**0**

votes

**0**answers

35 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 ...

**-1**

votes

**0**answers

8 views

### Partition of function into pieces for interpolation needs [migrated]

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...

**0**

votes

**1**answer

85 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 ...

**1**

vote

**0**answers

79 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

178 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}, ...

**-1**

votes

**1**answer

146 views

### What is the advantage of the knowledge of jumps for approximating a function with trigonometric polynomials?

Let $f:(a,b)\to\mathbb{R}$ be square integrable, bounded variation and piecewise continuous function. Let the points of jump be $\{x_i/a<x_i<b,i = 1,2,3,...n\}$. The goal is to approximate the ...

**6**

votes

**3**answers

422 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:
...

**1**

vote

**1**answer

102 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 $f$ function such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is interpolation polynomial on ...

**0**

votes

**1**answer

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

**3**

votes

**1**answer

99 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

68 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

377 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

39 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 ...

**0**

votes

**0**answers

17 views

### Estimating Missing Shortest-Path-Tree Distances from Point Coordinates

Problem:
given
a finite set of points $\mathcal{P}:=\lbrace p_i\in\mathbb{R}^2 \mid i\in \lbrace 0,...,n\rbrace\wedge i\neq j\Leftrightarrow p_i\neq p_j\rbrace$
a function ...

**2**

votes

**1**answer

112 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 ...

**2**

votes

**1**answer

208 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 ...

**3**

votes

**0**answers

76 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

125 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 ...

**2**

votes

**0**answers

135 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

224 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

274 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
...

**7**

votes

**3**answers

748 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 ...

**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 p₁, p₂, p₃, and p₄, a point B on that curve at some t value (where 0 ≤ t ≤ 1), a point A on the line (p₂ — p₃) at distance ratio t from p₂, and a point C ...

**4**

votes

**1**answer

335 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}
...

**0**

votes

**0**answers

143 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

191 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

196 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

424 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

318 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 ...

**4**

votes

**3**answers

460 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

107 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

467 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

214 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

397 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) = ...

**1**

vote

**0**answers

153 views

### One-sided version of the “best approximation polynomial” : Upper polynomial approximations

Let $X$ be a finite subset of $\mathbb R$ and let $f : X \to {\mathbb R}$. Suppose we want to approximate $f$ by a polynomial $g$ of fixed degree $d\geq 1$ with the additional condition
$g\geq f$. Let
...

**0**

votes

**0**answers

92 views

### Extrapolate trend in gridded data

I'm interested in extrapolating measured horizontal wind data of a vehicle into a forecast horizontal wind grid. Given the flight path you have a series of points that are "more accurate" than the ...

**4**

votes

**2**answers

501 views

### Directional Distortion of a Surface

Hi everyone,
I am facing a math road block.
I have two surfaces (3D) described by two functions $f_1$ and $f_2$ (known). I would like to create some sort of directional distortion along the loading ...

**15**

votes

**1**answer

1k views

### Interpolating between piecewise linear functions, with a family of smooth functions

Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that:
there is a number $k\in\mathbb ...

**4**

votes

**0**answers

276 views

### 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 - ...

**1**

vote

**5**answers

302 views

### Interpolating for particular coefficients

Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$.
One needs to evaluate $F(X)$ at $O(n)$ points to interpolate and get all the coefficients of $F(X)$.
However say I need ...

**3**

votes

**1**answer

374 views

### Fast approximation for local Delaunay simplex?

Consider a function $f(x)$ evaluated at a set of points $x_j\in\mathcal{D}\subset\mathbb{R}^d$. I'm working on the following type of low order interpolation method. Consider the Delaunay tesselation ...

**0**

votes

**1**answer

230 views

### Finding 3 dimentional B-spline controll points from given array of points from spline solution?

Wa are talking about Non-uniform rational B-spline. We have some simple 3 dimentional array like
{1,1,1}
{1,2,3}
{1,3,3}
{2,4,5}
{2,5,6}
{4,4,4}
Which are ...

**4**

votes

**1**answer

817 views

### How to determine the kernel of a Vandermonde matrix?

Given a Vandermonde matrix
$
V=
\begin{bmatrix}
1 & 1 & 1 & \ldots & 1 \\\\
x_1 & x_2 & x_3 & \ldots & x_n \\\\
x_1^2 & x_2^2 & x_3^2 & \ldots & x_n^2 ...

**0**

votes

**2**answers

251 views

### Finite interpolation by nondecreasing indefinitely differentiable functions in a finite-dimensional space

Some time ago, I asked about inite interpolation by
a nondecreasing polynomial here at Finite interpolation by a nondecreasing polynomial. This turned out to be an already solved problem; it also ...