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Questions tagged [interpolation]

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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Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
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Interpolation theory

Consider the interpolation space $Z=(X,Y)_{\theta,p}$, in the case $Y\subseteq X$ do we have that the following norm: $x\longrightarrow\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\...
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An open problem in Sobolev spaces

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator $$ E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n) \quad \text{and} \quad E:W^{1,q}(\Omega)\to ...
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For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
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For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
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Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
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Asymptotic behavior of sum linked with Lagrange interpolation

I already asked this a few weeks ago with no answer, so let me formulate differently. In performing Lagrange interpolation with nodes 1/n, one encounters the sum $$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{...
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Nonparametric Bayesian estimation of several black-box functions of different variables from their noisy sums

In order to introduce my problem, let’s start with the nonparametric estimation of a single unknown/black-box function $f:{\Omega _f} \to \mathbb{R}$ of a discrete variable $x$ in a finite domain ${\...
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Defining Boundary Conditions for Spline Interpolation via the Euler–Maclaurin Formula

The Euler–Maclaurin formula states an interdependency between \begin{align} I\quad:=&\quad\int_m^nf(x) \, dx;\ m,n\in\mathbb{Z}\\[6pt] S\quad:=&\quad\sum_{k=m}^n f(k) \\[6pt] D\quad:=&\...
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Chebyshev interpolation [closed]

Let's define the n-th degree Chebyshev polynomials by $$ T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomial $P$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...
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On different norms of the interpolating operator

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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Wasserstein interpolation between two probability measures on a metric space

Question 1 Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
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Reference to L^1 error for piecewise linear interpolation of Lipschitz functions

Let $f:[0,1]\to\mathbb{R}$ be a Lipschitz function, and $\pi f$ be its piecewise linear interpolant on an equispaced grid with $n$ points. It should be true (if I am not making mistakes with the ...
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Univariate polynomial interpolation with restricted degrees

Let $D=\{d_1, d_2, \ldots, d_n\}$ be an integer set. I'd like to know if I can interpolate any collection of $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{n}, y_{n})$ by a polynomial whose degree ...
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Interpolation space between $L^1\cap L^2$ and $L^1$

In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
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Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis

First, let us fix some Notation: Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let \begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
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Optimal $L^2$ bounds of cubic spline interpolation

Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
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Radial Basis Function Centers

Given $n$ inputs $x_1,...x_n$ and $n$ responses $f(x_1),...,f(x_n)$, a commonly used approach of estimating $f(\cdot)$, is by using a positive definite radial basis function $\phi(\cdot)$: \begin{...
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How to prove the this sobolev-like inequality presented in the paper “sobolev inequalities in disguise”

click to see the picture of one related page from the paper this is the link of the whole paper What I cannot really clearly understands is the content bellow: the shortcut of the inequality If ...
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Proof Reference - Polynomial interpolation at quadrature points

If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the ...
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Solutions to a certain Birkhoff-interpolation problem

$\newcommand{\CC}{\mathbb{C}}$ Let for $n > 1$ and $m = n-1$ $$ p = x^n + a_1 x^{n-1} + \cdots + a_m x $$ be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$. The ...
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Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter

Any map $f \colon \mathbb{R} \to \mathbb{R}$ induces a "composition map" $$f^\circ\colon \mathbb{R} \times \mathbb{N} \to \mathbb{R},$$ where $$f^{\circ n}(x) = \underbrace{f \circ \dotsb \circ f}_{...
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Do splines preserve monotonicity?

Start with a monotone nonincreasing function and sample it at finite set of points $x_0, ..., x_n$, $x_i<x_{i+1}$ so that $f(x_i)<f(x_{i+1})$. If you approximate $f$ with a linear spline then ...
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Cubic interpolating Spline - Number of extremum points

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum point ...
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inverse interpolation

Given data points $(x_i,y_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y_i=f (x_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ ...
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Identities for powers of functions based on generalization of Lagrange interpolation

Lagrange polynomial can be used to obtain an identity: $$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$ which holds for any integer $n>0$, any real ...
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A Taylor formula for a Vandermonde-like determinant

Let $f_0,\ldots,f_N$ be smooth functions over an interval $I\subset{\mathbb R}$. Let $x_0,\ldots,x_N\in I$ be given, and form the Vandermonde-like determinant $$\Delta_N=\det((f_i(x_j)))_{0\le i,j\le ...
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Variational Proof for minimum curvature of cubic splines

Background: Give an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)$ is a piecewise cubic polynomial with continuous second derivative. One can also prove, roughly,...
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How to choose contour for rational approximation

Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\...
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“Natural” ways of interpolating unitary matrices

Given two unitary matrices $A$ and $B$, that are "near" each other in some sense (perhaps $\left\lVert A-B\right\rVert <\epsilon$ for some norm, what are some sensible ways to interpolate between ...
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Interpolation in $X^{s,b}$ spaces

I've recently interested in the $X^{s,b}$ method to prove the well-posed results for dispersive equations. Recall that $X^{s,b}$ is the closure of Schwartz space with the norm $$ \|u\|_{X^{s,b}_{\tau=...
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Analytic interpolation with $k$-th derivative small on $\mathbb{R}$

I am looking for a condition such that the following theorem would hold: Suppose given an $\epsilon > 0$ and a natural number $k$. Suppose further given $x_{1}, ..., x_{n}$, $y^{0}_{1}, ..., y^{0}...
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Interpolation in Sobolev spaces

Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that $$ \hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2. $$ Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
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Interpolation by rational functions reference

I have been hearing a lot about a theory of interpolation using rational function, parallel to that of polynomial interpolation. I'm looking for a book chapter, or even short lecture notes, that will ...
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Polynomial interpolants in quadrature points and L2 convergence spectral rate

We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|...
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Increasing sequence of closed subspaces of $L^2$ and error estimate of a product of orthogonal projections

We define an increasing sequence of closed subspaces \begin{align*} V_{0} \subset V_{1} \subset V_{\ell} \subset \dots \end{align*} of $L^2(I)$ where $I=(0,x_{max})$, and each $V_{\ell}$ is equipped ...
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How to optimally choose points for multivariable Hermite interpolation?

I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input. I would like to create interpolation polynomial for it. The computation of $f$ is ...
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What can be said about moments of probability distribution if it satisifies hemholtz equation?

From physical considerations I have observed, that probability density in region of interest satisfies $$ \Delta u(x) + \phi(x)u(x) = f(x), $$ where $\phi(x)$ and $f(x)$ are both given functions and $...
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Maximum of a B-spline

Given $p+2$ nondecreasing (and not all identical) knots $t_0, \ldots, t_{p+1}$ on the real line, consider the normalized B-spline of degree $p$ defined over these knots. We know that the B-spline is ...
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Special Properties of “Vibrant” Spline-Functions

While checking an idea about knot-placement for spline interpolation, I needed to find a way to calculate splines, that are strictly monotone between adjacent pairs of knots and for which every knot ...
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A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof. Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...
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Who was first to use reproducing kernels in order to try to solve interpolation problems?

I understand that Sarason generalized the interpolation problem by taking it into the operator theoretic setting via reproducing kernels, but whose idea was it to use reproducing kernels such as the ...
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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 $|f|_r=\sup_{...
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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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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 (1,x,x^2,\cdots,x^{n-1})...
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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\\ ...