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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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Holomorphic interpolation of a square summable sequence

My question is as follows. For some $0 < R < 1$, consider the set $X = \{Re(z) > 0 \} \setminus D(0,R)$. Given at each integer $n \geq 1$ a value $a_n \in \mathbb{R}$, such that $(a_n)_n \in ...
user1274777's user avatar
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How to explain Runge Phenomenon for complex valued function? [closed]

I'm asked to explain Runge Phenomenon for the function $f(z)=\frac{1}{\alpha-z}$, where $\alpha$ is a constant complex number, i.e. I've to show that interpolating the above function at equally spaced ...
Deba4521's user avatar
-3 votes
0 answers
40 views

Is it ever useful in solving an equation to add the integer interpolation of a real number as a matrix product? [closed]

It is always possible to add a real number to one side of an equation without invalidating the inequality, as long as one adds an expression equal to that number to the other side of the equation. ...
virtuolie's user avatar
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How to understand 'the problem of determining the exact number of monomials in P(x) given by a black box is #P-Complete'

In this paper:https://dl.acm.org/doi/pdf/10.1145/62212.62241, what's given is $P(x)$ a (sparse) multivariate polynomial with real(or complex) coefficients. The author claimed two things. It is known ...
Youzhe Heng's user avatar
1 vote
2 answers
65 views

Why is interpolating $y=g(x)$ then applying $h(y)$ not equivalent to interpolating $h(g(x))$?

Say I have a table of voltage, current, and resistance values as so. V [V] I [A] R [$\Omega$] 1 2 0.5 3 5 0.6 The V and I columns are measurements, R is a simple calculation from Ohm's law (V=IR). ...
jrecord's user avatar
  • 13
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2 answers
76 views

Construct/prove existence of a function with given expansions at two different points

Consider two non-constant real polynomials $f(x)$ and $g(x)$: $$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N $$ $$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M $$ where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ ...
Quillo's user avatar
  • 2,101
2 votes
1 answer
65 views

Restrictions on Polynomial Interpolation Between Derivatives

If you have a a list if length $n$ of ordered triples of the form $(x, y, k)$ what are the restrictions on $x$, $y$, and $k$ such that there is exactly one polynomial $f(x)$ of degree $n - 1$ such ...
flakpm's user avatar
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1 answer
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How to change an iterative linear interpolation into a formula?

I'm programming a game which needs to perform a simple linear interpolation in an iterative manner on every frame the game draws the screen. See the below example: ...
user19179144's user avatar
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Difference between formula DFT (Discrete Fourier Transform) and trigonometric interpolation

Assume we have a discrete-real-valued scalar function $f[n]$, which is sampled equally spaced with distance $\Delta n$ using $M=2m-1$ samples. $f[n]$ is also periodic with respect to $2\pi$. My goal ...
Bastian's user avatar
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A function of specific form passing through two given points

Let $$s(t; a_0)=a_{0}t^{2}\left(\frac{1}{2}-\frac{t}{3T(a_0)}\right)$$ with $T(a_0)=\sqrt{\frac{6d}{a_{0}}}$ (where $d$ is some positive real constant). Then, let $$ s^*(t; a_0, t_w) = s\left(\frac{t-...
Airat Valiullin's user avatar
0 votes
1 answer
27 views

How can a polynomial function of degree k equal to it's interpolation of degree n (when n is greater than k) [closed]

Let $f∈Π_k$ (the set of all the polynomials of the degree of $k$ at last) and $x_0,x_1,…,x_n$ be the node points such that $k≤n$. Show that $$P_n (x)=f(x)$$ Where $P_n$ is the interpolation function ...
Overdeliver99's user avatar
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1 answer
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Oscillations of Lagrange interpolation polynomials

Let $I = [a,b]$ be a real closed interval. Let $n$ be a positive integer and let $x_i = a+i\frac{(b-a)}{n}$ for $i=0,...,n$. Let $p_j(x)$ be the Lagrange interpolating polynomial of the $n+1$ points ...
Alberto's user avatar
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Three dimensional spherical interpolation like trilinear interpolation

Say we have a 5x5x5 grid where a quaternion, q, exists at every point. The objective is to express the distribution of quaternions in a functional form in terms of the positions, i.e. q(x,y,z) = Ax + ...
Jesse Feng's user avatar
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How to optimize the number of data points used to interpolate a fit over an unknown polynomial (with noisy data)?

I'm trying to find the coefficients of an unknown polynomial given that I can choose to extract a coordinate at any point but at a cost. The cost goes up as more points are chosen. When a coordinate ...
Zinn's user avatar
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1 vote
1 answer
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For a bounded domain $\Omega \subset \mathbb{R}^3$, does the Besov space $B^{1/2,6/5}_{4/3}(\Omega)$ belong to $L^2$?

According to the formula, we have the real interpolation \begin{equation} (L^{6/5}(\Omega), W^{2,6/5}\Omega))_{1/4,4/3} = B^{1/2,6/5}_{4/3}(\Omega). \end{equation} for any bounded domain $\Omega \...
Keith's user avatar
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how to use a formula express this ? [closed]

Given two fixed points, the line connecting the two points is initially a straight line. Given a parameter t $\in$[0,1], when gradually changes from 0 to 1, the ...
Tom's user avatar
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Resources for learning about the theory behind interpolation

I would love to learn more about interpolation. However, most resources I have found just show the methods and do not give any theoretical justification. I am looking for a textbook/lecture notes on ...
user2316602's user avatar
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Do there exist any non-trivial "memoryless" interpolation functions

We shall define a memoryless interpolation function as a function $f:\Bbb{R}^2\times[0,1]\rightarrow\Bbb{R}$ which satisfies the following $3$ properties, $\begin{array}{lc} 1)& f(x,y,0)=x\\[2ex] ...
Fishbane's user avatar
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Optimal knot placement for approximating this function with B-spline.

I have the following data points, which all lie along a smooth, unknown, function (it looks sigmoidal but might not be): I want to approximate it rather accurately with B-splines, either quadratic or ...
Attack68's user avatar
  • 296
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1 answer
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Calculating an Inverse Matrix of a Matrix with variables

I am trying to understand a part of an article regarding quaternions spline interpolation, where the situation folded into the equation: $$ (\vec{a}\cdot\hat{e})\hat{e}+\frac{\sin\Delta\theta}{\Delta\...
BlueRevel 's user avatar
1 vote
1 answer
53 views

Lagrange interpolation and orthogonal polynomials

Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means, $$ \int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j} $$ wherein $p_i(x)$ for all $i$ is a ...
schneiderlog's user avatar
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0 answers
16 views

Is there an H^1 conforming interpolation with boundary condition with estimate?

Recently I have been researching Finite element analysis and I need an interpolation that satisfies (interpolation space $X_h\subset P^k$) boundary interpolation $(u,v_h)_{\partial\Omega}=(\Pi u,v_h)...
GeneLIU's user avatar
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0 answers
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Lagrange interpolation on a Galois Field in time $O(n \log (n))$

I have the values of a polynomial $p(x)$ defined on the Galois field mod $p$ (with $p$ prime) at the points zero to around two million. I need an algorithm to find the coefficients of the original ...
Cnoob's user avatar
  • 11
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0 answers
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How to control a circular interpolation using a reference point?

...
YGranja's user avatar
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1 answer
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Maximum degree of polynomial interpolating a two-degree polynomial with an additional condition

Let $f$ be a twice differentiable function over real line. Let $a\in \mathbb{R}$ and $h\in \mathbb{R}^{+}$. Let $P_f$ denote the interpolating polynomial of degree $2$ of $f$, i.e., $$f(a)=P_f(a), \...
PAMG's user avatar
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1 vote
2 answers
57 views

A pointwise Hadamard-Landau-Kolmogorov inequality for $C^2$ functions

A well-known interpolation inequality proved by Hadamard (1914) and was generalized by Landau (1913) and Kolmogorov (1939) asserts that $$ |f'(x)|^2 \leq 2 \sup_{x \in \mathbf R} |f(x)|\sup_{x \in \...
QA Ngô's user avatar
  • 496
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0 answers
37 views

Reducing the number of natural cubic spline interpolation points

Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is ...
Donatas Šimeliūnas's user avatar
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Interpolant from Dopri5 schema

Im trying to understand the linked paper (cf paper), please correct me if my understanding is wrong ! Im looking to get an interpolation function from the result of a Dormand Prince (RK45) integration ...
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0 answers
16 views

Trouble Understanding Tetrahedral Interpolation

(This is more specifically referring to 3D LUT creations, incase there are other uses of tetrahedral interpolation, which I'm sure there are) I've been reading about spline interpolation which ...
vannira's user avatar
  • 101
0 votes
1 answer
48 views

Why is the middle segment of a 4 points cubic spline not matching a 100 points cubic spline?

Let's say I have x0, x1, ..., x99 and y0, y1, ..., y99 ...
Jeffrey Chen's user avatar
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0 answers
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Exercise 8.15 Brezis - Interpolation inequality

I have a problem with this exercise (see the text in the following link). Interpolation like inequality ,Question from Brezis' book exercise 8.15 The link practically solves it. Only one last step ...
Seurat's user avatar
  • 63
0 votes
2 answers
98 views

Finding period of pendulum through interpolation

I’m looking for finding an efficient answer to this problem, which is to find the period time of a pendulum using interpolation. The pendulum behavior was given using the equations $\phi’’+\frac g L \...
albin's user avatar
  • 3
0 votes
0 answers
28 views

Interpolating polynomial for characteristic function of primitive Nth roots of unity among all Nth roots

In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose ...
user43208's user avatar
  • 8,659
3 votes
0 answers
73 views

How many points can be fit by a single sine function?

Consider a sine function parameterized by amplitude, frequency, phase, and constant offset. In other words, $h(x) = a\sin(fx + p) + c\tag*{}$ where $a$, $f$, $p$, and $o$ are arbitrary real numbers. ...
Ted Hopp's user avatar
  • 635
1 vote
2 answers
117 views

Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
Juan's user avatar
  • 33
0 votes
0 answers
15 views

Mapping Decreasing Volumes to Increasing Pixels in a Chart Application using Linear Interpolation

As a developer building a chart application, I'm using linear interpolation to map time values to pixel positions on the screen. My current approach utilizes the following function: ...
MohammadBaqer's user avatar
0 votes
1 answer
38 views

Lagrange Interpolation as generalized polynom

I need to write an algorithm that constructs a function of the form $f(x) = \sum_i^n q_i x^i$ that exactly goes through the points $p_i = (a_i, b_i)$. In general building such a function is not the ...
Sprinklerkopf's user avatar
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0 answers
15 views

Discrete to Continuous Sinc Filter.

I'd like to know, how to implement a curve interpolator using only past and current data samples. The goal is to get a smooth curve out of discrete samples that're fed in real-time. To this end, I ...
DannyNiu's user avatar
  • 211
1 vote
1 answer
48 views

Uncertainty Principle in Kernel-based Interpolation

If one wants to interpolate or reconstruct a function $f:\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$ on a finite Set $X_n:=\{x_1,\ldots,x_n\}\subset\Omega$ using translates of a ...
Max Stuthmann's user avatar
0 votes
1 answer
58 views

Interpolation spaces

Hy everybody I am curious about the definition; A compatible couple $(X_0, X_1)$ of Banach spaces consists of two Banach spaces $X_0$ and $X_1$ that are continuously embedded in the same Hausdorff ...
weymar andres's user avatar
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0 answers
23 views

find derivatives (hermite interpolation) via optimization: ensure uniqueness of optimization problem

Given a set of interpolation points $\boldsymbol x = \{x_1,...,x_n\}$ and interpolation values $y = \{y_1,...,y_n\}$. On interval $[x_i, x_{i+1}]$, we can write a C1 cubic hermite polynomial $$ p_i(...
Simon's user avatar
  • 185
1 vote
1 answer
53 views

Singular Matrix with Closed B-spline Interpolation when Degree and Number of Data Points are Both Even

I have written an algorithm to perform closed B-spline interpolation on a set of $N$ data points for a given degree $p$. I first generate a cyclic, uniform knot vector, and also use uniform ...
Gary Allen's user avatar
1 vote
1 answer
69 views

Finite differences of function $f(n)=1+2+3+\ldots+n$.

It is easy to compute that $\Delta f(n)=f(n+1)-f(n)=n+1,\;\Delta^2 f(n)=\Delta f(n+1)-\Delta f(n)=(n+2)-(n+1)=1$ and $\Delta^k f(n)=0$ for $k\geq 3$. Can we conclude that $f$ is polynomial of degree $...
Mark's user avatar
  • 354
0 votes
0 answers
53 views

cubic spline interpolation - spline interval

The k function values are to be interpolated by a piecewise cubic spline. The k-1 cubic polynomials are defined on the intervals [xi,xi+1],i ∈ {1,...,k-1}. Indicate which of the following statements ...
Adelhard's user avatar
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0 answers
23 views

Proving $\Phi_u(p)=|\Omega|^{-1 / p}\|u\|_{L^p(\Omega)}$ is Non-Decreasing for $u \in L^q(\Omega)$

Question: Let $\Omega \subset \mathbb{R}^d$ be measurable with $|\Omega|<\infty$, and let $u: \Omega \rightarrow \mathbb{R}$ be measurable. Define $$ \Phi_u:[1, \infty) \rightarrow[0, \infty], \...
CanDoMajoringMath's user avatar
0 votes
1 answer
173 views

Does Lagrange interpolation at Chebyshev points solve the Runge phenomenon?

I recently came across the concept of the Runge phenomenon while studying numerical methods for special functions in the book "Numerical Methods for Special Functions" by Amparo Gil, ...
Swakshar Deb's user avatar
1 vote
0 answers
66 views

Divided differences of polynomials

I am interested in computing divided differences of polynomials in a numerically stable way. Therefore I want to prove/disprove the following formula: $$\left(x^{n+m}\right)[x_1,\dots,x_n] = \sum_{1\...
Rasmus's user avatar
  • 514
15 votes
0 answers
272 views

Recovering a binary function on a lattice by studying its sum along closed paths

I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While I do not known $f$ explicitly, I have a "device" located at the origin $(1,1)$ which can do the following: Given an even ...
GSofer's user avatar
  • 4,333
1 vote
0 answers
41 views

How is this stability bound for the unique interpolant possible? $|s^*(x)|^2\le K(x,x)\|f\|_K\text{cond}_2(G_S)$

If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k....
Max Stuthmann's user avatar
0 votes
1 answer
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Prove $\max_{|x|\leq \frac13}|f(x)-p_2(x)|\leq \frac{1}{2\cdot 3^{\frac{11}{2}}}\,\max_{|x|\leq \frac13}|f^{(4)}(x)|$ for Runge's function.

Let $f(x)=\frac{1}{1+x^2}$ be Runge's function. If $p_2(x)$ is the interpolating polynomial of $f$ regarding the nodes $-\frac13,\,0,\frac13$, prove that: $$\max_{|x|\leqslant \frac13}|f(x)-p_2(x)|\...
Nikolaos Skout's user avatar

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