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TZID:America/Chicago
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DTSTART:20171105T020000
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TZOFFSETTO:-0600
RDATE:20181104T020000
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UID:calendar.20581.field_event_date_2.0@math.wustl.edu
CREATED:20180213T164016Z
DESCRIPTION:Abstract: The Fej\'er-Riesz theorem states that a non-negative
trigonometric polynomial is the hermitian square of an analytic polynomial
of the same degree. Rosenblum showed that the theorem is still valid if
the coefficents are Hilbert space operators. The speaker later extended t
his to strictly positive operator valued trigonometric polynomials in fini
tely many variables. Results in real algebra due to Scheiderer imply that
scalar valued non-negative trigonometric polynomials in two variables alw
ays factor as a finite sum of squares of analytic polynomials\, and that t
his fails in three or more variables. We discuss a purely analytic approa
ch\, using Schur complement techniques\, to showing that any non-negative
matrix valued trigonometric polynomial in two variables is a finite sum of
squares of analytic polynomials. In analogy with the Tarski transfer pri
nciple in real algebra\, the proof lifts the problem to an ultraproduct\,
solves it there\, and then shows that this implies the existence of a solu
tion in the original context. While the method is non-constructive\, it n
evertheless implies a concrete algorithm for such a factorization.\n\nHost
: Greg Knese
DTSTART;TZID=America/Chicago:20180409T160000
DTEND;TZID=America/Chicago:20180409T170000
LAST-MODIFIED:20180622T190149Z
SUMMARY:Analysis Seminar: 'Factoring non-negative matrix valued trigonometr
ic polynomials in two variables'
URL;TYPE=URI:https://math.wustl.edu/events/2707
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