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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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We consider the interpolation of the function $$f(w)=e^{2\pi \xi_0w}, 0<\xi_0<1$$ at well-separated real nodes $w_j$, $j=1,2,…,N$, satisfying $\sum\limits_{j\ge 1}\frac{1}{|w_j|}<\infty$. Let ...
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Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain and $ k_1 , k , k_2 \in \mathbb{N}, 0 \leq k_1 \leq k \leq k_2$. Let $f \in C^{k_2}(\bar{\Omega})$. Do we have $$[f]_k \leq C(k_1,k,k_2,\...
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Disclaimer: I am a first year mathematics student who is trying to write an applied math paper, so my question might seem trivial. Definitions: Let $N \in 2 \mathbb{N}$ and $u \in \mathbb{R}^N $ be a ...
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I met with a theorem while studying Pick Interpolation: Complete Pick Property is the same as $M_{\aleph_0\times\aleph_0}$-Pick Property It is trivial that $M_{\aleph_0\times\aleph_0}$-Pick Property ...
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This question is rummaging around in my head for quite some time. I will start with exposition on "model fitting" and then explain how Chebyshev polynomials are perfect on bounded intervals ...
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Let $f : [0, 1] \to \mathbb{R}$ be a smooth strictly convex function. Let $0 \leq x_1 < \dots < x_n \leq 1.$ I am interested in whether there exists a polynomial $p$ such that it is convex on $[...
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My questions: Is this sort of what Evans was referring to when he claims we can "arrange" for the support of $Eu$ to be compact? If not, what did he mean? How can I easily see that we can &...
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A basic kind of multivariate polynomial interpolation formula for degree-$d$ polynomials has the following shape: Theorem sketch. Let $z\in \mathbf{D}^n$ and $X=X_d\subset \mathbf{D}^n$ be the finite ...
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Suppose that $\lambda_1<\lambda_2<\ldots$ is a sequence of positive real numbers such that $$|\{n\in \mathbb N\,:\, \lambda_n \leq \lambda\}| \leq \sqrt{\lambda} \quad \forall\, \lambda>>1....
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Suppose that $0<b<1$ is a real number. I'm interested in whether there exists an entire function $f(z)$ which decays faster than any exponential $e^{c|\Re(z)|}$ along horizontal lines $\Re(z)\...
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Paterson-Stockmeyer algorithm If we need to compute a high-degree polynomial expression, such as: $$ P(y) = \sum_{k=0}^{B} a_k y^k $$ the Paterson-Stockmeyer algorithm can process the powers in ...
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Let $A$ be a commutative ring and $f\in A[x]$ a split monic. When $f$ is separable with roots $\mathrm Z(f)= \{ a_1,\dots ,a_k \}$, the Chinese remainder theorem (CRT) ensures that evaluation is an $A$...
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I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
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In this paper: Michael Ben-Or and Prasoon Tiwari. 1988. A deterministic algorithm for sparse multivariate polynomial interpolation, In: Proceedings of the twentieth annual ACM symposium on Theory of ...
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Let $V$ be a finite-dimensional complex vector space which is acted on faithfully by a finite group $\Gamma$. Suppose $\Gamma' \subset \Gamma$ is a proper subgroup. Fix a vector $v \in V$ such that ...
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Let two integers $L\leq N$, a sequence of $L+1$ distinct real numbers $\alpha_0,\dots,\alpha_L$, and integers $k_0,\dots,k_L\in\mathbb N$ such that $N=L+\sum_{i=0}^Lk_i$. I noticed that the two ...
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Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
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Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$. I am wondering if it there is a constant $C > 0$ such that for all ...
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In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
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Let $\{a_n\}_{n=1}^\infty$ be a sequence of complex numbers satisfying $|a_n|\le n^2$ and $|a_n|\to \infty$. I'm looking for a function $h(z)$ such that: (a) $h$ is holomorphic on a half-plane $\{\Re(...
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I recently posted this question on MSE. While it attracted interest, no answers were submitted, so I thought to try and post it here. I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While ...
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Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
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I was plotting 2-D shape functions for linear interpolation on a Smolyak sparse grid of level 2 associated to Gauss-Lobatto-Chebyshev nodes(cf https://en.wikipedia.org/wiki/Sparse_grid ), when I came ...
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I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
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This is a migrated question from: Thin-Plate-Spline understanding and solution. If I need to delete one of the questions let me know. I was suggested to post it here as well. As I understand it a Thin-...
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I want to know where can I find how to compute, given a function $f$ and $t>0$ $$K(t,f;L^{1,\infty};L^\infty)$$ where $L^{1,\infty}=\sup_{t>0}t f^*(t)$ and $K$ is the Petree K-functional ...
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L. Caffarelli, R. Kohn and L. Nirenberg showed in this article that, under some conditions, the following weighted interpolation inequality is valid THEOREM: There exists a positive constant $C$ such ...
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Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $C^\infty$ function satisfying the constraints $$ f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int_0^1 (f''(y))^2 \, dy \leq 1. $$ ...
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Question: What is known about the problem of covering a finite set of $\mathbb{P}$ of points in the plane with convex polygons that have the same number $m$ of points from $\mathbb{P}$ as corners and ...
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Although this is essentially a port of my MathSE question, I think the users there tend to not understand how to interpret the questions from a higher perspective (and often too literally). This is ...
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In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by $$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$ where $D^sf$ is defined by the Fourier transform $$(D^...
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Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
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Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space, \begin{multline} f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
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I thought I know the properties of polynomials quite thoroughly. But... I am looking for a recipe to construct, for any given finite set $S$ whose elements are sets of $N$ real vectors of length $n$ ...
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In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality: $$\...
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this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
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A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
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I've got a 3D figure created using around 30k points and has different regions colored in an specific way according to some unrelated variables that come from a project I'm creating. Taking in ...
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Let $\mathbb{F}$ be a finite field and let $\boldsymbol{x} = (x_1, x_2, \dots, x_n)$ be $n$ pairwise distinct points in $\mathbb{F}$. Given the vector $\boldsymbol{y} = (y_1, y_2, \dots, y_n)$, with $...
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For a vector $\vec{u}\in\mathbb{R}^N$ let's denote $\pi_N\left(\vec{u}\right)$ the unique piecwise linear and $1$-periodic function matching the components of $\vec{u}$ on the discretization $x_k = \...
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When can you reconstruct the power series of a function by taking the limits of the coefficients of the polynomials that interpolate its values at $0,1,2,\dots,j$? More precisely: Let $f\colon\mathbb{...
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Suppose I am working over a field $\mathbb{F}$ and have $n$ points in the point-value representation $(x_0,x_1,\cdots,x_{n-1})$. What is the fastest way to do polynomial interpolation and convert this ...
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For a certain interpolation problem, I'm looking into a sequence of functions of the form $$f_{m}(z) = \operatorname{sinc} \Bigg{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Bigg{)} . $$ Here, $m&...
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Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. ...
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Is there a formula expressing the top coeffient of the Lagrange interpolation polynomial for a function as an average of its ($n-1$)-st derivative (divided by $(n-1)!$)? I am looking for a reference; ...
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Question: what is known about the algorithmic construction of general interpolating spline functions with smoothness constraints at every knot? So far I could only find descriptions for splining ...
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I am new to Math Overflow, so I am not really sure whether this question fits the community standards. But, I posted this question in Stack Exchange and recieved no answers. Moreover, nothing even ...
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I am referencing a paper by CJC Kruger entitled "Constrained Cubic Spline Interpolation for Chemical Engineering Applications." In the paper he uses a the following formula to calculate ...
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Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold. Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function. Is there a Harmonic ...
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Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$. If $n=m=1$ then it's easy to see that: $$ ...
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