# Questions tagged [splines]

Splines and their properties and applications. A spline is a function defined piecewise by polynomials, and is typically used in interpolating problems.

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### Numerical solution to partially-free boundary optimization problem

Background First of all, I'm a PhD physicist working in numerical analysis, so I apologize for possible easy-to-spot mistakes (they're most likely not that easy for me). The problem I'm trying to ...
41 views

### Make number of B-spline basis functions match the number of abscissae for odd degree

For a given number of abscissae $x_1,\dots,x_i,\dots,x_n$, I am working with B-splines $B_{j,p}(x)$ of odd degree $p \in \mathbb{N}$. In the simplest case $p=1$, the functions are piecewise linear, ...
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### Thin-Plate-Spline understanding and solution

This is a migrated question from: Thin-Plate-Spline understanding and solution. If I need to delete one of the questions let me know. I was suggested to post it here as well. As I understand it a Thin-...
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1 vote
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### Special properties of "vibrant" spline-functions

While checking an idea about knot-placement for spline interpolation, I needed to find a way to calculate splines, that are strictly monotone between adjacent pairs of knots and for which every knot ...
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### Cubic splines convergence?

I am looking for a basic, classical, result on approximating a smooth function using cubic and linear splines. Is there a reference on the convergence, in some sense, of the splines to the function of ...
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### How to deduce the recursive derivative formula of B-spline basis?

Description Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denote a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$, and the $i$-th B-spline basis function of $p$-degree, ...
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### Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
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### Estimating overshoot in spline interpolation

Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...
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### General reparameterization of a B-spline

Say I have a B-spline function (or curve) of order $k_1$, defined over some knot vector $\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$ Do you know of a process of finding ...
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### Global approximation via convex combination of local approximations

I recently faced the problem of efficiently approximating a very large set of data points and, neither having a model of the empiric function, nor of the error distribution, my method of choice would ...
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### Norms of B-spline coefficients

In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...
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1 vote
199 views

### Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the definite integral $$\int_a^b f (\log x) \,\mathrm{d}x\,?$$
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