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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63 views

Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?)

Formally, let $K$ be a field, $a \in K[x]$, $D \subseteq K$, and $E = \{(x_i, a(x_i)) \mid x_i \in D\}$ be distinct evaluations of $a$ where $\lvert E\rvert > \operatorname{deg}(a)$ (so $E$ ...
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104 views

Cubic spline interpolation without a constant term

Two main questions: I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
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1answer
58 views

Spline Interpolation error of higher degree

It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$. Can I assume that, if one uses polynomials of degree $p$ and ...
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37 views

Interpolation estimate for trigonometric polynomial

Notations: $$ J^s u := \sum_{m \in \mathbb{N}} (1+\vert m \vert^2)^{s/2}\hat u_m e^{imt} $$ $$ \Vert \varphi \Vert_{W^{s,p}} := \Vert J^s u \Vert_{L^p} $$ $$ P_n u = \sum^{n-1}_{m=-n} \hat u_m e^{imt}...
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35 views

Parabolic Sobolev inequality in Sobolev mixed norm spaces

Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and $$ 1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}. $$ Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
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61 views

Can we improve the error bounds for spline interpolation if the interpolated function is smooth?

Let me first state the original problem I want to solve: Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
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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''(...
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117 views

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 ...
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23 views

Hermite interpolation with nested functions

Question: provided the existence of a unique solution, how to go about solving the following Hermite interpolation problem: given $f:y\in\mathbb{R}\mapsto z\in\mathbb{R},\ \boldsymbol{g}:x\in\mathbb{...
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117 views

L_q matrix inequality

The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
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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\|...
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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 $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
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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), ...
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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\...
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How to interpolate a vector field from random orientation projections only?

I want to obtain a smooth vector field from which I only have data on a set of scattered locations. I can face 3 different cases: The vector field is known at the given points: then I can apply some ...
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177 views

tetrahedral interpolation and integration along a segment

Let's say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$. Given a position $P$ inside the tetrahedron $...
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120 views

An interpolation inequality

I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has $$ \left(\int|\nabla f|^p dx\right)^\...
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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<...
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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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421 views

algorithm for convex $C^2$ interpolation

Let $x_0<x_1<\ldots<x_n$ and $f_0,f_1,\ldots,f_n$ be real numbers and $$s_i=(f_i-f_{i-1})/(x_i-x_{i-1}),~~~c_i=(s_{i+1}-s_i)/(x_{i+1}-x_{i-1}).$$ If $f$ is a convex function defined on $[x_0,...
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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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101 views

How to find the elliptical arc that corresponds to the cubic bezier curve

Let's assume I have a cubic bezier curve that is provided with A, B, C, D points, where A is the start of the curve B is the first control point C is the second control point D is the end of the ...
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1answer
67 views

Can this function be interpolated with a small power series

Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$? I feel the answer might be no but I'm not ...
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112 views

Interpolation of product spaces

Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space. Is it true that $X_{\theta}\times\mathcal{B}$ is an ...
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105 views

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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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^{...
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198 views

Non-polynomial splines, a non-linear problem

I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials. To be specific, given a class of functions such as "decaying ...
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2answers
135 views

Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$

I need to approximate $x \ln x$ on $[0,1]$ as a piecewise-linear function. If $P(x)$ is a piecewise-linear approximation, I want to minimize $$ \max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P....
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1answer
148 views

Convergence of Chebyshev interpolation in L^1

Let $f\in C^0([-1,1])$ and $P_n(f)$ its interpolation polynomial at the Chebyshev nodes. I would be interested to know about any existing results (positive or negative) about the convergence of $P_n(...
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$G1$ interpolating curves with symmetric slopes in ends of segments

given a set $\lbrace p_i| 1\le i \le n\rbrace =\lbrace(x_1,y_1),\,\cdots,\,(x_n,y_n)\rbrace$ of points , which method can be recommended to calculate a sequence of angles $\left(\varphi_1,\,\cdots,\,\...
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3answers
70 views

RKHS/non-parametric regression with missing response values

I am interested in doing RKHS regression with missing response variables. Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows \begin{equation}f(x)\approx u(x)=\...
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1answer
111 views

Polynomial interpolation, Chebyshev nodes, absolute continuity

How to prove that for an absolutely continuous function, the Lagrange interpolation polynomial at Chebyshev nodes converges uniformly to the function as the number of nodes goes to infinity?
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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 ...
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49 views

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 &...
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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 ...
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134 views

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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1answer
161 views

Polynomial-preserving boundary conditions for spline interpolation

Spline interpolation requires the definition of boundary conditions because the smoothness requirements do not yield enough conditions for a unique solution. Question: which kind of boundary ...
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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? ...
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1answer
101 views

Interpolating Maximum function with symmetric polynomials

Let $n$ and $p$ be two positive integers. Consider the function $$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$ that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are ...
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2answers
123 views

Comparison of methods to define a matrix function (Jordan canonical form, Hermite interpolation and Cauchy integral)? [closed]

There are many equivalent ways of defining a function $f(A)$ of a matrix $A$. We focus on Jordan canonical form, Hermite interpolation and Cauchy integral. What is the difference between methods for ...
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1answer
90 views

On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$. In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
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1answer
100 views

Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
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1answer
361 views

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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63 views

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 \...
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230 views

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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1answer
167 views

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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1answer
142 views

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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2answers
535 views

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 ...
14
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1answer
309 views

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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57 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:=&\...