  We study the $\ell^1$-summability of functions in the $d$-dimensional torus
$\mathbb{T}^d$ and so-called $\ell^1$-invariant functions. Those are functions
on the torus whose Fourier coefficients depend only on the $\ell^1$-norm of
their indices. Such functions are characterized as divided differences that
have $\cos \theta_1,\ldots,\cos\theta_d$ as knots for $(\theta_1\,\ldots,
\theta_d) \in \mathbb{T}^d$. It leads us to consider the $d$-dimensional
Fourier series of univariate B-splines with respect to its knots, which turns
out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal
series of the B-spline function.
