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.
169
questions
1
vote
0
answers
94
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 $$a<x_{n1}<x_{n2}<\cdots<...
- 281
3
votes
0
answers
130
views
Interpolating multivariate polynomials from their partial derivatives
Let $P(x_1,\dots,x_n)$ be a multivariate polynomial over a ground field $K$. For a multi-index $\alpha=(a_1,\dots,a_n)$ we denote the partial derivative $\frac{\partial^{a_1+\dots+a_n}P}{\partial x_1^{...
- 2,777
4
votes
4
answers
954
views
algorithm for convex $C^2$ interpolation
Let $x_0<x_1<\ldots<x_n$ and $f_0,f_1,\ldots,f_n$ be real numbers and
$$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}).$$
If $f$ is a convex function defined on $[x_0,...
- 3,771
3
votes
0
answers
167
views
Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$
Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.
...
- 230
3
votes
1
answer
320
views
How to find the elliptical arc that corresponds to the cubic bezier curve
Let's assume I have a cubic bezier curve that is provided with A, B, C, D points, where
A is the start of the curve
B is the first control point
C is the second control point
D is the end of the ...
- 131
0
votes
1
answer
72
views
Can this function be interpolated with a small power series
Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$?
I feel the answer might be no but I'm not ...
- 2,582
2
votes
1
answer
268
views
Interpolation of product spaces
Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space.
Is it true that $X_{\theta}\times\mathcal{B}$ is an ...
- 923
3
votes
0
answers
116
views
Hardness results for approximating Hölder continuous functions
Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems (Daubechies & Lagarias, SIAM J. Math. Anal. 22 (1991) ...
- 221
1
vote
0
answers
39
views
Properties of analytic "super-monomials"
Defining as monomials $m(x,n)\,:=\,x^n,\,n\in\mathbb{N}_0$, I denote by an "super-monomial" an analytic function of the form
$$ \overline{m}(x,n,(a))\ :=\ x^n+\sum\limits_{i=1}^\infty \frac{a_{n+i}x^{...
- 12.7k
6
votes
3
answers
436
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 ...
- 2,758
3
votes
2
answers
568
views
Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$
I need to approximate $x \ln x$ on $[0,1]$ as a piecewise-linear function. If $P(x)$ is a piecewise-linear approximation, I want to minimize
$$
\max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P....
- 325
0
votes
1
answer
407
views
Convergence of Chebyshev interpolation in L^1
Let $f\in C^0([-1,1])$ and $P_n(f)$ its interpolation polynomial at the Chebyshev nodes.
I would be interested to know about any existing results (positive or negative) about the convergence of $P_n(...
- 98
0
votes
0
answers
21
views
$G1$ interpolating curves with symmetric slopes in ends of segments
given a set $\lbrace p_i| 1\le i \le n\rbrace =\lbrace(x_1,y_1),\,\cdots,\,(x_n,y_n)\rbrace$ of points , which method can be recommended to calculate a sequence of angles $\left(\varphi_1,\,\cdots,\,\...
- 12.7k
1
vote
3
answers
86
views
RKHS/non-parametric regression with missing response values
I am interested in doing RKHS regression with missing response variables.
Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows
\begin{equation}f(x)\approx u(x)=\...
- 41
1
vote
1
answer
395
views
Polynomial interpolation, Chebyshev nodes, absolute continuity
How to prove that for an absolutely continuous function, the Lagrange interpolation polynomial at Chebyshev nodes converges uniformly to the function as the number of nodes goes to infinity?
- 281
2
votes
0
answers
101
views
How to evaluate an interpolation method, in terms of converging to the underlying function, as data points go to infinity?
I have an interpolation method, which takes function $f$ values at any given finite number $N$ of points in the domain and interpolate to get a function $f_{int}$.
I want to do some analysis on how ...
- 714
3
votes
0
answers
74
views
Solutions to a special confluent Vandermonde system
Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define
$$
M^{(0)} = \begin{pmatrix}
1 &...
- 593
2
votes
0
answers
114
views
Error bounds for spline interpolation. Hall and Meyer's conjecture
Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
- 151
3
votes
0
answers
201
views
The $L_\infty$ norm of the derivative of the $L_2$ spline projector
A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
- 151
1
vote
1
answer
295
views
Polynomial-preserving boundary conditions for spline interpolation
Spline interpolation requires the definition of boundary conditions because the smoothness requirements do not yield enough conditions for a unique solution.
Question:
which kind of boundary ...
- 12.7k
1
vote
0
answers
68
views
History of Underdetermined Interpolation
Are there any examples, earlier than spline-interpolation, of mathematical investigations of interpolation problems with more unknowns than conditions or are (polynomial) splines the earliest?
...
- 12.7k
1
vote
1
answer
144
views
Interpolating Maximum function with symmetric polynomials
Let $n$ and $p$ be two positive integers. Consider the function
$$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$
that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are ...
- 2,673
2
votes
2
answers
174
views
Comparison of methods to define a matrix function (Jordan canonical form, Hermite interpolation and Cauchy integral)? [closed]
There are many equivalent ways of defining a function $f(A)$ of a matrix $A$. We focus on Jordan canonical form, Hermite interpolation and Cauchy integral.
What is the difference between methods for ...
- 21
1
vote
1
answer
132
views
On a case of real-analytic interpolation
Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$.
In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
- 1,133
1
vote
1
answer
120
views
Interpolation of a trilinear functional
Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
- 203
4
votes
1
answer
680
views
Marsden's Identity and B-splines
Marsden's Identity states that for every $\tau$ in $\mathbb{R }$:
$$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$
with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$.
Following ...
- 485
1
vote
0
answers
84
views
Interpolation theory: equivalence of norms
Consider the interpolation space $Z=(X,Y)_{\theta,p}$. In the case $Y\subseteq X$ do we have that, for all $a>0$ the following norm:
$$N_a:x\mapsto\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \...
7
votes
0
answers
253
views
An open problem in Sobolev spaces
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator
$$
E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)
\quad
\text{and}
\quad
E:W^{1,q}(\Omega)\to ...
- 27.3k
2
votes
1
answer
177
views
For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often
Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
- 1,199
0
votes
1
answer
167
views
For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve
Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
- 1,199
22
votes
2
answers
647
views
Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
- 1,199
15
votes
1
answer
449
views
Asymptotic behavior of sum linked with Lagrange interpolation
I already asked this a few weeks ago with no answer, so let me formulate differently.
In performing Lagrange interpolation with nodes 1/n, one encounters the sum
$$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{...
- 11.7k
1
vote
0
answers
63
views
Defining boundary conditions for spline interpolation via the Euler–Maclaurin formula
The Euler–Maclaurin formula states an interdependency between
\begin{align}
I\quad:=&\quad\int_m^nf(x) \, dx,\ \ m,n\in\mathbb{Z},\\[6pt]
S\quad:=&\quad\sum_{k=m}^n f(k), \\[6pt]
D\quad:=&\...
- 12.7k
0
votes
1
answer
123
views
Chebyshev interpolation [closed]
Let's define the n-th degree Chebyshev polynomials by
$$ T_{n} (x)=\cos(n\arccos(x)).$$
Find a polynomial $P$ such that
$$\mid y- P (x) \mid$$
is minimal, using the first three Chebyshev ...
- 13
1
vote
0
answers
48
views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,199
4
votes
1
answer
148
views
Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,199
7
votes
2
answers
614
views
Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,199
2
votes
1
answer
1k
views
Wasserstein interpolation between two probability measures on a metric space
Question 1
Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
- 6,726
4
votes
2
answers
384
views
Reference to L^1 error for piecewise linear interpolation of Lipschitz functions
Let $f:[0,1]\to\mathbb{R}$ be a Lipschitz function, and $\pi f$ be its piecewise linear interpolant on an equispaced grid with $n$ points.
It should be true (if I am not making mistakes with the ...
- 19.4k
4
votes
2
answers
123
views
Univariate polynomial interpolation with restricted degrees
Let $D=\{d_1, d_2, \ldots, d_n\}$ be an integer set. I'd like to know if I can interpolate any collection of $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{n}, y_{n})$ by a polynomial whose degree ...
- 41
6
votes
2
answers
589
views
Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
- 193
3
votes
1
answer
765
views
Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis
First, let us fix some Notation:
Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let
\begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
- 75
2
votes
1
answer
249
views
Optimal $L^2$ bounds of cubic spline interpolation
Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
- 3,554
0
votes
2
answers
211
views
How to prove the this sobolev-like inequality presented in the paper “sobolev inequalities in disguise”
click to see the picture of one related
page from the paper
this is the link of the whole paper
What I cannot really clearly understands is the content bellow:
the shortcut of the inequality
If ...
1
vote
1
answer
104
views
Proof Reference - Polynomial interpolation at quadrature points
If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the ...
- 3,554
1
vote
0
answers
61
views
Solutions to a certain Birkhoff-interpolation problem
$\newcommand{\CC}{\mathbb{C}}$
Let for $n > 1$ and $m = n-1$
$$
p = x^n + a_1 x^{n-1} + \cdots + a_m x
$$
be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$.
The ...
- 630
0
votes
1
answer
119
views
Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter
Any map $f \colon \mathbb{R} \to \mathbb{R}$ induces a "composition map"
$$f^\circ\colon \mathbb{R} \times \mathbb{N} \to \mathbb{R},$$
where
$$f^{\circ n}(x) = \underbrace{f \circ \dotsb \circ f}_{...
- 155
2
votes
2
answers
339
views
Do splines preserve monotonicity?
Start with a monotone nonincreasing function and sample it at finite set of points $x_0, ..., x_n$, $x_i<x_{i+1}$ so that $f(x_i)<f(x_{i+1})$. If you approximate $f$ with a linear spline then ...
- 2,175
2
votes
2
answers
296
views
Cubic interpolating spline – number of extremum points
Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of ...
- 3,554
1
vote
1
answer
153
views
inverse interpolation
Given data points $(x_i,y_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y_i=f (x_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ ...
- 2,286