11
votes

Accepted

### 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.&...

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:

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 $\...

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 ...

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 ...

3
votes

Accepted

### 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) ...

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 ...

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 $(N-2) \times (N-2)$ ...

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 ...

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, ...

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 $y=\log x$, $dx = x dy = e^y dy$....

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 ...

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 ...

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 ...

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)}}-(...

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. ...

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 ...

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 ...

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 ...

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 ...

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}=\...

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 ...

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. ...

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 ...

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 ...

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'...

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 ...

1
vote

Accepted

### 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 ...

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