More on fast decreasing trigonometric polynomials

In a recent paper, for a fixed $m\in\mathbb N$, we introduced trigonometric polynomials$$L_n(x):=\frac1{h^m}\underbrace{\int_{-h/2}^{h/2}\dots\int_{-h/2}^{h/2}}_{m\,\text{times}}J_n(x+t_1+\cdots+t_m)\,dt_1\cdots\,dt_m,$$where $J_n$ is a Jackson-type kernel. In the current paper we show that $L_n$ and its first $m-1$ derivatives provide approximation to the $B$-spline of degree $m-1$ and its respective derivatives.