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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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 (...
joro's user avatar
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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 ...
Piotr Hajlasz's user avatar
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Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
Tutukeainie's user avatar
6 votes
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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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What are the eigenvectors of the Lagrange interpolation matrix?

Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - ...
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Recovering a binary function on a lattice by studying its sum along closed walks

I recently posted this question on MSE. While it attracted interest, no answers were submitted, so I thought to try and post it here. I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While ...
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Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
Tony419's user avatar
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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\\ ...
coccoinomane's user avatar
4 votes
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293 views

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 ...
Max Alekseyev's user avatar
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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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Interpolation between Sobolev spaces

In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by $$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$ where $D^sf$ is defined by the Fourier transform $$(D^...
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Bound of a regular function that cancels at some points

Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
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Do higher-order splines with Lipschitz derivatives exist on finite sets?

Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$. If $n=m=1$ then it's easy to see that: $$ ...
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Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
Vincent Granville's user avatar
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Interpolating multivariate polynomials from their partial derivatives

Let $P(x_1,\dots,x_n)$ be a multivariate polynomial over a ground field $K$. For a multi-index $\alpha=(a_1,\dots,a_n)$ we denote the partial derivative $\frac{\partial^{a_1+\dots+a_n}P}{\partial x_1^{...
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Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform. ...
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Hardness results for approximating Hölder continuous functions

Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems (Daubechies & Lagarias, SIAM J. Math. Anal. 22 (1991) ...
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Solutions to a special confluent Vandermonde system

Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define $$ M^{(0)} = \begin{pmatrix} 1 &...
Hamed's user avatar
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The $L_\infty$ norm of the derivative of the $L_2$ spline projector

A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
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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 ...
Victor Liu's user avatar
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117 views

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 ...
user144209's user avatar
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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 ...
Evgeniy Lokharu's user avatar
2 votes
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K functional of $L^{1,\infty}$ and $L^\infty$ real interpolation

I want to know where can I find how to compute, given a function $f$ and $t>0$ $$K(t,f;L^{1,\infty};L^\infty)$$ where $L^{1,\infty}=\sup_{t>0}t f^*(t)$ and $K$ is the Petree K-functional ...
User 2234's user avatar
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Top coefficient of the Lagrange polynomial as average of (n-1)-st derivative

Is there a formula expressing the top coeffient of the Lagrange interpolation polynomial for a function as an average of its ($n-1$)-st derivative (divided by $(n-1)!$)? I am looking for a reference; ...
user420620's user avatar
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Calculating non-polynomial spline functions

Question: what is known about the algorithmic construction of general interpolating spline functions with smoothness constraints at every knot? So far I could only find descriptions for splining ...
Manfred Weis's user avatar
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How to evaluate an interpolation method, in terms of converging to the underlying function, as data points go to infinity?

I have an interpolation method, which takes function $f$ values at any given finite number $N$ of points in the domain and interpolate to get a function $f_{int}$. I want to do some analysis on how ...
Rajesh D's user avatar
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Error bounds for spline interpolation. Hall and Meyer's conjecture

Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
Manuel's user avatar
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0 answers
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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 ...
Moonwalker's user avatar
2 votes
0 answers
56 views

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_{...
joaopa's user avatar
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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 ...
alext87's user avatar
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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 ...
Eckhard's user avatar
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First-order interpolation inequalities with weights by L.Caffarelli, R.Kohn and L.Nirenberg

L. Caffarelli, R. Kohn and L. Nirenberg showed in this article that, under some conditions, the following weighted interpolation inequality is valid THEOREM: There exists a positive constant $C$ such ...
Ilovemath's user avatar
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1 vote
0 answers
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Regular covering of planar pointsets with convex polygons

Question: What is known about the problem of covering a finite set of $\mathbb{P}$ of points in the plane with convex polygons that have the same number $m$ of points from $\mathbb{P}$ as corners and ...
Manfred Weis's user avatar
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1 vote
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Error bounds for Sobolev space norm approximation on a finite grid

Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space, \begin{multline} f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
Drew Brady's user avatar
1 vote
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A nonstandard polynomial interpolation problem

I thought I know the properties of polynomials quite thoroughly. But... I am looking for a recipe to construct, for any given finite set $S$ whose elements are sets of $N$ real vectors of length $n$ ...
Arnold Neumaier's user avatar
1 vote
0 answers
200 views

Function uniquely determined by its values at integer arguments

A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
Vincent Granville's user avatar
1 vote
0 answers
131 views

Polynomial interpolation of binary vectors

Let $\mathbb{F}$ be a finite field and let $\boldsymbol{x} = (x_1, x_2, \dots, x_n)$ be $n$ pairwise distinct points in $\mathbb{F}$. Given the vector $\boldsymbol{y} = (y_1, y_2, \dots, y_n)$, with $...
Bean Guy's user avatar
1 vote
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P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
Ayman Moussa's user avatar
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Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. ...
ccriscitiello's user avatar
1 vote
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60 views

2d interpolation minimizing the integral of the norm of the Hessian

It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative: $$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...
Bernard 's user avatar
1 vote
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184 views

Uniformly local Sobolev spaces and interpolation

Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space $$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
Capublanca's user avatar
1 vote
0 answers
33 views

Mismatching degrees and # derivatives in polynomial interpolation error formula

It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
gTcV's user avatar
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Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function

Is there any proof for this. Kindly request a reference in case available or any related documents towards this. PS : I am specifically interested in the case of scattered data (irregularly placed), ...
Rajesh D's user avatar
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0 answers
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Approximation to continuous functions over an closed interval

Let $$f\in C[a,b]$$ A triangular system is a series of numbers \begin{matrix} x_{11}\\ x_{21}&x_{22}\\ x_{31}&x_{32}&x_{33}\\ \cdots \end{matrix} that $$a<x_{n1}<x_{n2}<\cdots<...
Isomorphism's user avatar
1 vote
0 answers
39 views

Properties of analytic "super-monomials"

Defining as monomials $m(x,n)\,:=\,x^n,\,n\in\mathbb{N}_0$, I denote by an "super-monomial" an analytic function of the form $$ \overline{m}(x,n,(a))\ :=\ x^n+\sum\limits_{i=1}^\infty \frac{a_{n+i}x^{...
Manfred Weis's user avatar
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1 vote
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History of Underdetermined Interpolation

Are there any examples, earlier than spline-interpolation, of mathematical investigations of interpolation problems with more unknowns than conditions or are (polynomial) splines the earliest? ...
Manfred Weis's user avatar
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1 vote
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Interpolation theory: equivalence of norms

Consider the interpolation space $Z=(X,Y)_{\theta,p}$. In the case $Y\subseteq X$ do we have that, for all $a>0$ the following norm: $$N_a:x\mapsto\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \...
Samir Boujijane's user avatar
1 vote
0 answers
63 views

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:=&\...
Manfred Weis's user avatar
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1 vote
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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 $...
user521337's user avatar
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1 vote
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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 ...
Jürgen Böhm's user avatar