Questions tagged [interpolation]
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.
181 questions
1 vote
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Space of interpolating functions with constraints on interpolation
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 ...
1 vote
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Proving operator-valued Pick property [closed]
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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2 votes
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Orthogonal feature functions for model fitting on unbounded intervals (like Chebyshev polynomials for bounded intervals)
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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3 votes
1 answer
159 views
Hermite-type convex interpolation
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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1 vote
1 answer
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Sobolev extension: "we can arrange for the support of $\bar u$ to lie within $V\supset\supset U$". How, exactly?
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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74 views
$\ell_1$ norm of coefficients vector in polynomial interpolation (reference/history request)
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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2 votes
1 answer
227 views
On existence of a suitable density for a sequence
Suppose that $\lambda_1<\lambda_2<\ldots$ is a sequence of positive real numbers such that $$|\{n\in \mathbb N\,:\, \lambda_n \leq \lambda\}| \leq \sqrt{\lambda} \quad \forall\, \lambda>>1....
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2 votes
1 answer
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Interpolation by an entire function with decay requirement
Suppose that $0<b<1$ is a real number. I'm interested in whether there exists an entire function $f(z)$ which decays faster than any exponential $e^{c|\Re(z)|}$ along horizontal lines $\Re(z)\...
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0 votes
1 answer
200 views
How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm If we need to compute a high-degree polynomial expression, such as: $$ P(y) = \sum_{k=0}^{B} a_k y^k $$ the Paterson-Stockmeyer algorithm can process the powers in ...
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1 vote
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Universal formulas for polynomials with prescribed jets
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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4 votes
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Interpolation-extrapolation scales of H. Amann
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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3 votes
1 answer
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How to understand 'the problem of determining the exact number of monomials in P(x) given by a black box is #P-Complete'
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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4 votes
0 answers
241 views
A Lagrange interpolation question
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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1 vote
1 answer
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Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?
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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Interpolation polynomials with constraints
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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-1 votes
2 answers
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Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$
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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1 vote
1 answer
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How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?
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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1 vote
1 answer
153 views
Interpolation by holomorphic functions of small exponential type on a half-plane
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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5 votes
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179 views
Recovering a binary function on a lattice by studying its sum along closed walks
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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5 votes
1 answer
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Interpolation between two matrices so that $L^p$ norm is controlled
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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2 votes
1 answer
158 views
Surprising numerical coincidence while interpolating on Smolyak grid
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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1 vote
1 answer
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weakly separated sequences in RKHS are separated by Gleason metric
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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3 votes
1 answer
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Thin-Plate-Spline understanding and solution
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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2 votes
0 answers
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K functional of $L^{1,\infty}$ and $L^\infty$ real interpolation
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)$$ where $L^{1,\infty}=\sup_{t>0}t f^*(t)$ and $K$ is the Petree K-functional ...
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1 vote
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123 views
First-order interpolation inequalities with weights by L.Caffarelli, R.Kohn and L.Nirenberg
L. Caffarelli, R. Kohn and L. Nirenberg showed in this article that, under some conditions, the following weighted interpolation inequality is valid THEOREM: There exists a positive constant $C$ such ...
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5 votes
1 answer
498 views
Optimal constant to compare $L^2$ norm of smooth function on $[0, 1]$ to a grid
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $C^\infty$ function satisfying the constraints $$ f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int_0^1 (f''(y))^2 \, dy \leq 1. $$ ...
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1 vote
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Regular covering of planar pointsets with convex polygons
Question: What is known about the problem of covering a finite set of $\mathbb{P}$ of points in the plane with convex polygons that have the same number $m$ of points from $\mathbb{P}$ as corners and ...
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On "canonical" extensions of functions from integers to reals
Although this is essentially a port of my MathSE question, I think the users there tend to not understand how to interpret the questions from a higher perspective (and often too literally). This is ...
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3 votes
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Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by $$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$ where $D^sf$ is defined by the Fourier transform $$(D^...
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3 votes
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Bound of a regular function that cancels at some points
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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Error bounds for Sobolev space norm approximation on a finite grid
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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A nonstandard polynomial interpolation problem
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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Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
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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6 votes
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Equivalent forms of Fourier restriction conjecture
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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Function uniquely determined by its values at integer arguments
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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3D interpolation function
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 ...
1 vote
0 answers
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Polynomial interpolation of binary vectors
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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1 vote
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$\mathbb{P}_1$-finite element as convolution of $\mathbb{P}_0$-finite element
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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2 votes
1 answer
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Power series whose coefficients are limits of coefficients of polynomial interpolations
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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4 votes
3 answers
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Fastest Implementation of polynomial interpolation?
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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What is the inverse Fourier transform of $\operatorname{sinc} \Big{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Big{)} $?
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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1 vote
0 answers
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Maximal geodesically convex function interpolating three points on the hyperbolic plane
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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2 votes
0 answers
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Top coefficient of the Lagrange polynomial as average of (n-1)-st derivative
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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2 votes
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Calculating non-polynomial spline functions
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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2 votes
2 answers
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About Newton's forward and backward interpolation
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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1 vote
0 answers
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Cubic spline interpolation – slope approximation using adjacent points
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 ...
user206092
3 votes
1 answer
308 views
Harmonic interpolation with analytic initial condition
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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3 votes
0 answers
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Do higher-order splines with Lipschitz derivatives exist on finite sets?
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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1 vote
1 answer
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Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?)
Formally, let $K$ be a field, $a \in K[x]$, $D \subseteq K$, and $E = \{(x_i, a(x_i)) \mid x_i \in D\}$ be distinct evaluations of $a$ where $\lvert E\rvert > \operatorname{deg}(a)$ (so $E$ ...
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3 votes
1 answer
465 views
Cubic spline interpolation without a constant term
Two main questions: I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
user206092