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-1
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0answers
17 views

Gradient Estimation Using Bicubic Interpolation and Finite Differences [closed]

Suppose we know the values f(x,y) takes on in a 4x4 grid defined by all pairwise combinations of x={0,1,2,3} and y={0,1,2,3}. Bicubic interpolation using centered differences provides a way of ...
2
votes
1answer
117 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
0answers
58 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
1answer
107 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
0answers
112 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
2answers
213 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
1answer
263 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 ...
5
votes
2answers
401 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
2answers
949 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
1answer
299 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
0answers
137 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
1answer
180 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
1answer
148 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
2answers
341 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 ...
0
votes
0answers
224 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
3answers
426 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
0answers
78 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
1answer
382 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 ...
2
votes
0answers
196 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
1answer
388 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
0answers
137 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
0answers
84 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
2answers
494 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 ...
13
votes
1answer
976 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
0answers
261 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
5answers
291 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
1answer
317 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
1answer
221 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
1answer
706 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
2answers
243 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 ...