# 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.

125
questions

**1**

vote

**1**answer

40 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 $...

**-2**

votes

**0**answers

58 views

### Scilab polinomial interpolation of covid 19 day-by-day new infected [closed]

I am trying to understand how polinomial interpolation works and how I should implement it on Scilab. A friend suggested to see covid-19 day-by-day new infected and to compare the polinomial ...

**2**

votes

**1**answer

102 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)^\...

**1**

vote

**0**answers

36 views

### 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<...

**0**

votes

**0**answers

135 views

### Reference request: Band limited interpolation of data

I have come up with an interpolation method for irregularly placed data points on a square domain. The method assumes the data points are discrete, that is they coincide with nodes of a uniform ...

**3**

votes

**0**answers

67 views

### 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^{...

**4**

votes

**4**answers

221 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,...

**3**

votes

**0**answers

45 views

### 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.
...

**2**

votes

**1**answer

56 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 ...

**0**

votes

**1**answer

63 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 ...

**2**

votes

**1**answer

64 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 ...

**3**

votes

**0**answers

86 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 show that
\begin{align*}
\left\| f - f_{h} \right\|_{\infty} \...

**1**

vote

**0**answers

36 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^{...

**4**

votes

**1**answer

156 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 ...

**3**

votes

**2**answers

72 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....

**0**

votes

**1**answer

68 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(...

**0**

votes

**0**answers

20 views

### $G1$ 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,\,\...

**1**

vote

**3**answers

69 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)=\...

**1**

vote

**1**answer

84 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?

**2**

votes

**0**answers

46 views

### 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 ...

**3**

votes

**0**answers

40 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 &...

**1**

vote

**0**answers

39 views

### Error bounds for spline interpolation. Hall and Meyer's conjeture

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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh ...

**3**

votes

**0**answers

74 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 ...

**0**

votes

**0**answers

34 views

### What is the best approach for interpolation between a sets of points where each set is defined by two scalar parameters?

I have a collection of sets of points that are generated for me. Each set is generated in a way that assuming that the points follow a simple equation is not a valid assumption (a valid equation would ...

**0**

votes

**1**answer

61 views

### Polynomial-preserving boundary conditions for spline interpolation

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

**1**

vote

**0**answers

53 views

### 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?
...

**1**

vote

**1**answer

79 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 ...

**1**

vote

**2**answers

110 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 ...

**1**

vote

**1**answer

81 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 $f:U_\epsilon(0)\to\mathbb{R}$ ...

**1**

vote

**1**answer

95 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 ...

**3**

votes

**1**answer

229 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 ...

**1**

vote

**0**answers

61 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 \...

**7**

votes

**0**answers

221 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 ...

**2**

votes

**1**answer

162 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 ...

**0**

votes

**1**answer

126 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 ...

**22**

votes

**2**answers

480 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**

votes

**1**answer

292 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)^{...

**1**

vote

**0**answers

43 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:=&\...

**0**

votes

**1**answer

98 views

### 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 ...

**1**

vote

**0**answers

44 views

### 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 $...

**4**

votes

**1**answer

78 views

### 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 $...

**6**

votes

**2**answers

246 views

### 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 $...

**1**

vote

**1**answer

363 views

### 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 $X$ ...

**4**

votes

**2**answers

197 views

### 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 ...

**4**

votes

**2**answers

78 views

### 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 ...

**6**

votes

**2**answers

417 views

### 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 ...

**2**

votes

**1**answer

368 views

### 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\...

**2**

votes

**1**answer

106 views

### 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 & ...

**0**

votes

**2**answers

198 views

### 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 ...

**1**

vote

**1**answer

84 views

### 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 ...