18 problems

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a semialgebraic spline, meaning a continuous piecewise polynomial function on a semialgebraic partition of $\mathbb{R}^n$. Pierce–Birkhoff…

Strang's conjecture. For a generic triangulation $\Delta$, the dimension of the spline space is precisely

Let a smooth singularity curve intersect the boundary of $[0,1]^2$, and approximate its signed-distance function using a least-squares bicubic spline procedure. Least-squares appro…

Authors' conjecture. The superior performance of MTA for point clouds with data gaps is due to the closeness of the explicit MTA strategy to an L1-norm minimization.

The paper considers quantile regression series terms and test statistics under assumptions ensuring asymptotic results for additive univariate series terms, including univariate B-…

Let $\phi_n$ be a twisted $B$-spline of order $n$, and let $T_{(k,l)}^t$ denote the twisted translation indexed by $(k,l)\in\mathbb{Z}^2$. Riesz-sequence conjecture. The system … i…

Let $\Delta$ be a triangulation, let $r$ be the smoothness parameter, and let $d$ be the spline degree. Write $\mathcal{S}^r_d(\Delta)$ for the space of splines of degree $d$ and s…

Let $H$ be a graph, and let $e_1,e_2$ and $e_1',e_2'$ be two pairs of adjacent edges such that $H\cup\{e_1,e_2\}$ and $H\cup\{e_1',e_2'\}$ are both $C_2^1$-independent, with the co…

Let $\boldsymbol\beta_{\boldsymbol\beta}$ be the exponential B-spline associated with a list of roots $\boldsymbol\beta$, and let $\boldsymbol\beta$ be a vector whose elements are…

The proposed interpolators are compactly supported linear combinations of shifted exponential B-splines on the half-integer grid; their degree of regularity can be increased arbitr…

Let $P$ be a planar polyhedral complex, and let F=\max\{n\mid\text{\rm {there is an n-gon in }}P\}. Let $k$ and $r$ be the parameters in the spline-dimension formula of Theorem…

Let ${{\cal M}}$ be a Riemannian manifold, let $x\in{{\cal M}}$, and let $W_x\subset T_x{{\cal M}}$ be the subspace generated by the first-order tangent vectors of a coalescing con…

Consider the derived optimal quadrature rules for spline spaces of polynomial degree $d$ and continuity $c$ over finite domains, together with their corresponding asymptotic rules.…

Let an odd-degree source spline space have an optimal quadrature rule, and let a target spline space be obtained through an even-dimensional transition between the source and targe…

Foucart–Sorokina conjecture. The dimension is

Let $\mathcal{P}\subset\mathbb{R}^2$ be a hereditary polytopal complex, and let $F$ be the maximum length of the boundary of a polytope in $\mathcal{P}$. Regularity conjecture. … T…

Let $\mathcal{P}\subset\mathbb{R}^2$ be a hereditary polytopal complex, and let $F$ be the maximum length of the boundary of a polytope in $\mathcal{P}$. Regularity conjecture. … T…

Let $k$ be the order of a univariate B-spline space, let the knot sequence be arbitrary, and let the least-squares projector onto the corresponding spline space be defined with res…