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 \begin{equation} ||v-\Pi v||_0\leq Ch^{k}|v|_{k} \end{equation} where the ...
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Gagliardo diagram

Let $E_0,E_1$ a compatible couple of normed spaces. For every $x\in E_0+E_1$ we define the set $$\Gamma(x):=\{y=(y_0,y_1)\in\mathbb{R}^2: \exists(x_0,x_1)\in E_0\times E_1, x=x_0+x_1; ||x_i||_{E_i}\...
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Every interpolation pair is uniform for Banach category

I want to prove the following: If we restrict ourselves to the category $\mathcal{B}$(banach spaces), then every interpolation pair $(E,F)$ with respect to the compatible couples $(E_0,E_1),(F_0,F_1)$ ...
Gonzalo de Ulloa's user avatar
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Exercise 8.15 Brezis - Interpolation inequality

I have a problem with this exercise (see the text in the following link). Interpolation like inequality ,Question from Brezis' book exercise 8.15 The link practically solves it. Only one last step ...
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Does there exist $f \in \mathcal{C}^\infty\left([a, b]\right)$ such that $M_n \rightarrow \infty$ but $\left\|f - p_n\right\| \rightarrow 0$?

Let $f \in \mathcal{C}^\infty\left([a, b]\right)$ and $n \in \mathbb{N}$. Let $p_n$ be the Lagrange interpolating polynomial for a partition of equispaced points $a = x_0 < x_1 < \cdots < x_n ...
Cyclotomic Manolo's user avatar
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2 answers
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Gilbarg Trudinger Interpolation inequality in Sobolev spaces

I'm reading theorem 7.27 of Gilbarg and Trudinger's "Elliptic Partial Differential Equations of Second Order" It says Let $u\in W^{k,p}_0(\Omega)$, then for any $\epsilon>0,0<|\beta|&...
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Choosing appropriate interpoleting points

Suppose $f$ is a polynomial of degree $d$ and $B\subset\mathbb{R}$ is an open interval. Then let $A=\{x\in B:|f(x)|<\epsilon\}$ then can we always choose $d+1$ points $x_i$ from the set $A$ i.e $|f(...
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What exactly is the "real interpolation space" $(L^q, W^{2,q})_{1-p^{-1},p}$ for $1<p,q<\infty$?

I came across the notation https://en.wikipedia.org/wiki/Interpolation_space#Real_interpolation the real interpolation space is discussed for Bessel potential spaces. However, I do not see exactly how ...
Keith's user avatar
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Use interpolation inequality to improve strong convergence

I'm considering a sequence $f_{\epsilon}\in L^{1}(\Omega)\cap L^{\infty}(\Omega)$. Then by interpolation inequality, I have for all $1<p<\infty$ with $0<\theta<1$ such that $$\lVert f_{\...
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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 $p=\infty$. ...
ThreeEngineersInATrenchCoat's user avatar
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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 ...
Runyang Wang's user avatar
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operator norm calculation

I have to show that $\Vert{T_{N}(f)}\Vert_{L^p(\mathbb{T})}=1$ $\forall p\in[1,\infty]$ where $T_{N}(f)=\frac{1}{N}\sum_{n=0}^{N-1}f(x+n\alpha)$ with $\alpha\in\mathbb{R}$ and $f$ defined on $\mathbb{...
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Continuity of K function from real interpolation

When I learn "An introduction to Sobolev space and interpolation space" by Luc Tartar, Chapter 22"Real interpolation; K-method", I am confused by the continuity of $K(t;a)$, Let $...
monotone operator's user avatar
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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 ...
Lorenzo Pompili's user avatar
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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: ...
ly w's user avatar
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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 $(L^{p_0}(w_0), L^{p_1}(w_1))_{[θ]}$ ...
Carlos Esparza's user avatar
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Hermite interpolation formula [closed]

Now a very trivial question about this book Ralston: A first course in numerical analysis the Hermite formula example, how it follows $$\frac{-1}{32768}<E(0.60)<\frac{-1}{2^{23}}$$ from this: $...
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Show uniqueness of interpolation constructed with sum of exponentials

I'm trying to solve the following problem: Given $x_0,\ldots,x_n$ distinct real points. Choose a function $$P_n(x)=\sum_{j=0}^n c_j e^{jx}$$ s.t. $$P_n(x_i)=y_i \qquad i=0,\ldots,n $$ Show there's a ...
bobinthebox's user avatar
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Interpolation technique used in numpy

What is the technique used by the numpy interp() function? So using the following points ...
rafaelcb21's user avatar
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Motivation for real interpolation spaces

In the filed of interpolation spaces, given a compatible couple $(X_0,X_1)$ of Banach spaces i.e. $X_0$ and $X_1$ are embedded into a topological vector space, Peetre's $K$ and $J$ functionals are ...
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Analytic extension of $(-1)^n$ satisfying growth condition

Can the function $(-1)^n$, $n=1,2,...$ be extended to an analytic function $f(z)$ defined on the right half complex plane satisfying the growth condition $$|f(x+iy)|\le C e^{Px+A|y|},$$ with $A<\pi$...
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Uniform bound on product of Gamma functions in an article by Jerison and Kenig

Also asked on MathOverflow. I have been trying to read Jerison and Kenig's article Unique continuation and absence of positive eigenvalues for Schrödinger operators, and I am having difficulties ...
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Interpolation of Bochner-Sobolev spaces

Does anyone know how to prove or has an explicit reference for interpolation inequalities of the form $$ \|f\|_{H^{l}(0,T;H^{(1-l)}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^1(\...
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Aronszajn-Gagliardo Theorem: case of exponent $\theta$ (Exercise 2.8.4 from Bergh-Löfström)

I posted this question on MathOverflow but didn't get any answers, so I'm trying here . I'm working on the book "Interpolation Spaces" by Bergh and Löfström. I'm interested in solving the ...
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Favard spaces are complete

What seemed at first glance a straightforward verification had me completely lost, the definition I have of Favard space is: Let $X$ be a Banach space and $(D(A),A)$ be the infinitesimal generator of ...
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3D interpolation function

I've got a 3D figure created using around 30k points and has different regions colored in an specific way according to some unrelated variables that come from a project I'm creating. Taking in ...
Vako Galvan's user avatar
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Intersection of the kernel with the interpolation space

Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow Y’$ linear continuous, such ...
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Restriction of fractional Sobolev "function" of negative order to subset

Assume $ U\subset V\subset \mathbb{R}^n$ are bounded open subsets with smooth boundary. We define $H^{-s}(\Omega)=(H_0^{s}(\Omega))'$ for $s>0$. It is straightforward to show that $\left. v\right|_{...
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Interpolation Banach spaces

I am interested in the following. Given are two Banach spaces over $\mathbb{C}^n$ with respective norms $\|\cdot\|_0$ and $\|\cdot\|_1$. Suppose it holds that $\tfrac{1}{2}\left(\|x+y\|_0^p + \|x-y\|...
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Compact immersions of the bounded variation space $BV(\Omega)$

I have missed a detail attending a lesson, I would like at least some references to fill my gap. We were proving the compact immersion of $BV(\Omega)$ in $L^p(\Omega)$ with $1\leq p<1^{*}$ and $\...
Matteo Aldovardi's user avatar
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Interpolation spaces defined by singular value decomposition

Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is, $$ Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n $$ Since $\...
Yidong Luo's user avatar
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1 answer
107 views

Prove that $(A_0,A_1)_{\theta,q}$ is Banach.

Let $A_0$, $A_1$ be two Banach spaces, both embedded continuously in a Hausdorff topological vector space $\mathcal{A}$. Then we can consider the normed spaces $A_0 \cap A_1$ and $A_0 + A_1$ with the ...
Serjunpe's user avatar
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1 answer
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If $A$ and $B$ are Banach spaces, then $A+B$ is Banach

Let $A$ and $B$ be two Banach spaces continuously contained in a Hausdorff topological space $\mathcal{A}$. Then you can define the spaces $A \cap B$ and $A + B$, the latter being $$ A + B = \{a+b : a ...
Juan Redondo's user avatar
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Injection of the intersection of Banach spaces on the real interpolation space $(A_0,A_1)_{\theta,\infty}$.

Let $A_0$, $A_1$ be two Banach spaces, both injected continuously in a Hausdorff topological vector space $\mathcal{A}$. Then we can consider the normed spaces $A_0 \cap A_1$ and $A_0 + A_1$ with the ...
Serjunpe's user avatar
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Is there a name for a set of linear functionals $f_i$ that is "sufficiently rich" to uniquely identify a polynomial from the values $f_i p$

I have found this statement in some old lecture notes on interpolation in my lab. Let $\mathcal{P}_{n}(I)$ be the vector space of polynomials over some open interval $I\subset\mathbb{R}$. Suppose some ...
iamnotacrackpot's user avatar
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Conditioning of linear systems: Data interpolation

In a given problem, we known that the function $$ y = ae^x + be^{2x} + ce^{3x}$$ should interpolate a set of data points given by $(x_1, y_1), · · · ,(x_n, y_n)$. However, data is noisy and we have ...
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K- or J-interpolation of uniformly convex Banach spaces

Let $X_0, X_1$ be two real Banach spaces and let $X = (X_0, X_1)_{s, q}$ be the Banach space obtained by the K- or J-method of interpolation. Here, $0 < s < 1$, $1 < q < \infty$. If one of ...
Desura's user avatar
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Reference for interpolation theory

In the book "Interpolation theory, function spaces, differential operators" by Hans Triebel, I tried to understand the result Theorem~1(a) of section~2.4.2. In particular, I am interested in ...
kaushik's user avatar
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Sum of norms over elements is not equal to norm over the whole $\Omega$

In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$: $$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
bobinthebox's user avatar
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Interpolation of two inequalities

Let $X_0,X_1,Y_0,Y_1$ be Banach spaces and $T \colon (X_0+X_1) \times (Y_0+Y_1) \to \mathbb{R}$ such that $$| T(f,g) | \le \| f\|_{X_0} \|g\|_{Y_0}$$ and $$| T(f,g) | \le \| f\|_{X_1} \|g\|_{Y_1} + ...
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On a proof of Riesz-Thorin with the tensor power trick

In this post Terence Tao exposes the tensor power trick, and leaves as an exercise to use this tecnique to prove Riesz-Thorin. This is what I managed to do (I will use the same notation as here): ...
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Asking abount Finite element interpolation

At first, sorry if my question is a stupid thing. It is about mapping a function into semi-discrete space. Let's see, for the basis function $\{\phi_i(x)\}_{i = 1}^n $, the projection of function $u(x,...
Panasun's user avatar
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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 ...
Holmes's user avatar
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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 ...
user593295's user avatar
2 votes
2 answers
188 views

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. ...
KD10001's user avatar
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3 votes
1 answer
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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 ...
Three.OneFour's user avatar
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2 answers
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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 ...
andereBen's user avatar
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Inequality involving the Besov norm

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{...
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