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.

Filter by
Sorted by
Tagged with
9 votes
1 answer
853 views

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 ...
  • 11.7k
3 votes
1 answer
124 views

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{...
0 votes
0 answers
45 views

Is this spline with increasing polynomial degrees already known

Question: is the following method of interpolating a sequence of points $(x_0,y_0),\,\dots,\,(x_n,y_n),\ 0\lt x_{k+1}-x_k$ with a spline $S(x)$ defined as follows already known: $x_+^n := \begin{cases}...
  • 11.7k
1 vote
1 answer
647 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 ...
  • 13
1 vote
0 answers
54 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 ...
  • 11.7k
2 votes
0 answers
32 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 ...
  • 11.7k
5 votes
1 answer
249 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 ...
  • 151
1 vote
0 answers
58 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 ...
  • 123
0 votes
0 answers
84 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 ...
user avatar
3 votes
1 answer
195 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 $f(t) = bt + ct^2 + dt^3$, ...
user avatar
2 votes
1 answer
105 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
0 answers
154 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 $a=t_0<t_1<\cdots<t_n=b$,...
  • 273
1 vote
0 answers
64 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), ...
  • 704
2 votes
1 answer
534 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
0 answers
112 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) ...
  • 201
0 votes
1 answer
76 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 ...
  • 103
6 votes
3 answers
349 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 ...
  • 2,507
3 votes
2 answers
353 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 votes
0 answers
98 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 ...
  • 151
3 votes
0 answers
178 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 ...
  • 151
1 vote
1 answer
240 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 ...
  • 11.7k
4 votes
1 answer
557 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 ...
  • 475
1 vote
0 answers
58 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:=&\...
  • 11.7k
2 votes
1 answer
191 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 & ...
  • 3,240
2 votes
2 answers
255 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 ...
  • 1,952
2 votes
2 answers
232 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 ...
  • 3,240
1 vote
1 answer
177 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 ...
  • 11
1 vote
1 answer
86 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-...
  • 11.7k
2 votes
1 answer
901 views

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 ...
  • 3,240
6 votes
1 answer
504 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 ...
  • 268
2 votes
1 answer
862 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
0 answers
33 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 ...
  • 11.7k
3 votes
2 answers
605 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
1 answer
997 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, ...
  • 45
1 vote
1 answer
257 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 votes
0 answers
621 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 ...
  • 268
1 vote
1 answer
273 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 ...
  • 175
1 vote
0 answers
94 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 ...
  • 11.7k
7 votes
1 answer
580 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 ...
  • 166
1 vote
1 answer
188 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 $$\int_a^b f (\log x) \,\mathrm{d}x\,?$$
  • 119
1 vote
0 answers
514 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
1 answer
178 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
1 answer
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
2 answers
405 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'...
  • 163
11 votes
0 answers
574 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
0 answers
905 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} \...
  • 163
2 votes
1 answer
306 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
3 answers
469 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 ...
  • 177
1 vote
2 answers
349 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 ...
  • 15
3 votes
2 answers
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: (...
  • 607