# 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.

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### Proof of the equivalence between Triebel Spaces and Bessel Potential

I've encountered a question regarding the relationship between Triebel-Lizorkin spaces and Bessel potential spaces. Specifically, I understand that $F^s_{p,2} = H^{s,p}$, for $p \in (1,\infty)$. ...
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### Error on spline interpolation for integration

I am interested in studying the integral $$I \equiv \int_m^M dx \, f(x) d(x)$$ where $f(x)$ is an analitcally known function (mildly oscillating), while $d(x)$ is sampled at a finite, discrete set ...
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### Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?

Let two integers $L\leq N$, a sequence of $L+1$ distinct real numbers $\alpha_0,\dots,\alpha_L$, and integers $k_0,\dots,k_L\in\mathbb N$ such that $N=L+\sum_{i=0}^Lk_i$. I noticed that the two ...
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### Interpolation polynomials with constraints

Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
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### Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3}$ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$

Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$. I am wondering if it there is a constant $C > 0$ such that for all ...
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### How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
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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 $a=t_0<t_1<\cdots<t_n=b$,...
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### 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 $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
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### 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), ...
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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\...