# 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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### 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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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 $\mathscr{L}_i^n(x)\in C^{n-1}$ $... 0answers 29 views ### Polynomial Splines vs Whittacker-Shannon Interpolation for uniform data Question: (why) are polynomial splines preferred over Whittacker-Shannon interpolation? Is it for genuine mathematical reasons like numeric stability and/or precision, or for other reasons? Having ... 0answers 25 views ### Calculating non-polynomial spline functions Question: what is known about the algorithmic construction of general interpolating spline functions with smoothness constraints at every knot? So far I could only find descriptions for splining ... 1answer 175 views ### Best way to introduce B-splines? I have the option of mentoring pure math undergrads in a topic lying within Approximation Theory and I really want to do$B$-splines. Mostly because I have recently found applications of them in my ... 0answers 53 views ### Sensitivity of splines to control points Let$X\in\mathbb{R}^{n\times d}$be a set of$n$points in$\mathbb{R}^d$, and let$f(X)$be the operator that returns some spline interpolation of these points (say, cubic interpolation or Bezier ... 0answers 41 views ### Cubic spline interpolation – slope approximation using adjacent points I am referencing a paper by CJC Kruger entitled "Constrained Cubic Spline Interpolation for Chemical Engineering Applications." In the paper he uses a the following formula to calculate ... 1answer 136 views ### Cubic spline interpolation without a constant term Two main questions: I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term$a$. That is, the polynomial takes the form$f(t) = bt + ct^2 + dt^3$, ... 1answer 63 views ### Spline Interpolation error of higher degree It is well-known that the interpolation error of a cubic spline has at best order$O(h^4)$, which results from polynomials of degree$3$. Can I assume that, if one uses polynomials of degree$p$and ... 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$a=t_0<t_1<\cdots<t_n=b$,... 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), ... 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 ... 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) ... 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 ... 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 ... 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 $$\max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P.... 0answers 64 views ### Error bounds for spline interpolation. Hall and Meyer's conjecture Hall & Meyer, 1976, J. Approx. Theory, show for f \in C^4[a,b] and a mesh a = x_1, \ldots x_n = b with h = \max x_{j+1} - x_j, for \pi f a cubic spline interpolant over the mesh for some ... 0answers 145 views ### The L_\infty norm of the derivative of the L_2 spline projector A. Shadrin (Acta Mathematica, 2001) shows that the L_\infty norm of the L_2 projector P_\Delta onto the spline space S_k(\Delta) is bounded independently of the knot-sequence. I.e. for a ... 1answer 194 views ### Polynomial-preserving boundary conditions for spline interpolation Spline interpolation requires the definition of boundary conditions because the smoothness requirements do not yield enough conditions for a unique solution. Question: which kind of boundary ... 1answer 410 views ### Marsden's Identity and B-splines Marsden's Identity states that for every \tau in \mathbb{R }:$$(\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 ... 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:=&\... 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 & ... 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 ... 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 ... 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 ... 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-... 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,... 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 ... 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_{... 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 ... 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 ... 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, ... 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 ... 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 ... 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).$$Do you know of a process of finding ... 0answers 93 views ### Global approximation via convex combination of local approximations I recently faced the problem of efficiently approximating a very large set of data points and, neither having a model of the empiric function, nor of the error distribution, my method of choice would ... 1answer 440 views ### Norms of B-spline coefficients In Shumaker's book (Spline Functions: Basic Theory), we know that the l^\infty-norm of B-spline coefficients is bounded above and below by the L^\infty-norm of the spline itself. Are there similar ... 1answer 172 views ### Integrating B-Spline composed with log If f is a real B-Spline and a, b are real numbers, then is there a numerically stable way to evaluate the definite integral$$\int_a^b f (\log x) \,\mathrm{d}x\,?$$0answers 495 views ### How to find all the zeros of a cubic spline? Let's say I have a cubic spline represented piecewise by cubic polynomials. Do you know an efficient algorithm for computing all its zeros? Thank you. 1answer 170 views ### Spline fit with bounded derivations How can I do a Spline Fit with bounds on some derivations? Problem Given: Set of data points t_k, x_k Set of nodes n_i order D of the spline (in my case D=5) lower and upper bounds m_d,... 1answer 2k views ### Interpolating between piecewise linear functions, with a family of smooth functions Let [a,b)\subset\mathbb R, and F,G:[a,b)\to\mathbb R two decreasing piecewise linear functions so that F(x)\leq G(x) for any x\in[a,b). We assume that: there is a number k\in\mathbb N-\{0\} ... 2answers 404 views ### Which data structure should I use for hierarchical T-meshes and PHT-splines? Recently, I'm working on something about polynomial splines over hierarchical T-meshes, which is basically a rectangular grid that allows T-junctions. I want to do some numerical experiments but I don'... 0answers 570 views ### Once differentiable, piecewise degree three polynomials on triangulated planar domains Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of ... 0answers 898 views ### Comparison of cubic Hermite finite element and cubic B-spline finite element (regarding condition number of stiffness matrix) Background: Consider the one-dimensional second-order elliptic PDE,$$ \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} \... 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 ... 3answers 461 views ### Approximating derivatives between gridpoints Suppose we have a grid (possibly irregular) ofN$function/value pairs,$(x_i, f_i)$,$i=1...N$. The function is differentiable everywhere at least twice (perhaps more). What would be a good way to ... 2answers 332 views ### Finding 3 dimensional B-spline control points from given array of points from spline solution? Wa are talking about Non-uniform rational B-spline. We have some simple 3 dimensional array like {1,1,1} {1,2,3} {1,3,3} {2,4,5} {2,5,6} {4,4,4} Which are ... 2answers 2k views ### Interpolation splines of bounded curvature Given$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: (...
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 ...