d + 1} \rangle$.

Each of these B-splines can obtained, by translation and scaling, from the basic B-spline $Q^d$ with knots $i = 0, \ldots, d + 1$, $$Q^{d}(x) = \frac{1}{(d + 1)!} \sum_{i = 0}^{d + 1}
    (-1)^i {\genfrac{(}{)}{0pt}{}{d + 1}{i}} (x - i)_{+}^d.
  \label{2.1}$$ Here, $(x - i)_{+}^d$ denotes the truncated powers $(x - i)_{+}^d = (x - i)_{}^d (x - i)_{+}^0$, for $d > 0$, while $$(x - i)_{+}^0 = \begin{cases}
    0, & \hbox{$x < i$,} \\
    1, & \hbox{$x \geq i$.}
  \end{cases}$$ The normalized version of the basic spline is defined as $$N^d(x) = (d + 1) Q^d(x).
\label{2.2}$$ From (\[2.1\]) and (\[2.2\]), we obtain the *normalized* B-spline basis $\{N_i^d\}$: $$N_i^d(x) = N_{}^d \left( \frac{x - x_i}{h} \right).$$

If the interpolation nodes $\tau_i$ are located at the knot averages, i.e., $$x_i^{\ast} = \frac{x_{i + 1} + \cdots + x_{i + d}}{d}
    = x_i + h \frac{d + 1}{2}, \quad i = 0, \ldots, n - d - 1,
  \label{2.3}$$ then $$x_i^{} < x_i^{\ast} < x_{i + d + 1}^{},$$ and the Schönberg–Whitney theorem [@deBoor-2001] guarantees that the collocation matrix is nonsingular. Moreover, this matrix is totally nonnegative [@Karlin-68], i.e., all of its minors are nonnegative. Due 