BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS
Wayne M. Lawton
Department of Mathematics
National University of Singapore
Lower Kent Ridge Road
Singapore 119260Email [email protected]
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
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
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