More on fast decreasing trigonometric polynomials

Abstract

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.