As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but the more specific combinatorics may vary. What if the value of the dominating number $\frak d$ and ...

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following de Boor's notation we have that $B_{j,k,t}$ stands for the $j-th$ B-spline of order $k$ defined ...

Start with a monotone nonincreasing function and sample it at finite set of points $x_0, ..., x_n$, $x_i<x_{i+1}$ so that $f(x_i)<f(x_{i+1})$. If you approximate $f$ with a linear spline then the resulting piecewise-linear approximation will certainly preserve monotonicity. The question is: if y...

I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using splines (I am open to any choice at the moment). So, I want to create the interpolating function $S(x...