Extremal Trigonometric Polynomials

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

.