# Questions tagged [interpolation]

Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points. It is necessary because in science and engineering we often need to deal with discrete experimental data.

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### How to verify ihe Interpolation inequality for the weighted Bessel potential spaces?

I am trying to prove the following: Let $w$ be an admissible weight, $p_1,p_2\in[1,\infty)$, $\alpha_1,\alpha_2\in\mathbb{R}$, $\theta\in(0,1)$ and \begin{equation} \alpha=\theta\,\alpha_1+(1-\theta)\...
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### how to find out the matrix for parametric cubic spline for natural spline boundary condition.

Write Python programs to approximate the sine curve between 0 and 2 using curve fitting by (a) a cubic spline; For the case of the cubic spline, you may consider the use of “natural splines” which ...
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### Is there a driving noise such that it behaves ''Hölderly'' over a uniform partition?

It is well-known that in case of a linear parititon of $[0,1]$, $\{t_n\}_{n=1}^N = \{\frac{n}{N}\}_{n=1}^N$, we have $$\int_{t_n}^{t_{n+1}} dt = t_{n+1} - t_n = \frac{1}{N} \quad \forall n$$ But ...
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### Monotonic interpolation between 5 points

For a problem in my research, I found myself looking for an interpolant function between five points: $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $(x_4,y_4)$, and $(x_5,y_5)$. These points are ...
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### choosing between Multivariate, Univariate and Spline interpolation

I have dataset of points with coordinates and temperature measured at each point. I would like to interpolate the points to generate a continuous image. I have checked Scipy and I have seen that there ...
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### Approximate the gradient of a sample point [closed]

Suppose we are given a set of sample points in $\mathbb{R}^3$. I don't have any knowledge about the surface, but we may assume that it is smooth. I want to approximate the gradient of each sample ...
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### One step beyond cubic spline interpolation, a fourth-order problem?

I am trying to fit a polynomial through three points, where I also know the derivatives at the two endpoints. I don't need a truly general solution. My specific problem is constrained as follows: <...
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### how do i write down the matrix A and the right-hand side vector b corresponding to the system of equations

I need how to get to the answer for this question. I made this in a study group, but we didn't write down how we got to the answer and now i've been stuck on it for a long time. i cant seem to make ...
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### How to solve a Vandermonde-like linear system from an interpolation problem?

Let us consider the quadrature $Q_n(f)$ obtained by Lagrange interpolation to aproximate the integral $\int_{-1}^{1}f(x)dx$, using as nodes the $n+1$ roots of the Chebyshev's polynomial $T_{n+1}(x)$. ...
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### Best regression model for points that follow a sigmoidal pattern

I have the following list of points : I'm trying to find the best regression model to fit these points. The logistic regression is not close enough to the points : I guess I need something closer to ...
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### Cutoff 2D signed distance field

Summary: There are global parameters, $R$ - thickness of red area, $G$ - thickness of green area, $S=\frac{R}{R+G}$ - represents start signed distance of red color, $E=\frac{0.5}{R+G}$ - represents ...
1 vote
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### Fractional iteration of the exponential map z <- exp(lambda (z - w))

I want to use this map as a sort of chaotic oscillator for audio, where lambda and w are widgets you can control from something like a touch surface in real time. The map is from C to C, and lambda, z,...
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### Finding rational coefficients of a cubic polynomial that fits 4 data points that have been floored to an integer

I have 4 data points: (204, 5422892) (205, 5722486) (207, 6343357) (213, 8386502) I have information that these data points were generated with a cubic polynomial $y = ax ^ 3 + bx ^ 2 + cx + d$ with ...
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### Are B-spline basis functions also B-splines?

When you write a spline curve as a linear combination of b-spline basis functions, it's called a "b-spline". The basis functions are generated recursively by the deBoor-Cox algorithm, ...
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### Numerical Analysis - natural cubic spline and clamped cubin spline

a question from first exam period (A). True or false ( it is false, but I want to understand ). Given the following intersection points $x_0, x_1,...,x_n$ (interpolation nodes ) and the values of ...
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### What exactly is the basis Function for B-splines?

I am using splines for 1D model in a research project. I found a reference for how to write the basis function that I put into my code but can no longer find it. Most guides and references on splines ...
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### How to get enough data to draw an arc or a curve from 3 points?

Note: Although I want to accomplish this in java, I think the question is more suitable for this site since it is mostly mathematical. I am in the following scenario. I want to draw a curve and I have ...
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### How to show that $f \in L^p[0,1]$ for all $1 \leq p <2$ and $\lVert f \rVert_p$ uniformly bounded implies $f \in L^2[0,1]$?

I have searched for several posts and it seems to me that if $f \in L^p[0,1]$ for all $1 \leq p <2$ and $\lVert f \rVert_p$ is uniformly bounded with respect to $p$, then $f \in L^2[0,1]$. However,...
1 vote
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### interpolation between Bochner spaces involving $H^{-1}$

I am reading a paper in which they say that: Since $u_n$ is bounded in $L^2(0,T;H^2(\Omega))$ and due to the strong convergence of $u_n$ in $L^2(0,T;H^{-1}(\Omega))$ we conclude with an interpolation ...
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### Creating an (linear) interpolant of points on the surface of d-dimensional hyper-sphere.

Let's say we have a $d$-dimensional unit hypersphere. On the surface of the hypersphere, we have points that have a value of either $\{-1, 1\}$. I wish to create a function $\Phi$ that interpolates ...
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### Verify that the attractor of an IFS is the graph of an interpolation

I am going through exercises in Fractals Everywhere by Michael Barnsley. I have a confusion about Exercise 2.1 in Chapter 7: The function $f(x) = 1 + x$ is an interpolation function for the set of ...
I'm trying to reverse engineer an interpolation used for animation. It seems to use 5 coefficients of the taylor series of $f(x) =e^{x}$. ...