**4**

votes

**0**answers

95 views

### 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, \...

**0**

votes

**0**answers

49 views

### Who was first to use reproducing kernals 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 ...

**1**

vote

**0**answers

46 views

### 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} &\...

**2**

votes

**0**answers

44 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_{...

**0**

votes

**1**answer

116 views

### 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 ...

**6**

votes

**0**answers

246 views

### 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 ...

**0**

votes

**0**answers

49 views

### 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 ...

**0**

votes

**0**answers

26 views

### 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 ...

**8**

votes

**2**answers

206 views

### 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-...

**3**

votes

**0**answers

92 views

### 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\\
...

**2**

votes

**0**answers

67 views

### 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 $$P^*(x)=\sum_{...

**4**

votes

**1**answer

164 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\\
0&0&...

**1**

vote

**1**answer

162 views

### Reference for Bessel's interpolation formula

Please give me a reference for a standard, easy-to-find, textbook where I can find the full proof of Bessel's interpolation formula?
Thank you.

**0**

votes

**0**answers

33 views

### 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 ...

**1**

vote

**1**answer

71 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 ...

**1**

vote

**0**answers

89 views

### 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 ...

**2**

votes

**1**answer

257 views

### 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 (...

**2**

votes

**1**answer

196 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{ }f(S_1)\subseteq[1-\epsilon,1+...

**3**

votes

**1**answer

185 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 $L^p$,
$$|u|_{W^{...

**2**

votes

**1**answer

208 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 ...

**0**

votes

**1**answer

113 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 f....

**1**

vote

**0**answers

143 views

### 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 ...

**1**

vote

**0**answers

237 views

### 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}, ~~~r>0,-\frac{\pi}...

**6**

votes

**3**answers

442 views

### 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:
&...

**4**

votes

**2**answers

280 views

### 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 ...

**0**

votes

**1**answer

108 views

### 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 $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$.
$3)$ $R(x_g)=g+1$.
...

**4**

votes

**1**answer

463 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 ...

**0**

votes

**0**answers

76 views

### 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 ...

**7**

votes

**1**answer

419 views

### 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 $(x_i,y_i)$ ...

**4**

votes

**0**answers

73 views

### 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 (...

**3**

votes

**2**answers

354 views

### 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 ...

**3**

votes

**1**answer

369 views

### 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 ...

**2**

votes

**0**answers

91 views

### 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 ...

**4**

votes

**1**answer

144 views

### 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 $f$...

**2**

votes

**0**answers

167 views

### 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 ...

**3**

votes

**2**answers

254 views

### 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 ...

**6**

votes

**1**answer

291 views

### 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
$$\frac{...

**7**

votes

**3**answers

958 views

### 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 $(X_i^n,X_i^{n-1}...

**2**

votes

**2**answers

2k views

### 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 ...

**4**

votes

**1**answer

459 views

### 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}
v_{1,0}&v_{2,0}&\dots&...

**0**

votes

**0**answers

152 views

### 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) = \...

**-1**

votes

**1**answer

220 views

### 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 ...

**0**

votes

**1**answer

237 views

### 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 ...

**0**

votes

**2**answers

525 views

### 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 ...

**1**

vote

**0**answers

404 views

### 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 u\|_{L^{\...

**4**

votes

**3**answers

554 views

### 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 ...

**1**

vote

**0**answers

168 views

### 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

**0**

votes

**1**answer

579 views

### 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 ...

**3**

votes

**0**answers

225 views

### 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 ...

**5**

votes

**1**answer

400 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) = \...