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Abstract
For fixed
a
and
b , let
Q n be the family of polynomials
q ( x ) all of whose roots are real
numbers in
[ a , b ] (possibly
repeated), and such that
q ( a )
=
q ( b )
= 0 .
Since an element of
Q n
is completely determined by it roots (with multiplicity), we may ask how the polynomial is
sensitive to changes in the location of its roots. It has been shown that one of the Bernstein
polynomials
b i ( x )
= ( x
−
a ) n − i ( x
−
b ) i ,
i
= 1 , … , n
− 1 , is the member of
Q n with largest supremum
norm in
[ a , b ] . Here we
show that for
p
≥ 1 ,
b 1 ( x ) and
b n − 1 ( x ) are the members
of
Q n that
maximize the
L p
norm in
[ a , b ] .
We then find the associated maximum values.
Keywords
polynomial root dragging, $L^p$ norm, Bernstein polynomial
Mathematical Subject Classification 2000
Primary: 30C15
Milestones
Received: 5 May 2010
Revised: 4 May 2011
Accepted: 12 July 2011
Published: 21 March 2012
Communicated by Martin Bohner