Interpolation is the theory of constructing smooth functions, usually polynomials or trigonometric polynomials, whose graph passes through a number of given points in the plane. Splines and Bézier curves, piecewise linear or cubic interpolation, Lagrange and Hermite interpolation are example topics.

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Multilinear Interpolation

Suppose I have a multilinear map $\Gamma(u,v)$ satisfying \begin{align} \big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\ \big\| \Gamma(u,v)\big\|_{L^\infty} ...
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Interpolation polynomial smaller than its function?

Let $q$ be a real number such that $q>1$ and $f$ be an entire function on $\mathbb C$ such that $\overline{\lim}_{r\to+\infty}\limits\frac{\ln|f|_r}{\ln^2r}<\frac{1}{2\ln q}$, where ...
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Interpolation of a series of data points via Chebyshev approximation?

first of all: english is not my native language, so there might be differences between what I meant and what you understood. Sorry for that in advance. As a research project, I try to comprehend and ...
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Interpolation between $H^1$ and $H^1\cap L^1$

Suppose that $T:H^1(\mathbb{R}^3)\rightarrow\mathbb{R}$ is a linear bounded operator, with operator norm $M_2$. In particular, given $1\leq p\leq2$, there exist optimal constants $M_p\leq M_2$ such ...
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How to take partial derivative of spherical interpolation of quaternions?

Using the standard definition of quaternionic spherical linear interpolation (slerp): $$ Q(q_0,q_1,t) := q_0(q_0^{-1}q_1)^t, $$ how can I take each partial derivative? Actually, I'm confident how to ...
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Monotonicity per dimension of multivariate scattered data

For my thesis, I am working on interpolation using the RBF method (Radial Basis Functions). Before interpolating, I want some a priori insight into the data, for example check in which dimensions it ...
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generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto ...
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Branches of 3j symbols

Question Is there a quick way to identify the branches in a 3J symbol? Context I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients, $$ \begin{pmatrix} \ell_1 &\ell_2 &\ell_3\\ ...
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Natural neighbor interpolation

Recently I am interested in Natural neighbor interpolation, that is : Given a function $P(x)$ and some interpolation points $\{x_i,P(x_i)\}_{i=1}^N$, we have the interpolation function ...
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161 views

Sign of 3j symbol (in view of interpolation)

Question Is there a closed formula for the sign of a 3j symbol? Context I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients, $$ \begin{pmatrix} \ell_1 &\ell_2 &\ell_3\\ ...
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113 views

Reference for Bessel's interpolation formula

may you please give me a reference for a standard, easy-to-find textbook where I can find the full proof of Bessel's interpolation formula (see ...
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Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...
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61 views

Finding t vlaue in Bezier curve [closed]

According to this question, I'm looking for some method to find the t value in Quadratic bezier curve equation: $$ B(t)=P_0+t(1-t)P_1+t^2P_2 \space \space where \space 0 ≤ t ≤ 1 $$ In this ...
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Equivalence of Sobolev--Slobodeckii and interpolation space on boundaries

Let $s \in (0,1)$. Given a sufficiently smooth hypersurface $\Gamma$ in $\mathbb{R}^n$, one can define the Sobolev--Slobodeckii space with the norm $$|u|_{H^s(\Gamma)}^2 = \int_\Gamma |u|^2 + ...
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About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order) Can ...
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Interpolation between weighted $L^p$ spaces

Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded. Let $T$ be the ...
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193 views

Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$. Let $\epsilon\in[\frac{1}2,1)$. Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...
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171 views

Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, ...
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189 views

Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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112 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
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Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
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Estimating the kernel of Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for complex $z$

Let $f(z,a)$ be an analytic function on $C^+=\{\Re z>0\}$ for each fixed $a>0$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
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Polynomial threading through a monotone corridor

I have a need to find a polynomial of minimal degree that connects two points and stays within a given "corridor," by which I mean an $x$-monotone polygon. Here is an example:       ...
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Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous function $f$ such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is an interpolation polynomial ...
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Minimum degree rational function interpolation

Find a rational function $R(x)$ such that: $1)$ For $i\in\{1,\dots,g\}$, $x_{i}=x_{i-1}+g$ with $x_0=0$. $2)$ For $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$. $3)$ $R(x_g)=g+1$. ...
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390 views

Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...
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Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem: Is there a method to extend the notion ...
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Integer valued polynomial through some points with rational coordinates

I asked this question on MSE about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO. If we have a set of points $(x_i,y_i)$ ...
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What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even when restricted to ...
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Lagrange Interpolation and integer polynomials

Suppose that there is a polynomial $P$ with integer coefficients such that $P(x_i)=y_i$ for $i=1,\ldots,n$. Is it true that the result of Lagrange interpolation through the data $(x_i,y_i)$ is a ...
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Estimate infinity norm with Lp and W1p norm

Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have $$\|f\|_{L^\infty}\leq C\|f\|_{L^p}^{1-\frac{1}{p}}\|f\|_{W^{1,p}}^{\frac{1}{p}}?$$ My ...
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Greedy interpolation of functions

Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation: Set $r_0 = f$. for $k = 0,1,\ldots,$ Find the location of the global ...
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Do interpolation nodes have to be dense?

Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval. For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates ...
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Rational interpolation: Error bounds for coefficients

The following question was asked on MSE, but might be more suitable here. Assume there is a rational function $$ f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}} $$ of type $(m,n)$ with ...
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zeta(3) in Euler's Section 153

Jeffery Lagarias, in his recent article Euler's constant: Euler's work and modern developments in the AMS Bulletin, mentions that Euler obtained $\zeta(3)={{2\pi^3 b(3/2)}\over 3}$ for some "Bernoulli ...
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interpolation with derivative of rational fraction

Studying a problem in conformal geometry, I am facing to the following interpolation problem. Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that ...
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Multivariate analogue of Vandermonde determinant

Dear all, Consider the $(n+1)\times (n+1)$ matrix $A$ with indeterminates $X_i, Y_i$, $0\leq i\leq n$ such that the $(i,j)$-th entry is given by $X_i^jY_i^{n-j}$. The $i$-th row is ...
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Finding the formula for Bezier curve ratios (hull/point : point/baseline)

Given a cubic Bezier curve defined by points $p_1$, $p_2$, $p_3$, and $p_4$, a point $B$ on that curve at some $t$ value (where $0 \leq t \leq 1$), a point $A$ on the line $(p_2 - p_3)$ at distance ...
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Norm of inverse confluent Vandermonde matrix

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} ...
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How to interpolate in 3-D non-euclidean space?

Assume, one has a 3-D non-euclidean space of points $p_i = \left(x_i, y_i, z_i\right) \in \mathcal{R}^2 \times \mathcal{R}_{> 0}$ with the following "distance" function $d\left(p_1, p_2\right) = ...
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A special polynomial interpolation

Let $λ_1,\ldots,λ_m$ real numbers pairwise distinct and $μ_1,\ldots,μ_m$ real numbers all nonzero. We know from polynomial interpolation that for a given $r$ such that $1\leq r\leq m$, there exists ...
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Polynomial interpolation whose roots are real and simple

Let $\lambda_1,\ldots,\lambda_m$ real numbers pairwise distinct and $\mu_1,\ldots,\mu_m$ real numbers all nonzero. We know from the Lagrange polynomial interpolation that there exists an unique ...
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Interpolating a “manifold” between two points

Edit: I have reworded the question. This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional ...
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Relation between interpolation spaces and besov spaces

Consider the following two norms: The interpolation norm: 1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...
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Inequality of von Neumann for more than two contractions

Good morning, I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
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How to find all the zeros of a cubic spline ?

Let's say I have a cubic spline represented as a piecewise cubic polynomials? Do you know an efficient algorithm for computing all its zeros ? Thank you
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Interpolation of derivatives

If $U$ is an open interval of $\mathbb{R}$ and $f : U \to \mathbb{R}$ is an $L^2(U)$ function with second derivative $f'' \in L^2(U)$ (in the weak sense), is $f \in W^{1,1}(U)$? EDIT: Removed false ...
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Where can I find interpolation inequalities for derivatives of the following form?

Here they are: $$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$ and $$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...
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399 views

Polynomials with prescribed points to match prescribed bounds

Consider real polynomials on the interval $I=[-1,1]$. It is easy to see that the smallest degree for a non-negative polynomial with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g. $P(x) = ...
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One-sided version of the “best approximation polynomial” : Upper polynomial approximations

Let $X$ be a finite subset of $\mathbb R$ and let $f : X \to {\mathbb R}$. Suppose we want to approximate $f$ by a polynomial $g$ of fixed degree $d\geq 1$ with the additional condition $g\geq f$. Let ...