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.
159
questions
3
votes
1
answer
93
views
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-...
- 49
2
votes
0
answers
39
views
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 ...
- 21
1
vote
0
answers
63
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 ...
- 337
5
votes
1
answer
210
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
$$
f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int_0^1 (f''(y))^2 \, dy \leq 1.
$$
...
- 353
1
vote
0
answers
16
views
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 ...
- 12k
0
votes
0
answers
55
views
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 ...
- 109
3
votes
0
answers
113
views
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
$$(D^...
- 71
3
votes
0
answers
38
views
Bound of a regular function that cancels at some points
Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
- 111
1
vote
0
answers
35
views
Error bounds for Sobolev space norm approximation on a finite grid
Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space,
\begin{multline}
f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
- 353
1
vote
0
answers
61
views
A nonstandard polynomial interpolation problem
I thought I know the properties of polynomials quite thoroughly. But...
I am looking for a recipe to construct, for any given finite set $S$ whose elements are sets of $N$ real vectors of length $n$ ...
- 3,671
0
votes
0
answers
58
views
Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality:
$$\...
6
votes
0
answers
118
views
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 ...
- 86
1
vote
0
answers
189
views
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. ...
- 2,986
0
votes
0
answers
66
views
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
116
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 $...
- 61
1
vote
0
answers
32
views
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 ...
- 2,410
1
vote
1
answer
71
views
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{...
- 145
2
votes
2
answers
351
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 ...
- 63
0
votes
0
answers
129
views
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&...
- 3,939
1
vote
0
answers
73
views
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. ...
- 479
2
votes
0
answers
66
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; ...
- 317
2
votes
0
answers
39
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 ...
- 12k
3
votes
2
answers
461
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 ...
- 383
0
votes
0
answers
94
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 ...
user206092
3
votes
1
answer
240
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 ...
- 139
3
votes
0
answers
29
views
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:
$$
...
- 4,883
1
vote
1
answer
85
views
Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?)
Formally, let $K$ be a field, $a \in K[x]$, $D \subseteq K$, and $E = \{(x_i, a(x_i)) \mid x_i \in D\}$ be distinct evaluations of $a$ where $\lvert E\rvert > \operatorname{deg}(a)$ (so $E$ ...
- 539
3
votes
1
answer
202
views
Cubic spline interpolation without a constant term
Two main questions:
I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
user206092
2
votes
1
answer
113
views
Spline Interpolation error of higher degree
It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.
Can I assume that, if one uses polynomials of degree $p$ and ...
- 23
1
vote
1
answer
128
views
Parabolic Sobolev inequality in Sobolev mixed norm spaces
Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and
$$
1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}.
$$
Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
- 475
0
votes
0
answers
176
views
Can we improve the error bounds for spline interpolation if the interpolated function is smooth?
Let me first state the original problem I want to solve:
Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
- 273
0
votes
0
answers
48
views
2d interpolation minimizing the integral of the norm of the Hessian
It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative:
$$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...
- 397
3
votes
0
answers
189
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 ...
- 2,986
0
votes
0
answers
139
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 ...
- 101
1
vote
0
answers
157
views
Uniformly local Sobolev spaces and interpolation
Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space
$$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
- 923
1
vote
0
answers
28
views
Mismatching degrees and # derivatives in polynomial interpolation error formula
It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
- 233
1
vote
0
answers
66
views
Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function
Is there any proof for this. Kindly request a reference in case available or any related documents towards this.
PS : I am specifically interested in the case of scattered data (irregularly placed), ...
- 704
5
votes
0
answers
401
views
Vector-valued interpolation for sublinear operators
Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\...
- 359
1
vote
1
answer
530
views
tetrahedral interpolation and integration along a segment
Let's say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$.
Given a position $P$ inside the tetrahedron $...
- 13
2
votes
1
answer
138
views
An interpolation inequality
I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has
$$ \left(\int|\nabla f|^p dx\right)^\...
- 39
1
vote
0
answers
84
views
Approximation to continuous functions over an closed interval
Let $$f\in C[a,b]$$ A triangular system is a series of numbers
\begin{matrix}
x_{11}\\
x_{21}&x_{22}\\
x_{31}&x_{32}&x_{33}\\
\cdots
\end{matrix}
that $$a<x_{n1}<x_{n2}<\cdots<...
- 241
3
votes
0
answers
103
views
Interpolating multivariate polynomials from their partial derivatives
Let $P(x_1,\dots,x_n)$ be a multivariate polynomial over a ground field $K$. For a multi-index $\alpha=(a_1,\dots,a_n)$ we denote the partial derivative $\frac{\partial^{a_1+\dots+a_n}P}{\partial x_1^{...
- 2,227
4
votes
4
answers
788
views
algorithm for convex $C^2$ interpolation
Let $x_0<x_1<\ldots<x_n$ and $f_0,f_1,\ldots,f_n$ be real numbers and
$$s_i=(f_i-f_{i-1})/(x_i-x_{i-1}),~~~c_i=(s_{i+1}-s_i)/(x_{i+1}-x_{i-1}).$$
If $f$ is a convex function defined on $[x_0,...
- 3,671
3
votes
0
answers
105
views
Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$
Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.
...
- 212
3
votes
1
answer
246
views
How to find the elliptical arc that corresponds to the cubic bezier curve
Let's assume I have a cubic bezier curve that is provided with A, B, C, D points, where
A is the start of the curve
B is the first control point
C is the second control point
D is the end of the ...
- 131
0
votes
1
answer
69
views
Can this function be interpolated with a small power series
Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$?
I feel the answer might be no but I'm not ...
- 2,524
2
votes
1
answer
227
views
Interpolation of product spaces
Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space.
Is it true that $X_{\theta}\times\mathcal{B}$ is an ...
- 923
3
votes
0
answers
113
views
Hardness results for approximating Hölder continuous functions
Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems (Daubechies & Lagarias, SIAM J. Math. Anal. 22 (1991) ...
- 211
1
vote
0
answers
38
views
Properties of analytic "super-monomials"
Defining as monomials $m(x,n)\,:=\,x^n,\,n\in\mathbb{N}_0$, I denote by an "super-monomial" an analytic function of the form
$$ \overline{m}(x,n,(a))\ :=\ x^n+\sum\limits_{i=1}^\infty \frac{a_{n+i}x^{...
- 12k
6
votes
3
answers
359
views
Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
- 2,526