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Extremal Trigonometric Polynomials

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N
4
Consider the trigonometric polynomial
T
n
(ϕ)
n
∑
k=1
a
k
cos(kϕ)+
b
k
sin(kϕ)
of degree
n
such that not all of the
a
k
and
b
k
are zero. Babenko's theorem (1984) states that the measure of the subset of
-π<ϕ<π
for which
T
n
(ϕ)>0
is at least
2π/(n+1)
. The unique extremal polynomial, positive exactly on the interval
2π/(n+1)
and normalized so that
T
n
(0)=1
, is constructed and shown for small values of
n
.
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