# Questions tagged [interpolation-theory]

For questions about interpolation of operators. This includes: real and complex interpolation, interpolation estimates, interpolation spaces. Questions about the estimation of a function from a given input should be asked under the [interpolation] tag instead.

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### lagrange interpolation estimate in finite element analysis

It suddenly occurred to me that when we apply the Bramble-Hilbert lemma to estimate the error, we have something like $$||v-\Pi v||_0\leq Ch^{k}|v|_{k}$$ where the ...
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### Help with a Poincare type inequality: Interpolation in Sobolev spaces

I have a bit of an an odd question with this poincare-type inequality. I have been able to come up with something, given below, for $k=0,l=1$ but am unsure what to do and where to go when $p=\infty$. ...
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### What is the best bound for $f'$ knowing bounds on $f$ and $f'''$?

Let $f:\mathbb R\to \mathbb R$ be a three-times differentiable function. Suppose $|f(x)|\le 1$ and $|f'''(x)|\le 3$ for any $x\in \mathbb R$. Show that $|f'(x)|\le 1$ for any $x\in \mathbb R$. The ...
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### Interpolation between $L^p(\mathbb R^n)$ and $\operatorname{BMO}(\mathbb R^n)$

Consider a measurable, real function $f$ defined on $\mathbb R^n$ which belongs to $L^p(\mathbb R^n)\cap \operatorname{BMO}(\mathbb R^n)$, for some $1\leq p<\infty$. An interpolation inequality ...
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### On $\varepsilon-$interpolating and $\varepsilon-$integral preserving functions

Let us consider a function $f \in \mathcal{C}^{2n+1}(\mathbb{R})$ and a fixed $\varepsilon$>0. We say that a function $\phi \in \mathcal{C}^{2n+1}(\mathbb{R})$ is an $\varepsilon-$interpolating ...
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### How would a better mathematician than I complete this half-finished definition of what it means for a curve to be smooth?

Once upon a time, I was taught how to play connect the dots. Some years later, I was given pseudo-code for an algorithm which would compute a polynomial of minimum degree passing through some points. ...
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### What is formulation of Linear Bspline Interpolation ？

In Julia language, Linear interpolation can be performed by specifying the parameters of the function:(http://juliamath.github.io/Interpolations.jl/latest/devdocs/) For example: ...
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### Complex interpolation of weighted $L^p$ spaces

Bergh and Löfstrom's book Interpolation spaces - An introduction gives a short proof sketch (Thm. 5.5.3, p. 120) of the fact that the complex interpolation space $(L^{p_0}(w_0), L^{p_1}(w_1))_{[θ]}$ ...
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### Complex interpolation of spaces defined by operator with bounded imaginary powers

In Corollary 4.6 of this paper https://core.ac.uk/download/pdf/81991766.pdf they seem use a result that if $\Delta : D \to H$ is essentially self adjoint, such that $D \subseteq H$ is a dense subset ...
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### Interpolation theorems

I am trying to understand a result from the Marcinkiewics interpolation theorem. It goes as follows. If $T$ is sublinear (1,1)-weak type and bounded in $L^\infty(R^n)$ with norm 1. I am trying to show ...
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### Why doesn't Lagrange Interpolation work well in this case? [closed]

So I was asked to use Lagrange interpolation to find the an approximation for the population at a given year. ...
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### Elementary interpolation inequality between Lebesgue and Sobolev-Slobodeckij spaces

Let $W^{s, 2}$ for $0 < s < 1$ denote the Sobolev-Slobodeckij spaces on the interval $(0, 1)$ and $L^2$ the Lebesgue space on the same interval. I'm interested in an elementary proof that there ...
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### Polynomial interpolant as $\varepsilon \rightarrow 0$

I need a check on the following exercise: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ a smooth function and consider the data $(0,f(0)), (\epsilon,f(\epsilon))$ and $(1,f(1))$. Let $p(x)$ the ...
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The $L^p$ norm of a function $f$ is given by $$\|f\|_{L^p(\mathbb{R}^d)}=\left(\int_{\mathbb{R}^d}|f(x)|^p\,dx\right)^{1/p}$$ Let $\phi$ be a non-negative smooth function supported in \$\{x\in\mathbb{...