# Tag Info

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.&...
• 37.3k
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:
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• 701

### 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 ...
• 701
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 $(N-2) \times (N-2)$ ...
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### 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 ...
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### 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, ...
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### 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 $y=\log x$, $dx = x dy = e^y dy$....
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### 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 ...
• 12.7k

### 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 ...
• 701

### 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 ...
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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. ...
• 701
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 ...
• 701
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 ...
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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'...
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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 ...
• 649
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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