12 problems

For each integer $m\geq 3$, let the degree-$(2m-1)$ biharmonic stencil be the unique symmetric $2m$-point interpolatory mask whose coefficients satisfy the polynomial reproduction…

Consider $d53d^n$ endowed with the metric $d(x,y)=\lVert x-y\rVert$, where the norm is any norm on $d53d^n$. Let $S_2$ be a subdivision scheme on $d53d^n$ satisfying, for…

Let $I=\{i:c_i>0\}$ be the support of the mask of a dyadic subdivision scheme with non-negative coefficients. Convergence conjecture. The subdivision scheme converges unless the gr…

Conjecture on the standard parametrization. Thanks to the averaging step, the standard parametrization is $C^1$.

Let a Hermite subdivision scheme act on vector data in $\mathbb{R}^{d+1}$, with the components interpreted as consecutive derivatives, and suppose the scheme converges to a vector-…

The nonlinear subdivision schemes $S_\epsilon$ and $T_{2,2}$ generate limit functions from the initial data considered in Figures 2 and 3, and the smoothness exponent is estimated…

Direct recurrence conjecture. The coefficients can be computed directly by

Let $m$ be the arity, let $l'\in\mathbb{N}_0$, and set $l=n=2l'+1$. Let $\sigma_m^n(z)$ denote the corresponding symmetry factor, and let $\delta(z)$ be the variable used in the ge…

Let $\omega$ be the parameter of the generalized Butterfly subdivision scheme, with the parameter range considered in the source given by $\omega\in\left[0,\frac14\right)$. The sch…

Consider the generalized Butterfly subdivision scheme at the exceptional parameter value $\omega=1/16$. Its limit function is denoted by $\varphi$, and its Hölder exponent is…

Let $n\ge 2$ and let $\Lambda_n=\{\lambda_0,\ldots,\lambda_{n-1}\}$ be an arbitrary set of real numbers. Let $\psi^{\Lambda_n}$ be the generalized Daubechies wavelet associated wit…

The paper studies a geometric subdivision scheme with associated normal scheme ${\bf B}$ and tangential scheme ${\bf A}$. Its regularity refers to the Hölder regularity of the limi…