Convex functions and Fourier coefficients

Abstract:

Let $f$ be a continuous function defined on the interval $(0,1)$. For $n = 1,2, \ldots$ and $0 < s < t < 1$, denote by ${a_n}(f;s,t),{b_n}(f;s,t)$ the $n$th Fourier coefficients of $f|(s,t)$. It is shown that the following statements are equivalent: (i) $f$ is strictly convex on $(0,1)$. (ii) ${b_n}(f;s,t) < (2/n\pi )[f(s) - f((s + t))/2]$ for all $n = 1,2, \ldots$ and whenever $0 < s < t < 1$. (iii) ${b_n}(f;s,t) > (2/n\pi )[f((s + t)/2) - f(t)]$ for all $n = 1,2, \ldots$ and whenever $0 < s < t < 1$. If, in addition, $f$ is twice differentiable, then (i) and the following statement are also equivalent: (iv) ${a_n}(f;s,t) > 0$ for all $n = 1,2, \ldots$ and whenever $0 < s < t < 1$.