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.
52
questions
1
vote
1answer
103 views
Integrating a B-Spline basis function with respect to the standard normal PDF
I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:
$$
\int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,
$$
where $B_i^k$ is a ...
0
votes
0answers
25 views
Polynomial Lagrange splines
Question:
if $x_0 \lt x_1\lt\cdots\lt x_{N-1}\lt x_N$, how can the spline-analogues $\mathscr{L}_i(x)$ of Lagrange polynomials be calculated when they are defined via
$\mathscr{L}_i^n(x)\in C^{n-1}$
$...
1
vote
0answers
29 views
Polynomial Splines vs Whittacker-Shannon Interpolation for uniform data
Question:
(why) are polynomial splines preferred over Whittacker-Shannon interpolation?
Is it for genuine mathematical reasons like numeric stability and/or precision, or for other reasons?
Having ...
2
votes
0answers
25 views
Calculating non-polynomial spline functions
Question:
what is known about the algorithmic construction of general interpolating spline functions with smoothness constraints at every knot?
So far I could only find descriptions for splining ...
5
votes
1answer
175 views
Best way to introduce B-splines?
I have the option of mentoring pure math undergrads in a topic lying within Approximation Theory and I really want to do $B$-splines. Mostly because I have recently found applications of them in my ...
1
vote
0answers
53 views
Sensitivity of splines to control points
Let $X\in\mathbb{R}^{n\times d}$ be a set of $n$ points in $\mathbb{R}^d$, and let $f(X)$ be the operator that returns some spline interpolation of these points (say, cubic interpolation or Bezier ...
0
votes
0answers
41 views
Cubic spline interpolation – slope approximation using adjacent points
I am referencing a paper by CJC Kruger entitled "Constrained Cubic Spline Interpolation for Chemical Engineering Applications." In the paper he uses a the following formula to calculate ...
3
votes
1answer
136 views
Cubic spline interpolation without a constant term
Two main questions:
I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
2
votes
1answer
63 views
Spline Interpolation error of higher degree
It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.
Can I assume that, if one uses polynomials of degree $p$ and ...
0
votes
0answers
79 views
Can we improve the error bounds for spline interpolation if the interpolated function is smooth?
Let me first state the original problem I want to solve:
Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
1
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0answers
59 views
Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function
Is there any proof for this. Kindly request a reference in case available or any related documents towards this.
PS : I am specifically interested in the case of scattered data (irregularly placed), ...
2
votes
1answer
232 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 ...
3
votes
0answers
108 views
Hardness results for approximating Hölder continuous functions
Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems (Daubechies & Lagarias, SIAM J. Math. Anal. 22 (1991) ...
0
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1answer
56 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 ...
4
votes
1answer
228 views
Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
3
votes
2answers
192 views
Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$
I need to approximate $x \ln x$ on $[0,1]$ as a piecewise-linear function. If $P(x)$ is a piecewise-linear approximation, I want to minimize
$$
\max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P....
2
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0answers
64 views
Error bounds for spline interpolation. Hall and Meyer's conjecture
Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
3
votes
0answers
145 views
The $L_\infty$ norm of the derivative of the $L_2$ spline projector
A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
1
vote
1answer
194 views
Polynomial-preserving boundary conditions for spline interpolation
Spline interpolation requires the definition of boundary conditions because the smoothness requirements do not yield enough conditions for a unique solution.
Question:
which kind of boundary ...
4
votes
1answer
410 views
Marsden's Identity and B-splines
Marsden's Identity states that for every $\tau$ in $\mathbb{R }$:
$$(\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 ...
1
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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
1answer
151 views
Optimal $L^2$ bounds of cubic spline interpolation
Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
2
votes
2answers
177 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 ...
2
votes
2answers
181 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 ...
1
vote
1answer
119 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
74 views
Smoothness Conditions for Planar "Mock-parametric" Spline Interpolation
By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-...
1
vote
1answer
656 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,...
6
votes
1answer
391 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 ...
2
votes
1answer
728 views
Relation between Cox-deBoor recursion and Convolution (b-spline basis)
Consider the Cox-deBoor recursion formula for producing b-spline basis functions given a knot vector:
$N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$
otherwise, $=0$
$N_{i,p}(u)=\frac{u-u_{i}}{u_{...
1
vote
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
494 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 ...
4
votes
1answer
918 views
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, ...
1
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1answer
251 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 ...
4
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0answers
498 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 ...
1
vote
1answer
205 views
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).$$
Do you know of a process of finding ...
1
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0answers
93 views
Global approximation via convex combination of local approximations
I recently faced the problem of efficiently approximating a very large set of data points and, neither having a model of the empiric function, nor of the error distribution, my method of choice would ...
7
votes
1answer
440 views
Norms of B-spline coefficients
In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...
1
vote
1answer
172 views
Integrating B-Spline composed with log
If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the definite integral
$$\int_a^b f (\log x) \,\mathrm{d}x\,?$$
1
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0answers
495 views
How to find all the zeros of a cubic spline?
Let's say I have a cubic spline represented piecewise by cubic polynomials. Do you know an efficient algorithm for computing all its zeros?
Thank you.
2
votes
1answer
170 views
Spline fit with bounded derivations
How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds $m_d$,$...
15
votes
1answer
2k views
Interpolating between piecewise linear functions, with a family of smooth functions
Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that:
there is a number $k\in\mathbb N-\{0\}$ ...
1
vote
2answers
404 views
Which data structure should I use for hierarchical T-meshes and PHT-splines?
Recently, I'm working on something about polynomial splines over hierarchical T-meshes, which is basically a rectangular grid that allows T-junctions. I want to do some numerical experiments but I don'...
11
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0answers
570 views
Once differentiable, piecewise degree three polynomials on triangulated planar domains
Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...
2
votes
0answers
898 views
Comparison of cubic Hermite finite element and cubic B-spline finite element (regarding condition number of stiffness matrix)
Background:
Consider the one-dimensional second-order elliptic PDE,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
\...
2
votes
1answer
280 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 ...
6
votes
3answers
461 views
Approximating derivatives between gridpoints
Suppose we have a grid (possibly irregular) of $N$ function/value pairs, $(x_i, f_i)$, $i=1...N$. The function is differentiable everywhere at least twice (perhaps more).
What would be a good way to ...
1
vote
2answers
332 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 ...
3
votes
3answers
7k views
Finding the length of a cubic B-spline
Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are chorded approximations and the opinion that the analytic solution is "unbearably ...
