Let $k>0$ be a positive integer. Set $n=4k.$ Let $R(t)$ be a polynomial of degree $n-1$ with coefficients in $\lbrace -1,1 \rbrace$.

Consider the discrete average

$$ D(n,R) = \frac{\sum_{j=0}^{n-1} \vert R(exp(2\pi i j/n)) \vert}{n} $$

and the average

$$ A(n,R) = \frac{\int_{0}^{2\pi} \vert R(exp(it) \vert dt}{2\pi}. $$

When $k=1$ so that $n=4$ we have that for one half of the possible polynomials $R(t)$ $$ D(n,R) \leq A(n,R). $$

Question: What happens when $k>1.$

`big`

(so that computations may become more complicated). Can be useful to observe (besides nice gerry's observation) that $$ A(n,R) = \int_{0}^{1} \vert R(exp(2\pi i t) \vert dt $$ so that seems that some kind of`Riemann sums`

are involved ? – Luis H Gallardo Feb 6 '11 at 19:42`integral`

from the`Riemann Sum`

. – Luis H Gallardo Feb 7 '11 at 18:24`Littlewood polynomials`

See the nice paper of Peter Borwein and Michael Mossinghoff (available on the net): The $L_1$ norm of polynomials. – Luis H Gallardo Feb 7 '11 at 19:15