Tuesday, 16 June 2009

cv.complex variables - Minimizing the modulus of a polynomial around a circle

There is another way.
Every non-negative trigonometric polynomial f on the circle
is of the form |q|2, where q is an analytic polynomial.
(I mean by this that f is of form sumNNanzn and
q(z)=sumN0bnzn).
This is called the Fejer-Riesz theorem.



So, you guess a minimum for |p|2, call it m, and then see
whether f=|p|2m is the modulus squared of a polynomial
(an algebraic identity).
If it is, try again with larger m; if not, reduce m.



For a fuller account, see the survey article by Helton and Putinar:



@incollection {MR2389626,
AUTHOR = {Helton, J. William and Putinar, Mihai},
TITLE = {Positive polynomials in scalar and matrix variables, the
spectral theorem, and optimization},
BOOKTITLE = {Operator theory, structured matrices, and dilations},
SERIES = {Theta Ser. Adv. Math.},
VOLUME = {7},
PAGES = {229--306},
PUBLISHER = {Theta, Bucharest},
YEAR = {2007},
MRCLASS = {47-02 (14P10 47A13 47A57 47A63 90C22)},
MRNUMBER = {MR2389626 (2009i:47001)},
MRREVIEWER = {Joseph A. Ball},
}



-John E. McCarthy

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