Strong converse inequalities and quantitative Voronovskaya-type theorems for trigonometric Fej\'er sums

Let $\sigma_n$ denotes the classical Fej\'er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $(I-\sigma_n)^r(f)$ in terms of the modulus of continuity of order $r$ (with specific constants) in all $\mathbb{L}^p$ spaces $1\leq p \leq \infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $(I-\sigma_n)^r(f)$.