All Questions
14 questions
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
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
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
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 $$|...
- 3,574
3
votes
2
answers
623
views
Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$
I need to approximate $x \ln x$ on $[0,1]$ as a piecewise-linear function. If $P(x)$ is a piecewise-linear approximation, I want to minimize
$$
\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
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
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
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