13 problems

Let $k\geq 1$ be an integer, and consider a $k$-monotone density whose $k$th derivative satisfies $(-1)^k g_0^{(k)}(x_0)>0$ at the point $x_0$. In the corresponding regression-spli…

Spline approximation conjecture. The assumption is satisfied for all $C^{k_t}$ splines with polynomial degree $p_t\geq 2$ such that $p_t-k_t$ is odd. This would extend the stated v…

Quasi-optimal convergence conjecture. Based on numerical findings, for splines, quasi-optimal convergence occurs whenever the difference between degree and regularity is odd.

Quasi-optimal convergence conjecture. Based on numerical findings, for splines on uniform meshes, quasi-optimal convergence occurs whenever the difference between polynomial degree…

Let ${\boldsymbol \tau}$ be an arbitrary knot sequence, let $h$ be the mesh parameter, and let $\mathcal{S}_{p,{\boldsymbol \tau}}$ be the associated spline space. Universal sharpn…

Let $p$ be a given spline degree, let ${\boldsymbol \tau}$ be a knot sequence, and let $h$ denote the mesh parameter appearing in the spline approximation estimate. Simultaneous sh…

Let ${\boldsymbol \tau}$ be a uniform knot sequence such that the dimension of $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ is $2m$. For any $q\geq 0$, let $Q_p^q$ be th…

Let $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ be the periodic spline space of degree $p$ associated with a uniform knot sequence ${\boldsymbol \tau}$, and let $Q_p^q$…

Schoenberg eigenvalue conjecture. For fixed $k>0$,

Let $k$ be an integer, set $a=0$ and $b=1$, and let $\mathcal H_k f$ denote the Hermite spline interpolant of a function $f\in C^{(2k)}[0,1]$ at knots $0<y_1<\cdots<y_{2k-4}<1$. Wr…

Let $k$ be an integer, set $a=0$ and $b=1$, and define the truncated power function … For interpolation knots $0<y_1<\cdots<y_{2k-4}<1$, let $\mathcal H_k b_t$ denote the associate…

A generalized additive partial linear model may include interactions between its nonparametric components, and tensor polynomial splines can be used to approximate such interaction…

Conjecture on extending the main theorem. The main result of the theorem is valid for a wider class of processes and generating functions than those covered by condition (C). The c…