All Questions
Tagged with approximation-theory interpolation
38 questions
22
votes
2
answers
652
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 ...
- 1,209
8
votes
2
answers
644
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 $...
- 1,209
6
votes
2
answers
976
views
Divergence of the Lagrange interpolation on the Chebyshev nodes
Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous function $f$ such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is an interpolation polynomial ...
- 109
6
votes
1
answer
640
views
Maximum of a B-spline
Given $p+2$ nondecreasing (and not all identical) knots $t_0, \ldots, t_{p+1}$ on the real line, consider the normalized B-spline of degree $p$ defined over these knots.
We know that the B-spline is ...
- 278
6
votes
1
answer
2k
views
Multivariate polynomial interpolations
I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...
- 195
5
votes
1
answer
188
views
Do interpolation nodes have to be dense?
Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval.
For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates $f$...
- 85
5
votes
1
answer
203
views
How to choose contour for rational approximation
Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\...
- 243
4
votes
1
answer
1k
views
Variational proof for minimum curvature of cubic splines
Background: Given an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)\in C^2([a,b])$ is a piecewise cubic polynomial on each subinterval $(x_i, x_{i+1})$.
Given a ...
- 3,574
4
votes
1
answer
747
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 ...
- 515
4
votes
1
answer
151
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 $...
- 1,209
4
votes
1
answer
469
views
Polynomial interpolants in quadrature points and L2 convergence spectral rate
We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|...
- 3,574
3
votes
2
answers
622
views
Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$
I need to approximate $x \ln x$ on $[0,1]$ as a piecewise-linear function. If $P(x)$ is a piecewise-linear approximation, I want to minimize
$$
\max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P....
- 367
3
votes
1
answer
1k
views
Relation between Chebyshev Interpolation and Expansion
I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
Pointwise Lagrange ...
- 178
3
votes
0
answers
38
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:
$$
...
- 5,405
3
votes
0
answers
118
views
Hardness results for approximating Hölder continuous functions
Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems (Daubechies & Lagarias, SIAM J. Math. Anal. 22 (1991) ...
- 221
3
votes
0
answers
216
views
The $L_\infty$ norm of the derivative of the $L_2$ spline projector
A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
- 151
2
votes
2
answers
315
views
Cubic interpolating spline – number of extremum points
Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of ...
- 3,574
2
votes
1
answer
312
views
Optimal $L^2$ bounds of cubic spline interpolation
Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
- 3,574
2
votes
1
answer
177
views
For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often
Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
- 1,209
2
votes
1
answer
230
views
Smallest degree of approximating polynomial
Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$.
Let $\epsilon\in[\frac{1}2,1)$.
Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ }f(S_1)\subseteq[1-\epsilon,1+...
- 13.9k
2
votes
1
answer
737
views
Interpolation Operator Bounded in Sobolev Norm
Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$,
$$|u|_{W^{...
- 2,286
2
votes
0
answers
124
views
Error bounds for spline interpolation. Hall and Meyer's conjecture
Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
- 151
2
votes
0
answers
197
views
How to optimally choose points for multivariable Hermite interpolation?
I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input.
I would like to create interpolation polynomial for it.
The computation of $f$ is ...
- 355
2
votes
0
answers
124
views
Greedy interpolation of functions
Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation:
Set $r_0 = f$.
for $k = 0,1,\ldots,$
Find the location of the global ...
- 3,217
2
votes
0
answers
296
views
Rational interpolation: Error bounds for coefficients
The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...
- 656
1
vote
1
answer
165
views
inverse interpolation
Given data points $(x_i,y_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y_i=f (x_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ ...
- 2,286
1
vote
1
answer
931
views
Interpolation of a series of data points via Chebyshev approximation?
first of all: english is not my native language, so there might be differences between what I meant and what you understood. Sorry for that in advance.
As a research project, I try to comprehend and ...
- 13
1
vote
0
answers
56
views
Error bounds for Sobolev space norm approximation on a finite grid
Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space,
\begin{multline}
f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
- 380
1
vote
0
answers
58
views
$\mathbb{P}_1$-finite element as convolution of $\mathbb{P}_0$-finite element
For a vector $\vec{u}\in\mathbb{R}^N$ let's denote $\pi_N\left(\vec{u}\right)$ the unique piecwise linear and $1$-periodic function matching the components of $\vec{u}$ on the discretization $x_k = \...
- 3,425
1
vote
0
answers
35
views
Mismatching degrees and # derivatives in polynomial interpolation error formula
It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
- 243
1
vote
0
answers
99
views
Approximation to continuous functions over an closed interval
Let $$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
1
vote
0
answers
49
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 $...
- 1,209
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 ...
- 1,209
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 ...
- 2,772
0
votes
1
answer
488
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
83
views
Interpolating points and orthogonal polynomials on varying intervals
In general, a Lebesgue-Stieltjes integral, $\int (\cdot) \, d \alpha(x)$, that defines an inner product on the space of polynomials establishes a notion of orthogonality.
Suppose we have a sequence of ...
- 13
0
votes
0
answers
267
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 $a=t_0<t_1<\cdots<t_n=b$,...
- 283
0
votes
1
answer
125
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