**2**

votes

**1**answer

218 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 ...

**0**

votes

**0**answers

29 views

### Caputo fractional derivative

We define the Caputo fractional derivative $\partial_t^\alpha$ as follows:
$$
\partial_t^\alpha f(t)
=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-r)^{-\alpha} f'(r) dr,
\quad 0<\alpha<1.
$$
For ...

**2**

votes

**1**answer

176 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{ ...

**3**

votes

**1**answer

98 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 $L^p$,
...

**1**

vote

**0**answers

245 views

### Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**2**

votes

**1**answer

82 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

97 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

87 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

182 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

161 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

424 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

126 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

73 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

139 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

70 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

381 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

41 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

134 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

222 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

79 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

127 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

142 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

228 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

780 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

339 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

195 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

202 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

433 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

333 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

470 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

109 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

477 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

159 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

503 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

283 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

303 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

381 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

836 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

254 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 ...