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.
27
questions
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28 views
Eigenvalues associated to differential opertator over natural spline spaces
Let $S_n^k$ denote de linear space of natural splines of degree $k$ and uniform grid $a < t_1 < \ldots < t_n < b$ over the bounded interval $[a,b] \subseteq \mathbb R$.
For any $m \in \...
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0answers
26 views
approximation power of tensor-product splines
Let $\Omega$ be a rectangle in 2-D, and $f \in W^s_p(\Omega)$. I am looking for a result on how well a tensor product spline quasi-interpolation $Qf$ approximates $f$. In fact, I need to know how well ...
2
votes
1answer
70 views
Bounds on the second derivative of a natural cubic spline in terms of the data
Suppose we have real numbers $x_1 < \cdots < x_n$ and $v_1, \ldots, v_n$. Let $f$ be the natural cubic spline such that $f(x_i) = v_i$. Is there a simple explicit bound on $\|f''\|_\infty$ in ...
0
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1answer
47 views
Rule to determine rotationally invariant orders of the points of arbitrary 2d splines
I would like to find a rule to determine the order of the points of arbitrary 2d splines, which should be invariant with respect to rotation (as far as possible).
To illustrate the problem, let us ...
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0answers
54 views
Error bounds for spline interpolation. Hall and Meyer's conjecture
Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
1
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1answer
96 views
Polynomial-preserving boundary conditions for spline interpolation
Spline interpolation requires the definition of boundary conditions because the smoothness requirements do not yield enough conditions for a unique solution.
Question:
which kind of boundary ...
3
votes
1answer
264 views
Marsden's Identity and B-splines
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 ...
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0answers
57 views
Defining boundary conditions for spline interpolation via the Euler–Maclaurin formula
The Euler–Maclaurin formula states an interdependency between
\begin{align}
I\quad:=&\quad\int_m^nf(x) \, dx,\ \ m,n\in\mathbb{Z},\\[6pt]
S\quad:=&\quad\sum_{k=m}^n f(k), \\[6pt]
D\quad:=&\...
2
votes
2answers
157 views
Do splines preserve monotonicity?
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 ...
1
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2answers
148 views
Cubic interpolating spline – number of extremum points
Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of ...
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1answer
104 views
Splines with bounded first derivative?
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 ...
1
vote
1answer
489 views
Variational proof for minimum curvature of cubic splines
Background: Give an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)$ is a piecewise cubic polynomial with continuous second derivative. One can also prove, roughly,...
5
votes
1answer
240 views
Maximum of a B-spline
Given $p+2$ nondecreasing (and not all identical) knots $t_0, \ldots, t_{p+1}$ on the real line, consider the normalized B-spline of degree $p$ defined over these knots.
We know that the B-spline is ...
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0answers
32 views
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 ...
3
votes
2answers
434 views
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 ...
1
vote
1answer
236 views
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 ...
3
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0answers
399 views
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 ...
6
votes
1answer
364 views
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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0answers
494 views
How to find all the zeros of a cubic spline?
Let's say I have a cubic spline represented piecewise by cubic polynomials. Do you know an efficient algorithm for computing all its zeros?
Thank you.
2
votes
1answer
169 views
Spline fit with bounded derivations
How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds $m_d$,$...
1
vote
2answers
397 views
Which data structure should I use for hierarchical T-meshes and PHT-splines?
Recently, I'm working on something about polynomial splines over hierarchical T-meshes, which is basically a rectangular grid that allows T-junctions. I want to do some numerical experiments but I don'...
11
votes
0answers
566 views
Once differentiable, piecewise degree three polynomials on triangulated planar domains
Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...
2
votes
1answer
272 views
Cubic spline smoothing question
I came across this link when searching for an algorithm for spline smoothing. Though I understand basically what I have to do, I need further clarifications on the formula chosen for curvature ...
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vote
2answers
326 views
Finding 3 dimensional B-spline control points from given array of points from spline solution?
Wa are talking about Non-uniform rational B-spline. We have some simple 3 dimensional array like
{1,1,1}
{1,2,3}
{1,3,3}
{2,4,5}
{2,5,6}
{4,4,4}
Which are ...
2
votes
2answers
2k views
Interpolation splines of bounded curvature
Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Can we have a polynomial spline (e.g., natural cubic spline) passing through all these points, such that: (...
1
vote
2answers
2k views
Fourier series of B-spline
The Fourier series of a function (B-spline) is given by:
$$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$
But the B-spline has only finite support. How ...
2
votes
1answer
549 views
Cubic spline of a two-variable function
So, I am aware of how to (both iteratively and using a linear equation) compute the cubic spline of a one-variable function with $m$ control points. However, I am not sure how to do any type of spline ...
