CONTOUR INTEGRALS AND THEIR APPLICATIONS

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CONTOUR INTEGRALS AND THEIR APPLICATIONS
Wayne Lawton Department of Mathematics National
University of Singapore S14-04-04,
matwml_at_nus.edu.sg http//math.nus.edu.sg/matwml
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ARYABHATA

• characterized the set (x, y) of integer
  solutions of the equation

where a and b are integers. Clearly this equation
admits a solution if and only if a and b have no
common factors other than 1, -1 (are relatively
prime) and then Euclids algorithm gives a
solution. Furthermore, if (x,y) is a solution
then the set of solutions is the infinite set
Van der Warden, Geometry and Algebra in Ancient
Civilizations, Springer-Verlag, New York, 1984.
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BEZOUT

• investigated the polynomial version of this
  equation

Clearly this equation has a solution iff
have no
and
common roots and then Euclids algorithm gives a
solution.
Bezout identities in general rings arise in
numerous areas of mathematics and its application
to science and engineering
Algebraic Polynomials control, Quillen-Suslin
Theorem Laurent Polynomials wavelet, splines,
Swans Theorem H_infinity the Corona
Theorem Entire Functions distributional
solutions of systems of PDEs Matrix Rings
control, signal processing
E. Bezout, Theorie Generale des Equations
Algebriques, Paris, 1769.
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INEQUALITY CONSTRAINTS
are
Theorem If RPLP
on the unit circle
LP
then
and
on
Proof. Let LP
real on
with
that is real on
with
Choose a LP
then choose
W.Lawton C.Micchelli, Construction of conjugate
quadrature filters with specified zeros,
Numerical Algorithms, 144 (1997) 383-399
W.Lawton C.Micchelli, Bezout identities with
inequality constraints, Vietnam J. Math.
282(2000) 97-126
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UPPER LENGTH BOUNDS
Theorem
There exists
with
Furthermore, for fixed
and for fixed L
Proof Uses resultants.
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LOWER LENGTH BOUNDS
Theorem
For any positive integer n, there exist LP
and
with
Proof See VJM paper.
Question Are there better ways to obtain bounds
that bridge the gap between the upper and lower
bounds
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CONTOUR INTEGRAL
representation for the Bezout identity is given by
Theorem Let
are a disjoint contours and the
contains all roots of
and
interior
of
then for
excludes all roots of
where
are LP, real on T, and satisfy the Bezout
identity.
Proof Follows from the residue calculus.
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SOLUTION BOUNDS
Lemma
where
a contour that is disjoint from
and whose (annular) interior contains T
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CONTOUR CONSTRUCTION
on T,
Since
hence if
-invariant contours then it suffices
are
to consider these quantities inside of the unit
disk D.
For k1,2 let
union of open disks of radius
centered at zeros of
in D
and
be the disk of this radius centered at 0.
if
else
Theorem
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CONCLUSIONS AND EXTENSIONS
The contour integral method provide sharper
bounds for
and therefore for B than the resultant method but
sharper bounds are required to bridge the gap.
Contour integrals for BI with n gt terms are given
by
where
encloses all zeros of T except for those of
Residue current integrals give multivariate
versions