A comment on this problem would be that if f^(n) are monotone ( here f any continuous function, not necessarily odd or even, also I assume that f^(n) is monotone not |f^(n)| ) then one can assume that they are positive. And if Fourier coefficients are real and positive then they must be absolutely convergent, that is {f^(n)} is in l_1. This follows easily from property of Fejer's kernel, i.e. that it is positive operator with integral 1:

sum over k of (1-|k|/n)f^(k)*exp(ikt) = integral of F(t-s)f(s) <= sup|f| so 1/2( sum over -n/2, n/2) of f^(k)) <= sup|f|

A comment on this problem would be that if $\hat f(n)$ are monotone ( here $f$ any continuous function, not necessarily odd or even, also I assume that $\hat f(n)$ is monotone not $|\hat f(n)|$ ) then one can assume that they are positive. And if Fourier coefficients are real and positive then they must be absolutely convergent, that is $\{\hat f(n)\} \in l_1$. This follows easily from property of Fejer's kernel, i.e. that it is positive operator with integral 1:

$$ \sum_k (1-|k|/n)\hat f(k)exp(ikt) = \int F(t-s)f(s) \le \sup|f|$$ so $1/2 \sum_{k \in (-n/2, n/2)} \hat f(k)) \le \sup|f|$.