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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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 ...
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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 $\mathscr{L}_i^n(x)\in C^{n-1}$ $...
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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 ...
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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 ...
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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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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 ...
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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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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 $f(t) = bt + ct^2 + dt^3$, ...
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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 ...
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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 $a=t_0<t_1<\cdots<t_n=b$,...
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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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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 ...
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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) ...
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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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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 ...
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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....
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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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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 ...
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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 ...
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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 ...
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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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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 & ...
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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 ...
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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 ...
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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 ...
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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-...
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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,...
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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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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_{...
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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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 $$\int_a^b f (\log x) \,\mathrm{d}x\,?$$
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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.
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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 $m_d$,$...
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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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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 ...
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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} \...
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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 ...
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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 ...
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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 ...
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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: (...
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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 ...
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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 ...