# Tagged Questions

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.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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&... 0answers 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) = \... 1answer 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 ... 1answer 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 ... 2answers 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 ... 0answers 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^{\... 3answers 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 ... 0answers 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 1answer 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 ... 0answers 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 ...
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) = \...