# 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 ...

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### 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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### What's a good approximation for the first derivative at the endpoints of given datapoints for a cubic spline interpolation?

I'm using a cubic spline interpolation for given data points. The boundary condition for the spline is that $f'(a)$ and $f'(b)$ are given (I'm using a finite difference formula $\frac{y_1-y_0}{x_1-x_0}...

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### Curvature of randomly generated B-spline curve

I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...

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### True origin of the term "Spline"

In mathematical contexts the term spline essentially refers to interpolating or approximating piecewise functions with continuity constraints.
According to the history of mathematical splines
In the ...

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### Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?

Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $f(t) = bt + ct^2 + dt^3$, ...

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### 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 ...

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### 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 $a=t_0<t_1<\cdots<t_n=b$,...

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### 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), ...

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### 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 ...

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### 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) ...

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### 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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### 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 ...

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### 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....

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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:=&\...

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### 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 & ...

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### 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 ...

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### 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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### 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 ...

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### 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-...

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### Variational proof for minimum curvature of cubic splines

Background: Given an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)\in C^2([a,b])$ is a piecewise cubic polynomial on each subinterval $(x_i, x_{i+1})$.
Given a ...

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### 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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### 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_{...

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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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### 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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### 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.

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### 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 $m_d$,$...

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### 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 $k\in\mathbb N-\{0\}$ ...

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### 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'...

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### 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 ...

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### 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}
\...