <*)(l-2°0 + <x 2n-l (0 < *

< J)

On the other hand we have again from (X,8) in using (IX, 3) and (X,U)

nTT\X«)

X (<* ) <

l

v ' n v=l cK 2n

TT(v„(o)+

y-2

- <=*) n=l

28

and therefore by (Xj£) applied to n - 1,

2n

(x,io) ^

1 (^)< T?T

IT ^(1-2 <*)

2

+U <*(l~ oC) Sin2

(0 <«f < |) .

If we make now the hypothesis

(X,ll) °< < 0 ,

we have then

Vy(0) + ®< 2 - c* > (°< ) > VyC-oo) + - <* {)} > 1) 9

where

v^(-co)

h COS

2

IT

(V - 2, •••

, n)

and. all terms are positive! therefore, in multiplying and in using (X,U)

(x,i2) - =* 2)> TT V°< ) > TT (<*

2 - * + V-2 y-1 It cos

2 ££•

It follows now from (IX ,3)

(X,13)

2n

0n„i ( ** " * )

“ I (^s )' “ n-1T

2n

IT /*-<*+ i—7 v=l 1 U cos

29

On the other hand, if we put in (ill, 9) v =

2

, we obtain

«

2

) <X n-1 (* - n-1

- TT

v=l

cK +

I|. COS 2 11

and therefore by (X,2)

n-1/ , (x,ih) rrU2

9=1 \

c< + h

2 ^-Tt , cos

2 n-1

+ (l - ®< ) - «

2

)

We have, since the roots of are x l» j

(X,150

n-1

^n-1^ ” ' * TT (xj + °< ^ - °< ) , 9=1

and this is in virtue of ( VII ,10) > 1 and tends to 1 if <=* 0 o We

obtain finally

(X,16)

n-1

CK(-

2n ^2

) X

l

( " ) * - g

gnT (°< < 0) ,

2n

where both denominators are > 1 and tend to 1 with <*• —> 0.

XI. A(D^)AW A(D^ )

We have by definition

(XI, 1) A(=<) = A(d19) = l/A n

(<x) »

3

K

30

(XI, 2) >(<*)» A(d£?}

) =

For A (=<) we obtain at once from (IX, 2) and (vn,U) the values

A(-oo) - oo , A (0) =

'Tf

2 sin

TJxfc

A(|) = t cot ^ ,

(XI,3)

A(l) - 1

2 sin IT

TJnb?

A ( oo ) - oo

The asymptotic behavior for =< oo is easily obtained from (IX, 10)

and (IX, 12) . From (IX,10) we obtain, in developing the square root of

the right hand expression,

/i + + (n-l)(2n-l)

, 2

6 04 + 0

d\

i + ^ + o ri

2* 2l|o<2 Vex

(XI, u) A (o< ) = + 0 ( ok •»-$> oo)

From (IX,12) we obtain in the same wgy

« +

T

Jb

+ rb cot? &

+ 0

\\c&.

1 Qi)

— 1 - _2_ + TMi_ ^

8-

1 - —L~ cot

2 ££ - + o (

8e^. 2 2n 8«^ w3

y

31

(XI,S) A(^) 1

2 + 1

cot

2 rf

Tn

+ 0 (or - oo)

On the other hand it is easy to see that A(=<) is monotonic ally

2 increasing with « from <=< - 0 to ^ } oo „ Indeed A (°<) = A. (« )

is, as we have used in the section V, the maximum of the quadratic form

o o 2 2

(°< x^)~+ (<* x.2 + + (cx x^+ x^ + + ° ° ° + (®< x

n

+ x^ + ° » • + x

n-l'

2 2 on the unit sphere x^ + <• <* ° + x

n

- 1„ This maximum is obviously assumed for non negative values of the variables and is therefore not

decreasing with - 0 „

We consider now X (°< ) „ Here we obtain immediately from (VTI,3)

and (IX, 2) the three special values %

(XI, 6) X(o) = 0 , X(g-) - 2 tan 4— , A(l)-i 1 _

cos TOT

Further, for -4 oo we have from ( VIII ,8) by (IX, 2)

*!<*<>

>(<* ) - o<

e*. + i + i tan

2 + 0 oo ) ,

^ he*

2

T

hi?

Wl2 ^ + 0

C^)

11 1

1 “= —• + ——w *3“ ——— tan

{

= o,ji « l.i 1 . 2 ir i . n

-i 1 - 7^- + ——5 + ——5 tan ' 7-2 + 0

O cK o'* 0

(XI, 7) *(*) = «*-! + 1 , 2 'Tf r

. OT tan Tn 0 fe) ( c< =*> 00 )
>
32

Finally,, for - co we have from (7111,11) and (IX.2)

A(«) 1 + illi + (2n-l)(n-l)

+ Q ^ s 2 6^ &)

2

(XI s 8) A(<*) — + 75^TM + ® (®^ “ oo ) 0

The discussion of the monotopy properties of A(e*) for <=« < 0

and A(<=*0 for c< > 0 appears to present considerable difficulties 0 On the other hand, we can prove that A( «*) is monotonically decreasing

for <p< < 0 o Indeed, if we put

x- - ft , ft > o , 7=jg- »

( n) the transformation with the matrix DjL is CK

yx

= - ft

^

y2

= *i - ft x

2

yn *1 + x

2

+ ••• + V-l “ ft x

n

If we write here jy - ~z ^

- l s

* •

, n) , we obtain the inverse of

the above transformation in eliminating successively x^,Xg, °** from

the right side expression in

33

= r ®i x

2 =

If z 2 + ^ X

1

e o x

n

' ^ zn

+ ^ (x

l

+

*2 + ••• + x

n-l )

The result mil be a transformation S

(S) x v = £ A (/) z (9 - 1.

pi r r

n>

where A (Y) are polynomials in y" with non negative coefficients, and

r we have

(XI, 9) X(D^)

A(S)

On the other hand, the maximum of the quadratic form

n

/

yf \ 2

E [ 2 Vr)

y =1 u=l r r )

,n on the unit sphere 2 z 3 1 is obviously attained for non negative

p. jl

z and is therefore mono tonically increasing with y . By (XI, 9) A(®0

is therefore monotonically decreasing with y and monotonically increasing with {3 o We see that if goes from 0 to - oo, X ( ®0

monotonically increases from 0 to oo.

August 7, 19!?

2

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