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11 votes
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True origin of the term "Spline"

The Oxford English Dictionary doesn't necessarily give the earliest uses of a word. But spline with the meaning "A long, narrow, and relatively thin piece or strip of wood, metal, etc.; a slat.&...
Brendan McKay's user avatar
10 votes
Accepted

Do splines preserve monotonicity?

I assume you mean nondecreasing, i.e., monotonically increasing. If so, the answer is "no." For example, interpolate (0,0), (1,10) and (2,11) with a natural cubic spline:
Dustin G. Mixon's user avatar
4 votes

Variational proof for minimum curvature of cubic splines

See pp.87~107 of Prenter, Paddy M. Splines and variational methods. Courier Corporation, 2008. especially p.100 where "uniqueness theorem" is proved and spline is defined as minimizer to $\...
Henry.L's user avatar
  • 8,011
4 votes

Maximum of a B-spline

Using the recursive derivative formula (see for example here): $$N'_{i,p}(t) = \frac{p}{t_{i+p}-t_i} N_{i,p-1}(t) - \frac{p}{t_{i+p+1}-t_{i+1}} N_{i+1,p-1}(t)$$ We get that the maximum is achieved ...
Iddo Hanniel's user avatar
3 votes

Best way to introduce B-splines?

The undergrad course I took which included B-splines spent a lot of time first on Bézier curves. You might not necessarily want to spend much time on them, but I think they can motivate a variant on ...
Peter Taylor's user avatar
  • 6,776
3 votes
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Optimal $L^2$ bounds of cubic spline interpolation

From the properties of integrals it is not hard to derive an upper bound on the $L^2$ norm from the upper bound on the $L^\infty$ norm. Since you have (from Hall & Meyer) a bound $\left| f(x)-s(x) ...
Iddo Hanniel's user avatar
3 votes

Do splines preserve monotonicity?

Not only is the answer "no", but for any number $N$ you can construct a monotone function and sample it such that the natural spline approximation will have $N$ extremum points. See the figure (and ...
Iddo Hanniel's user avatar
3 votes
Accepted

Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$

There is an analytical solution to the problem in the following sense: Given a number $N$, the optimal interpolation points $x_0=0, x_1, ..., x_{N-1}=1$ are the roots of an $(N-2) \times (N-2)$ ...
Iddo Hanniel's user avatar
3 votes

Cubic interpolating spline – number of extremum points

Can we use the number of local extremum points of $f$ to bound the number of local extremum points of $s$? No. See https://en.m.wikipedia.org/wiki/Monotone_cubic_interpolation for a counterexample ...
David Ketcheson's user avatar
3 votes

Cubic splines convergence?

A nice and classical reference for splines (at third-year undergraduate or graduate level) is Powell, M. J. D. Approximation theory and methods. Cambridge University Press, Cambridge-New York, ...
user111's user avatar
  • 3,854
3 votes

Integrating B-Spline composed with log

The $\int_a^b f(\log x) dx$ expression actually has a nice analytical form, which enables to evaluate it in an elegant manner. First, we perform a change of variables $y=\log x$, $dx = x dy = e^y dy$....
Iddo Hanniel's user avatar
2 votes

Finding 3 dimensional B-spline control points from given array of points from spline solution?

What the PO actually is asking for, is how obtain the control points (or equivalently the wireframe) of a NURBS surface (the PO says "plane" instead of surface) from points on the surface (the black ...
Manfred Weis's user avatar
  • 12.7k
2 votes

Relation between Cox-deBoor recursion and Convolution (b-spline basis)

The B-Spline basis functions as defined by the Cox-DeBoor formula cannot, in general, be constructed with convolution. The convolution construction, as I'll explain below, only works for the special ...
Iddo Hanniel's user avatar
2 votes

Splines with bounded first derivative?

This is an interesting question and as @user100927 has correctly commented, a necessary condition for this to be possible is: $$\int_{x_i}^{x_{i+1}}f(x)dx ≤ y_{i+1}−y_i$$ for all $i$. To prove this ...
Iddo Hanniel's user avatar
2 votes

Non-polynomial splines, a non-linear problem

The interpolant $$A\left(e^{\frac{a}{A}(x-x_i)}-1\right)+B\cdot\left(\cosh\left(\frac{a}{A}(x-x_i\right)-1\right)+y_i\ =\ (A+B)\mathbf{e^{\frac{a}{A}(x-x_i)}}+B\cdot \mathbf{e^{-\frac{a}{A}(x-x_i)}}-(...
Manfred Weis's user avatar
  • 12.7k
2 votes

Non-polynomial splines, a non-linear problem

This is not a full answer, but I guess it is more than a comment. One way to reduce the task is to apply the spirit of VARPRO to separate the linear coefficients and the non-linear parameters, i.e. ...
gmvh's user avatar
  • 2,788
2 votes

Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?

Well, after some more thinking I'm going to answer my own question. It was pretty much just a matter of linking all elements together. Here are my notations, in $\mathbb R_N[X]$. Note $\partial$ the ...
Adrien Wohrer's user avatar
2 votes

Thin-Plate-Spline understanding and solution

Not sure what should be the ultimate standard reference, but one book often referred to, Greenberg. Applications of Green's Functions in Science and Engineering 1971, 2015, in turn refers to even ...
rych's user avatar
  • 281
2 votes
Accepted

Cubic spline interpolation without a constant term

It depends on what level of smoothness you require at your knots. The question only says that the spline has to interpolate data (= each piece must match the given values at the two knots that it ...
Jukka Kohonen's user avatar
2 votes
Accepted

Integrating a B-Spline basis function with respect to the standard normal PDF

Since a B-spline is a piecewise polynomial function, the question is whether there exists an exact equality formula for the integral $\int_{-a}^{b}u^pe^{-u^2/2}du$. This integral equals an elementary ...
Carlo Beenakker's user avatar
2 votes
Accepted

Marsden's Identity and B-splines

I found the solution after some research, hence I'll post it here in case anyone have curiosity: Marsden's Identity states that for all $\tau$ in $\mathbb{R}$ it holds that: $$(\cdot -\tau)^{k-1}=\...
Chaos's user avatar
  • 485
1 vote

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

After some more thought, I gave up Meinguet's approach above (based on sums of the form $\sum_j c_j\tau_{x_j}f$) for a more direct approach in order to prove continuity of the elements of $H$. Any ...
Adrien Wohrer's user avatar
1 vote
Accepted

Spline Interpolation error of higher degree

The following paper by de Boor suggests that this is the case, although he develops the proof only up to degree 6 splines. de Boor, C., On the convergence of odd-degree spline interpolation, J. ...
Iddo Hanniel's user avatar
1 vote

Bounds on the second derivative of a natural cubic spline in terms of the data

The first thing to notice is that the second derivative of a cubic spline is itself a degree-1 spline i.e., a piecewise linear function. Therefore, the maximal $|f''|$ value is attained at one of the ...
Iddo Hanniel's user avatar
1 vote

Polynomial-preserving boundary conditions for spline interpolation

There are no generic boundary conditions that guarantee that the interpolating spline reproduces a sampled polynomial. On the other hand, any $p-1$ conditions (where $p$ is the spline polynomial ...
Iddo Hanniel's user avatar
1 vote

Cubic interpolating spline – number of extremum points

The number of local extremum points of $s(t)$ can be bound as a function of $n$ (where $n+1$ is the number of interpolation points). The final bound, which I will develop using B-Splines, is $n-1$. I'...
Iddo Hanniel's user avatar
1 vote
Accepted

Smoothness Conditions for Planar "Mock-parametric" Spline Interpolation

The "Wilson-Fowler" spline has the properties that you describe. Each segment is defined by a cubic polynomial in a rotated coordinate system. These splines were quite common in the days before ...
bubba's user avatar
  • 649
1 vote
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General reparameterization of a B-spline

B-splines are a basis-function representation for piecwise polynomial functions. Therefore, if the reparameterization you seek cannot be represented as a piecewise polynomial it cannot, in general, be ...
Iddo Hanniel's user avatar

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