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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10 views

Determine spline coefficients by linear interpolation between 2 other splines

I am looking for a way to calculate a spline function from 2 previously calculated spline functions by linear interpolation. I have these functions (590.33, 911.4, 1192.51 -> linear interpolation ...
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22 views

Prove that p (x) is the interpolation polynomial on the points ...

Prove that p (x) is the interpolation polynomial on the points $ \begin{bmatrix} 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 1\\ 0\\ \end{bmatrix}\begin{bmatrix} 2\\ 0\\ \end{...
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Deriving interpolation polynomials on regular grids

Given a fine grid $\mathbb{Z}_h = \{kh\,:\,k\in \mathbb{Z}\}$ and a coarse grid $h+\mathbb{Z}_{2h} = \{h+k2h\,:\,k\in\mathbb{Z}\}$ I want to interpolate a function $f:h+\mathbb{Z}_{2h} \rightarrow \...
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How are interpolation operators derived for multigrid

I am trying to construct transfer operators $I^H_h, \, I^h_H$ for multigrid where $H \ne 2h$. I have gone through Briggs' tutorial, Hemker's paper, Hackbush's book, Trottenberg's book, but the details ...
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What happens to the degree of the quotient polynomial if the divison is not clear?

Let's say that I have one polynomial $a(x)$ of degree $n$ with coefficients over $\mathbb{F}_p$, where $\mathbb{F}_p$ is a finite field of size $p$. Asuming that $(x - r) \mid a(x)$ (i.e., that $r \in ...
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sample a grid on a surface represented by mesh data

I have the 3D coordinates of some points that represent a surface. these points are originally from a mesh data, that is, each 3 points represent a triangular surface to represent the complete surface ...
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Interpolation problem (recursive interpolation)

Let $x_0<x_1<...<x_K$ be points on the real line. Let $P$ be the polynomial of degree $K$ such that $P(x_i)=(-1)^i$ for all $i$. Then there exists points $y_0<y_1<...<y_K$ such that ...
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1D Interpolating subdivision for lifting schemes

I am looking into wavelet lifting methods first introduced by Swelden, and explained in this paper: Build your own wavelets at home. In this paper (in chapter 2 specifically), they discuss ...
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39 views

Interpolation inequality of fourier transformation

Let $1<p<2$ and $\frac1p+\frac{1}{q}=1.$ Show that for any $r\in(p,q)$ and for any $u\in L^p(\mathbb{R})$, we have $$\int_\mathbb{R}\left||y|^\frac{1}{q}\widehat{u}(y)\right|^r\frac{dy}{|y|}\leq ...
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find min, max ratio to interpolate triangle points by two angles based on arc

Summary: We have parametric $j$ - mininal angle in radians, $k$ - maximal angle in radians and $r$ - arc radius, $d$ - arc diameter, $C$ - center Green $45\deg$ arc: $Ax=Cx+cos(j)*r$, $Ay=Cy+sin(j)*r$...
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Derivation and Integration in polynomial spaces

My german numerical-calculus-book gives an example of integration and derivation in polynomial-spaces. But I do not understand the approach. Question 14.2 We have a polynomial with degree n and ...
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Smooth function passing through a countable number of points

Given a sequence $(y_n)_n$ of real numbers, can we find a smooth ($C^1$, $C^2$ or even $C^\infty$) real function $\phi$ such that $\phi(2^n)=y_n$ for all $n\in\mathbb{N}$ ? It's clear if we want a ...
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Obtain z value from x and y input

I wonder if there are tools, that can generate an equation from my (x, y, z) values. this is a sample from the data that I am using: ...
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How does changing the period affect the coefficients of the fourier series?

0 In a textbook exercise (without answers unfortunately), we were asked to change the following inerpolant from 1-periodic to T-periodic. $q(t)=a_{0}+\sum_{j=1}^{n} a_{j} \cos (2 \pi j t)+b_{j} \sin (...
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Offset bell curves that sum to a fixed value

I'm in need of a well studied bell curve function $f(x)$ that satisfies the following equation: $\exists P,\!C \; \forall x \:\left(\sum_{i=-\infty}^{\infty}{f(x+i P)}\right)=C$ That is; an infinite ...
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34 views

Restrict f(x) to pass through (0, 0) and (1, 1)

I know there are ways to "restrict" a function, f(x), so that it is forced to go through certain points, I'm not really sure what it's called though. Simply multiplying it by x will make the ...
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Linear interpolation based on closeness to a value between a minimum and maximum

Say I have the minimum integer value 0 and the maximum value of 10. How would I return a percentage based on the closeness to a given number within that range? Eg. If I gave it the number 5 to test ...
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Show that the sum of Lagrange Polynomials $\sum_{i=0}^{n} L_{i}(t)=1 \quad \forall t \in R$ [duplicate]

I am reviewing a homework problem that is supposed to be really easy but I have trouble wrapping my head around it. For $j=0, \ldots, n \quad t_{j} \neq t_{i}$ if $ i \neq j $ we define the $n$ ...
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how many control points are needed if n+1 data points are interpolated by p degree(or p+1 order) B-Spline?

Exactly, I suppose that both the number of control points and knot points are decided by the number of data points to be interpolated and the degree of B-Spline basis. However, I found two different ...
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Find a basis for $S(1,0)h[a,b]=\{p∈C0[a,b],p|Ii∈P1(Ii)\}$

On the intervall $[a,b]$ let $a = x_o< x_1 < x_2 < ... < x_n = b$ be a decomposition. Consider the Vectorspace of the piecewise linear functions $$S_h^{(1,0)}[a,b] = \{p \in C^0[a,b], p|_{{...
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Solve for x given formula for cubic interpolation

Given formula for a linear interpolation, I can solve for $x$ as follows: \begin{align} y = a(1-x)+bx\\ y = a+x(b-a)\\ x = \frac{y-a}{b-a} \end{align} How do I solve for $x$ given formula for cubic ...
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Why does inverse quadratic interpolation converge quicker than inverse linear interpolation

I have used linear inverse interpolation and quadratic inverse interpolation to estimate the root of a function. I found that my linear interpolation procedure required 9 iterations to achieve ...
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Minimum error linear interpolation of arbitrary curve using fixed number of points

Problem A finite curve, $r$, defined by $y = f(x)$, is defined by the linear interpolation between the 2 closest points (immediately higher and immediately lower) of a set points $\{(x_0,y_0), (x_1,...
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Range $u_{\mathrm{max}}-u_{\mathrm{min}}$ of the solution to graph Poisson equation $Lu=b$

I found out an interesting phenomenon when trying to solve the linear equation $Lu=b$, and I don't know how to interpret such phenomenon. Goal: Fit $u(x, y)=\cos(x)$ for $x,y \in [0,\pi]$. We draw ...
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Help find any information about strange trigonometric interpolation

I need to find any information about this interpolation method. It is labeled "trigonometric interpolation" and is placed in the chapter "Hermite interpolation". My final goal is ...
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Error in first derivative of cubic spline interpolant

Let $f: [a,b] \rightarrow \mathbb{R}$ be a $C^{\infty}$ function, and let $a = x_0 < x_1 < \cdots < x_n = b$ be a partition of the interval $[a,b]$. Let $s(x)$ be a piecewise polynomial ...
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orthogonality of Lagrange basis with legend nodes

I came across the following statement and I don't know how to justify it. If $L_i$ is a Lagrange basis, and $x$ is a zero of Legendre polynomial, then $$ \int_{-1}^{1} L_i(x)L_j(x) dx = \delta_{ij}w_j$...
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Binary polynomial evaluation

Let $p(x) = a_0 + a_1x + a_2x^2 + \dots + a_{n-1}x^{n-1}$ be a polynomial in $\mathbb{Z}_p[x]$ with binary coefficients, i.e., such that $a_i \in \{0,1\}$ for all $i = 0,1,\dots,n-1$. I like to refer ...
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Estimate an upper bound for interpolation polynomial

Given a function f and an unique interpolation polynomial P, we can say that for every x there is a r so that $f(x)-P(x)=\frac{\omega(x)f^{n+1}(r)}{(n+1)!}$ where r is in the smallest interval $[x_0,.....
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How to interpolate (polynomial) a matrix with matrix multiplication?

I would like to know how to do polynomial interpolation on a matrix. I have a feeling that it involves the Vandermonde matrix, but that's about it. I don't want to use a computer algorithm to do it, ...
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how big should n be so that the error be less than a certain number - Lagrange Interpolation

I am having trouble understanding how to solve this question: if $f(x) = e^x$, then how big must $n$ be, so that $\vert p_n(x)-e^x \vert \leq 10^{-6}$ for all $x \in [-1,1]$ The interpolation points $...
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Prove that there is exactly one element for interpolation in a Chebyshev system

Consider a Chebyshev system $g_0,...,g_n \in C[a,b]$ and $(n+1)$ value pairs where $x_i\neq x_j$ for $i\neq j$ that are all in $[a,b]$. Prove that there is exactly one element $g \in span(g_0,...,g_n)$...
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Why is there no unique solution for this interpolation task?

Consider the space $span(g_0,g_1)$ with $g_0(x)=1,g_1(x)=x^2$. Look at an interpolation task for the following pair of values. $(x_0, y_0) = (−1/2, 1); (x_1, y_1) := (1, 2)$ Why is there not always an ...
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FUN with f̶̶l̶̶a̶̶g̶̶s̶ Newton Cotes Quadrature formula and Bernoulli polynomials of the second kind

I was told to phrase my question in a more exciting way when I asked it last time. The following is a preliminary consideration. If you don't need it, just scroll down to START HERE. Here we go then: ...
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Estimate error in integration with interpolation polynomial

let $f(x)=\ln\left(1+\frac{x}{2}\right)$ and let $P_6(x)$ be the 6th order interpolating polynomial. We are given points $x_0,...,x_6$, and need to estimate the error $\displaystyle\left|\int_\limits{...
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Alternatives to spherical linear interpolation?

I need to calculate a 3d vector between to other vectors. I want to avoid the "scaling" factor that comes with linear interpolation, so I am looking at https://en.wikipedia.org/wiki/Slerp . ...
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Is there a convex function satisfying certain growth and smothness conditions?

I would like find (if it exists) a function $F:\mathbb R^2\to\mathbb R$ such that the following conditions hold true: $F\in C^\infty(\mathbb R^2)$ $F$ is radial, that is $F(x,y)=f(\sqrt{x^2+y^2}\,)$ $...
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Is the wikipedia bilinear interpolation example wrong?

On the Wikipedia Bilinear interpolation page there is this numerical example. I am talking about the example with this image. There I see $I_{20,14.5}=\frac{15-14.5}{15-14} \cdot 91 + \frac{14.5-14}{...
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Properties of Finite Differences

I have seen this statement in different variants, but could not find a proof: If the $n$-th order differences of equally spaced data are non-zero and constant then the data can be represented by a ...
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B-Spline Basis Function Discrepancy in Definition

Introduction The standard recursive definition of the B-Spline basis, given a knot vector $U = u_0 , ... ,u_n$ is: $$B_{i,0}(u)= \begin{cases} 1, & u_i \le u< u_{i+1} \\ 0, & \text{else} \...
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Error for Quadrature (Interpolation Type)

An exercise asks me to prove this: Numerically integrating $$I=\int_a^b f(x)\ \text{d}x \ \ \text{or}\ \ \int_a^b \rho(x)f(x)\ \text{d}x$$ using any formula of interpolation type $$L_n = \sum_{k=0}^n ...
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How to deal with this quadratic program?

Consider the quadratic program Suppose the initial estimate $x(0)$ of the solution is $[0, 0]^T$. Write down the system of linear equations you would solve to get the next estimate of the solution. ...
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86 views

Defining a log spiral from three points

Let us have three points: $p_0 = (0,0)$, $p_1 = (a,0)$ and $p_2=(b,c)$. (We can assume that $b<a$.) I want to define a log spiral (in polar space) of the form $r=r_0e^{k\theta}$ from some $center=(...
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Can I join smoothly (and convexly) two pieces of function?

Let $$f(x)=(x^2-1)^2\,,\quad g(x)=x^2$$ for $x>0$. Their plots cross at two points, the largest is $x_0=\frac{1+\sqrt 5}{2}$. $g(x)$, in purple: $f(x)$, black dot: $x_0$" /> I would like to build a ...
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38 views

Linear extrapolation

I have a very basic question, I normally do linear interpolation between 2 endpoints $a$ and $b$, where both $a$ and $b$ $\in R^n , n>10$ as below: $ (1-t)*a + t * b $ , as $t$ moves from $0$ to $1$...
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30 views

Integral of $n$-th Bernoulli polynomial of the second kind

We have \begin{align} \int x(x−1)(x−2)...(x−n)\,dx=(n+1)!\cdot\psi_{n+2}(x), \end{align} where, $\psi_n(x)$ is the $n$-th Bernoulli polynomial of the second kind. We have \begin{align} I &= \...
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Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MO: https://mathoverflow.net/questions/406610/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. Consider ...
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35 views

Interpolation over more than 2 points

Suppose I have 2 points $A$ and $B$. Doing an interpolation $Lerp(A,B,t)$ is pretty easy: $$Lerp(A,B,t) = A + (B-A)t$$ My question is: given I have $n$ points $P_1, P_2, \dots, P_n$, how would I ...
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24 views

Is there a name for a set of linear functionals $f_i$ that is "sufficiently rich" to uniquely identify a polynomial from the values $f_i p$

I have found this statement in some old lecture notes on interpolation in my lab. Let $\mathcal{P}_{n}(I)$ be the vector space of polynomials over some open interval $I\subset\mathbb{R}$. Suppose some ...
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What is this function called? Interpolation or Mapping?

First to non math people I'm great at math but to anyone actually good I'm kinda bad. Second I code alot of geometric style animations lately and one common set of code I've been using I want to look ...

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