The $ZB$-Splinets package offers a comprehensive framework for building $ZB$-splines, their orthogonalization, and evaluating or projecting data onto a $ZB$-splines basis. It supports users in visualizing $ZB$-splines by providing intuitive plotting functions. This package is designed to facilitate advanced functional data analysis tasks by combining robust mathematical tools and user-friendly functionality, ensuring seamless integration into your analysis workflow.
The $ZB$-spline of degree $k \in \mathbb{N}_0$ is defined as the first derivative of a $B$-spline:
$$
Z_{i}^{k+1}(x) := \frac{\mathrm{d}}{\mathrm{d}x} B_{i}^{k+2}(x).
$$
From this definition and the differentiation formula for $B$-splines, the relation for $k \geq 0$ is:
$$
Z_i^{k+1}(x) = (k+1) \left( \frac{B_i^{k+1}(x)}{\lambda_{i+k+1} - \lambda_i} - \frac{B_{i+1}^{k+1}(x)}{\lambda_{i+k+2} - \lambda_{i+1}} \right),
$$
with the local support:
$$
\text{supp};Z_i^{k+1}(x) = \text{supp};B_i^{k+2}(x) = [\lambda_i, \lambda_{i+k+2}).
$$
For more details see the
arxiv link for the paper, Efficient spline orthogonal basis for representation of density functions
You can install the development version of Z-Splinets from GitHub using the devtools package:
# Install devtools if you haven't already
install.packages("devtools")
# Install Z-Splinets package from GitHub
devtools::install_github("HibaNassarDTU/Z-Splinets")