All Questions
7 questions with no upvoted or accepted answers
3
votes
0
answers
38
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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:
$$
...
- 5,405
3
votes
0
answers
118
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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
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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
57
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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 &...
- 380
1
vote
0
answers
58
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$\mathbb{P}_1$-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
0
answers
83
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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
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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