BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

Wayne M. Lawton

Department of Mathematics

National University of Singapore

Lower Kent Ridge Road

Singapore 119260Email wlawton@math.nus.sg

Tel (65) 772-3337Fax (65) 779-5452

Bezout Identities

where products belong to a ring R with identity 1

1... BABA mm11

Origin in diophantine problems, where p’s are coprime integers, gave rise to the Euclidean Algorithm and the Chinese Remainder Theorem

Related to Corona Theorem, transcendental numbers, wavelets, deconvolution, interpolation, and control

Interpolation Concepts

Lattice Subroupsd

21RLL

Restriction Operator

Inclusion Operator

Convolution Rings kkck

,*,LCR

12RR

2

T

1RR

Interpolation Method

Interpolatory Filter 12

p,Rp The Interpolation Operator

2

p

1RR

I

defined by fpfI T

p satisfies

fffpfI1p

Desired Properties

Accuracy: if is a low-degree polynomial

then ,ffIp

f

Positivity: if fthen fI

p

is positive-definite

is positive-definite

Remark: positivity is required for interpolation of statistical autocovariance functions, occurs if and only if the interpolatory filter p is positive-definite

Z-TransformConstruct an isomorphism

onto the ring of Laurent polynomials, from a basis for

}C)e,...,e(t{T diid 11

2L

),z,..,z(R:d12

and define the torus group

A Laurent polynomial is determined by the trigonometricpolynomial defined by its values on the torus group.

Denote Pp

Stability Group

Torus group acts as a group of transformations on

}PgPRP:Tg{G 1d

Define the stability group

Then

tzPztP

)L,L(indexG 21

Equivalent Properties

Accuracy: zero, to specified order, onP

Positivity: P dTis nonnegative-valued on

}1{\G

Interpolatory

Gg

GgP(Poisson)

(Poisson)

(Bochner)

Filter Design Approach

Step 3. Define

is coprime}Gg:gA{

dT}1{\G

Step 1. Construct 0A on

Step 2. Solve Bezout 1B)gA(Gg

g

0A on

0Bg dTon

Gg

g

1BgB,ABP

Step 1

Step 1.3 Define

are distinct }Gg:gw{ Step 1.1 Select monomial w

Step 1.2 Construct

gwwgwwzA 1g

}1\{Gg

gAA

Step 2

Use Theorem 1 (Lawton-Micchelli) to solve Bezout

1B)gA(Gg

g

0Bg dTon

are coprime and

AA m1,...,If

0A j dT

Theorem 1

onthen there exist 0B j dT

on

that solve the Bezout identity

1... BABA mm11

mmir iMM

dT

Proof of Theorem 1

where

Step 1. Use Quillen-Suslin to compute invertible matrix

m2m2R

ri

ir

MM

MM~

]A,..,A[ m1whose first row equals

rM and iM belong to the subring R

Step 2. Form the invertible matrix

of Laurent polynomials that are real-valued on

1m2T11 ]s,...,s[r T

m21mm2 ]f,...,f,1,f,...,f,1[r~

f

Proof of Theorem 1

Use this fact to define vectors having continuous entries that are positive, real-valued, respectively

m

1kkAs dT

Step 3. Since have no common zeros in kA }0{\C

is strictly positive on

1m2T]m2F,...,1mF,1,mF,...,2F,1[F

R

Proof of Theorem 1

so the entries of

Step 3. Approximate

are positive on

f by

1m21R

F~

b dTThen the Bezout identity with positive constraints is solved by entries of

Tm1 ,...]B,...,B[b