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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.

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Proof of the equivalence between Triebel Spaces and Bessel Potential

I've encountered a question regarding the relationship between Triebel-Lizorkin spaces and Bessel potential spaces. Specifically, I understand that $F^s_{p,2} = H^{s,p}$, for $p \in (1,\infty)$. ...
Kilian Koch's user avatar
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Error on spline interpolation for integration

I am interested in studying the integral $$ I \equiv \int_m^M dx \, f(x) d(x) $$ where $f(x)$ is an analitcally known function (mildly oscillating), while $d(x)$ is sampled at a finite, discrete set ...
knuth's user avatar
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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 ...
Adrien Wohrer's user avatar
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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 ...
JustSomeGuy's user avatar
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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 ...
Drew Brady's user avatar
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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 ...
Elvis's user avatar
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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(...
Claudio's user avatar
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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 ...
GSofer's user avatar
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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 $||Bx||_\infty \leq ||x||_\infty$....
tommy1996q's user avatar
2 votes
1 answer
108 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 ...
G. Fougeron's user avatar
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the design of kernel function and integral transform

I read a solution of an integral inequality. The solution uses condition $$f(1)=f(0)=f'(0)=0$$ to derive that $$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\...
Hao Huang's user avatar
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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 ...
ashK's user avatar
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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-...
user8469759's user avatar
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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 ...
User 2234's user avatar
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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 ...
Ilovemath's user avatar
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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 $$ f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int_0^1 (f''(y))^2 \, dy \leq 1. $$ ...
Drew Brady's user avatar
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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 ...
Manfred Weis's user avatar
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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 ...
Graviton's user avatar
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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^...
kuuga's user avatar
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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 $n=1$, ...
Pii_jhi's user avatar
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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 &...
Drew Brady's user avatar
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67 views

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 $n$ ...
Arnold Neumaier's user avatar
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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: $$\...
Jingeon An-Lacroix's user avatar
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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 ...
Tutukeainie's user avatar
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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. ...
Vincent Granville's user avatar
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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 ...
Vako Galvan's user avatar
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0 answers
132 views

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 $...
Bean Guy's user avatar
1 vote
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P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
Ayman Moussa's user avatar
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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{...
Mike Krebs's user avatar
2 votes
3 answers
1k views

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 ...
Ordinary's user avatar
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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&...
Max Muller's user avatar
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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. ...
ccriscitiello's user avatar
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73 views

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; ...
user420620's user avatar
2 votes
0 answers
42 views

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 ...
Manfred Weis's user avatar
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3 votes
2 answers
2k views

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 ...
Sayan Dutta's user avatar
1 vote
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119 views

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 ...
user avatar
3 votes
1 answer
266 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 ...
Ali Taghavi's user avatar
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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: $$ ...
ABIM's user avatar
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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 $E$ ...
Joe Bebel's user avatar
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3 votes
1 answer
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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 $f(t) = bt + ct^2 + dt^3$, ...
user avatar
2 votes
1 answer
241 views

Spline Interpolation error of higher degree

It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$. Can I assume that, if one uses polynomials of degree $p$ and ...
Astraeus's user avatar
3 votes
1 answer
217 views

Parabolic Sobolev inequality in Sobolev mixed norm spaces

Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and $$ 1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}. $$ Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
Guohuan Zhao's user avatar
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0 answers
246 views

Can we improve the error bounds for spline interpolation if the interpolated function is smooth?

Let me first state the original problem I want to solve: Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
trisct's user avatar
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2d interpolation minimizing the integral of the norm of the Hessian

It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative: $$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...
Bernard 's user avatar
3 votes
0 answers
199 views

Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
Vincent Granville's user avatar
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147 views

L_q matrix inequality

The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
yangpliu's user avatar
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Uniformly local Sobolev spaces and interpolation

Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space $$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
Capublanca's user avatar
1 vote
0 answers
34 views

Mismatching degrees and # derivatives in polynomial interpolation error formula

It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
gTcV's user avatar
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Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function

Is there any proof for this. Kindly request a reference in case available or any related documents towards this. PS : I am specifically interested in the case of scattered data (irregularly placed), ...
Rajesh D's user avatar
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432 views

Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
Tony419's user avatar
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