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# Questions tagged [interpolation]

Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points. It is necessary because in science and engineering we often need to deal with discrete experimental data.

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### Any employment for the Varignon parallelogram?

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram, which is called the Varignon parallelogram of the quad. While answering a question about Quadrilateral Interpolation it ...
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The simplest finite element shape in two dimensions is a triangle. In a finite element context, any geometrical shape is endowed with an interpolation, which is linear for triangles (most of the time),...
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### Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
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### How interpret $\||x|^\gamma u\|_{L^r}\leq C\|\nabla u\|_{L^p}^{a}\||x|^\beta u\|_{L^q}^{1-a}$ for $u\in\mathcal C_0^1(\mathbb R^d)$?

Let $$\frac{1}{r}+\frac{\gamma }{d}=a\left(\frac{1}{p}+\frac{\alpha -1}{d}\right)+(1-a)\left(\frac{1}{q}+\frac{\beta }{d}\right),$$ where $d\geq 1$, $a\in [0,1]$, $\alpha ,\beta,\gamma \in \mathbb R$...
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### Lagrange linear, quadratic, and cubic interpolations maximum interpolation error functions comparison

Using Theorem 1, calculate that the maximum interpolation error that is bounded for linear, quadratic, and cubic interpolations. Then compare the found error to the bounds given by Theorem 2. The ...
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### Smooth transition between two lines (2d)

I have function that is defined as $$Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30$$ $$Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30$$ The ...
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### A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} \...
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### Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
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### Determining Coefficients of a Finite Degree Polynomial $f$ from the Sequence $\{f(k)\}_{k \in \mathbb{N}}$

Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ ...
I have 2 questions, but I'll put both of them here since they are closely related: An integer valued polynomials $P(x)$ is a polynomial whose value $P(n)\in\mathbb{N}$ for every $n\in\mathbb{N}$. 1-...
### Polynomial Interpolation When part of $y_i$'s are Shuffled
Hypothesis: 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 ...