# Extremal Trigonometric Polynomials

Extremal Trigonometric Polynomials

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

T

n

n

∑

k=1

a

k

b

k

n

a

k

b

k

-π<ϕ<π

T

n

2π/(n+1)

2π/(n+1)

T

n

n