17 problems

Let $m\in\mathbb N$ be an arbitrary derivative order, and consider B-spline estimators of a univariate function subject to an $m$th-order nonnegative derivative constraint. The ass…

Let $k$ denote the order of the univariate B-spline basis, and let its relevant condition number be the quantity measuring the stability of that basis as $k$ increases. de Boor's c…

Asymptotic B-spline conjecture. Relations of these types should also hold when the normalized Chebyshev polynomials are replaced by the normalized Gegenbauer and Hermite polynomial…

Define the B-splines by $Q_1(x):=\chi_{[-1/2,1/2]}(x)$ and $Q_{n+1}(x):=(Q_n*Q_1)(x)$ for $n\geq 1$. Let $\mathcal{F}(Q_n)$ be the set of $(a,b)\in\mathbb{R}_+^2$ for which…

Let $Q_2=Q_1*Q_1$, where $Q_1(x):=\chi_{[-1/2,1/2]}(x)$, and let $\mathcal{G}(Q_2,a,b)$ denote the associated Gabor system. Nielsen's conjecture. The Gabor system…

Lemvig and Nielsen's conjecture. The Gabor system $\mathcal{G}(Q_2,a,b)$ is not a frame along the hyperbolas

Positive lower-bound conjecture. There exists a positive lower bound for the operator inequality in (31); equivalently, there is a constant $A>0$ such that

Let $(U,V)$ be distributed according to the maximum-correlation distribution $C^*$ constructed from the B-spline basis … on $[0,1]$, where the degree satisfies $d\geq 0$ and the kn…

Refinement conjecture. Every possible configuration of linear dependency is a refinement of one of such cases.

Let $n$, $k$, and $j$ be the parameters of the Baskakov-type linking operators, let $c>0$, and let $p_{n\rho-c(k-2),j\rho+i_1+ots+i_k+k-1}(t)$ and $N_{n,k,j}(t)$ denote the corresp…

Let ${\boldsymbol{l}}$ be a column of ${\boldsymbol{L}}$, and let $\bar{{\boldsymbol{H}}}$ be the sparse matrix constructed as its left null-space operator, with each column summin…

Exponential-decay conjecture. The ratio

Non-frame conjecture. The Gabor system $\mathcal{G}(B_2,a_0,b_0)$ is not a frame. Furthermore, it is not a frame along the hyperbolas

Frame set conjecture. The frame set of $B_n$ is the maximal set of parameters $(a,b)$ that avoids the known obstructions.

Let $C_m^d$ be the circulant associated with the degree-$d$ B-spline collocation matrix, with Toeplitz symbol coefficients $t_0^d,\ldots,t_r^d$, and let $\lambda_\mu(C_m^d)$ denote…

Let $d$ be the degree of polynomial B-splines, let the interpolation nodes be the knot averages (Greville sites), and let $A$ be the corresponding B-spline collocation matrix. Writ…

Let $g$ be a function on $D=[x_0,x_1]$ that is known only up to a parametric form near the boundaries, and represent it using a combination of that parametric form and B-splines wi…