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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Would this be an example of interpolation?
I've never used interpolation in any way before, so I wanted to think up an example where it might apply.
Let's say I have the vertical advection of vertical lapse rates in temperature, $\frac{\...
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How to fill the temporal gaps in ESA(European Space Agency) CCI soil moisture data. [closed]
I downloaded the soil moisture data from ESA Soil Moisture Climate Change Initiative (Soil_Moisture_cci): Version 07.1 (Link:https://catalogue.ceda.ac.uk/uuid/ea3eb0714dc6402b905fe9f7ee50dbbc?jump=...
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Uniqueness of interpolation for distinct positive real numbers by non-negative coefficients $x_i$ and $\sum_{i=1}^n x_i =1$
Let $a_1,a_2,\dots,a_n>0$ be distinct positive real numbers and let $x_1,x_2,\dots,x_n \ge 0$ be non-negative real coefficients such that $\sum_{i=1}^n x_i = 1$.
Is it possible to find another set ...
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Interpolation, advantages and disadvantages
So during our numerics course we learned a few interpolation methods Aitken/Neville, divided differences,Lagrange and the Vandermonde matrix. How these work is clear to me for the most part, I'm just ...
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Choosing appropriate interpoleting points
Suppose $f$ is a polynomial of degree $d$ and $B\subset\mathbb{R}$ is an open interval. Then let $A=\{x\in B:|f(x)|<\epsilon\}$ then can we always choose $d+1$ points $x_i$ from the set $A$ i.e $|f(...
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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 ...
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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,...
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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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Exchange case in proving interpolation theorem by induction on the length of proof tree
I'm trying to prove Craig's interpolation theorem of propositional logic using Maehara's method by induction on the length of proof tree using sequent calculus. the theorem is as stated below:
$$ \...
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Robust polynomial interpolation with 1/x distribution?
I have a function $f(x), x\in[0,1],$ which I can sample at points $x = 1/k$ for integer $k$, $k<K$ where $K$ is some sufficiently large integer that can be chosen depending on the error. I want to ...
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Quadrature Formula and Interpolation
Suppose we want to integrate a function from $x_0$ to $x_0 + 3h$, and we know the value of this function at $x_0$, $x_1 = x_0 + h$, and $x_2 = x_0 + 2h$. Write a quadrature formula to approximate the ...
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How to smoothly remove boundary discontinuities in functions and their derivatives at the edges of their defined domain?
If I want to Fourier transform a function
$$t \in [-1,1] \to f $$
but this function and it's first $n$ differentials are not equal at the edges : $$f^{(k)}(-1) \neq f^{(k)}(1) , k \in \{0,1,\cdots,n\}$...
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Keeping the amplitude of a formula constant through a change of frequency
There is a function $f(x)=\frac{1}{\lfloor x \rfloor}\times x$ where the amplitude changes and goes to zero for large positive values, while the frequency remains the same for the appearance of the ...
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underdetermined polynomial interpolation for a function $f$
I want to find a cubic polynomial $p$ approximating a continuous function $f$ over $[a,b]$. A critical property of $f$ is that $f(a)=a$ and $f(b)=b$, so $p$ must also satisfy $p(a)=a=f(a)$ and $ p(b)=...
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Finding a function that connects 3 points on a graph in the shortest path.
I have $3$ points on a $2$d graph. Say, $(X_1, Y_1)$, $(X_2, Y_2)$, $(X_3, Y_3)$. I want to find a function which is sum of several modulus functions which can achieve this .
For example,
$(-1,1)$, $...
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A two dimensional interpolation problem given data along one dimension only
This is a question from a physics experiment I am currently working on and I would try to translate it to a mathematical form to explain certain problems/doubts I am facing and ask the community to ...
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Having interpolated $y=f(x)$ find a point with a given $y$ coordinate
I have a set of points $(x_1, y_1), (x_2, y_2), ... (x_n, y_n)$ and a function $y=f(x)$ where $f$ is an interpolation of the dataset. Given $y_t$ - target $y$ coordinate and $\epsilon$ - required ...
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Proof of the error bound of the Barnes interpolation method
Can someone help me with the proof of the Error bound of the Barnes interpolation method if the equation is given as follows:
D(a,k) = exp(-ak^2)
where a = wave number
This formula is found in the ...
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Show that $\exists\alpha>0$ so that $|f(x_i)- \frac{1}{h}\int _0^1 f(x)\Phi_i (x)\,dx |\le\alpha h^2$.
I am trying to prove the following statement.
Let $f\in C^2([0,1])$ and let $\Phi _i\ for\ i=1,…N-1$ be the family of hat functions. Show that there exists a $\alpha >0$ such that
\begin{equation}
\...
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Interpolation: cubic splines
Cubic splines
Given $n+1$ data points $\left(x_i, f_i\right)$ such that: $x_i<x_{i+1}$
We want a function $y(x)$ such that this function $f(x)$ interpolates continuously (up to and including the ...
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What is the name of this type of interpolation/approximation? Piecewise tangent lines' sections / Fabius function example?
What is the name of this type of interpolation/approximation? Piecewise tangent lines' sections / Fabius function example?
Let $f(x)$ be a differentiable function, and let $\{x_i\}$ be a set of ...
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Interpolation with constant rate of increase
Let's assume we have data for two points ( X_Left -> Y_Left, X_Right -> Y_Right ), and we break the interval between X_Left, and X_Right, into equaly distant sub intervals.
In case I wanted to ...
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Does Riesz-Thorin Theorem hold for Bochner spaces?
Following the book by Bennett and Sharpley, "Interpolation of Operators" (1988, Academic Press), I have found the following result, which is a version of the Riesz-Thorin Convexity Theorem ...
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Largest Step size in Quadratic Interpolation ( Numerical Interpolation )
Question: The function $f(x) =(1+x)^6$ is to be tabulated at equispaced points in the interval $[0.1]$ using quadratic interpolation. Find the largest step size that can be used used so that the ...
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Inclusion of Sobolev spaces in $L^p$
I want to show that for $s\in[0,n/2)$ and $p \in [2,2n/(n-2s))$ we have the following (continuous) inclusion
$$
H^s(\mathbb{R}^n) \subset L^p(\mathbb{R}^n).
$$
I have tried to find two interpolation ...
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Thin-Plate-Spline understanding and solution.
As I understand it a Thin-Plate-Spline in 2D is an interpolant function that minimizes
$$
\left\{
\begin{array}{ll}
\int_{\mathbb{R}^2} \left(f_{xx}^2 + 2f_{xy}^2 + f_{yy}^2\right)dxdy & \\
\text{...
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How do I "stabilize" the interpolator that uses the roots?
In a previous question, I have been instructed that, in order to pass through all and only the distinct points $(x_{1}, 0) ... (x_{k}, 0)$ with a non-constant continuous function, I can simply use ...
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Triangulations of square grids
If you have a square two-dimensional grid, that are lots of ways it could be triangulated -- for an individual square one could add an edge from upper-right to lower-left, or from lower-right to upper-...
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Approach for efficient interpolation of sum of polynomials?
I'm trying to find an approach for recovering the coefficients of a polynomial (the "sum polynomial") that is the sum of a bunch of polynomials. The application is to build a secure protocol ...
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Smooth interpolation between two sinusoidal functions, with constraints
I am trying to create a function $f(t)$ that smoothly interpolates between $\sin(tg)$ and $\sin(t(g+1))$, where $g$ is a small integer. Additionally, I want the function to satisfy the constraint $f(t)...
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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 ...
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What is this hermite-like spline that uses taylor series of exp?
I'm trying to reverse engineer an interpolation used for animation. It seems to use 5 coefficients of the taylor series of $f(x) =e^{x}$.
...
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Largest singular value in geometry (3D line fitting)
I have a question about 3D line fitting algorithm as described here:
https://www.codefull.net/2015/06/3d-line-fitting/
Please tell me, is the largest singular value somehow related to length of the ...
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