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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4 views

1D Interpolating subdivision for lifting schemes

I am looking into wavelet lifting methods first introduced by Swelden, and explained in this paper: Build your own wavelets at home. In this paper (in chapter 2 specifically), they discuss ...
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Compare error bound to theoretical error bound

$P_3(x) = 3x^3 +3$ is an interpolating Lagrange polynomial for $\widetilde{P(x)} = x^4-2x^3-x^2+2x$ generated from the data points $$(-1, 0), (0, 3), (1, 6), (2, 27)$$ $\widetilde{P(x)}$ is itself a ...
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39 views

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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15 views

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 $\...
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19 views

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 $\...
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70 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 ...
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94 views

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 ...
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52 views

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 ...
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24 views

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 ...
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30 views

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 ...
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Showing that a sequence does not have an analytic extension of exponential type

Problem: Let $H(\delta)$ denote the right half plane given by $Re(z)>-\delta$ for some $0<\delta<1$Suppose I have a sequence, which I will write as a function $\tilde f:\mathbb{N}\to\mathbb{C}...
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44 views

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 ...
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63 views

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}...
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29 views

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,...
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61 views

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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64 views

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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81 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. ...
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Riesz Interpolation for Bourgain space, and a question about Schwartz space

1)If I have an operator $T$ s.t $||Tv||_{X_{-s,0}} \le c_1 ||v||_{X_{-s,1-b}}$ and $||Tv||_{X_{-s,1-b}} \le c_2 ||v||_{X_{-s,1-b}}$. Where $X_{s,b}$ is the Bourgain space .Then how to apply Riesz ...
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142 views

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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53 views

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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57 views

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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59 views

Modified toom-cook algorithm

For these transformation matrices generated using modified toom-cook algorithm , why are they different compared to the following transformation matrices in the picture below ? We can solve the ...
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23 views

toom-cook algorithm matrix G

For this toom-cook algorithm at https://arxiv.org/pdf/1803.10986v1.pdf#page=6 , how do I get the value 4/2 in the matrix G ?
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What algorithms exist for Polynomial Interpolation of arbitrary degree in two variables.

Is there a way to take a set of points $((x_0, x_1), y)$ of arbitrary length and interpolated them into a polynomial of pre-defined degree. So we would have a function $y = f(x_0, x_1) = (a_0) + (a_1 ...
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221 views

Interchange Of Infimum and Integral

The following question is in regard to a line from the textbook `Interpolation Spaces' by Bergh and Lofstrom. More precisely, it is in regard to the last line on page 110, in the proof of Theorem 5.2....
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Complex interpolation between $L^2$ and $H_0^1$

Let $\Omega \subset \mathbb{R}^n$ be any smooth bounded domain and consider the complex interpolation space $X_\theta=[L^2(\Omega), H_0^1(\Omega)]_\theta$, $0<\theta<1$. I want to show the ...
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63 views

Besov or Triebel-Lizorkin spaces versus Lorentz spaces

At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$ $$ \...
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Equivalence between norms on interpolation spaces

Let $X_1,X_2$ be disjoint (except for the $0$ vector) Banach subspaces of a topological vector space $X$, i.e.: $$ X_1\cap X_2 = \{0\}. $$ Define a norms on the space $ X_1 + X_2 $ by $$ \|f\|_A\...
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99 views

Prove the following interpolation inequality

Let $f \in L^1(\mathbf R^N \times \mathbf R^N) \cap L^p(\mathbf R^N \times \mathbf R^N)$. Let $\rho(x) = \int_{\mathbf R^N} f(x,v) dv$. Prove that \begin{equation} \|\rho\|_{L^q(\mathbf R^N)} \le C \|...
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33 views

Estimate the maximum interpolation error $\tan(\sin(x^3))$

I have proposed to solve this problem but I am already giving up because I cannot find a way out. If someone has come across it and solved it, I appreciate how to face the solution. I want to ...
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Is there something like an interpolation theorem for Fredholm operators?

Given three Banach spaces $X,Y,Z$ we say that $Z$ is an interpolation space between $X$ and $Y$ if the following holds (Where $B(X)$ stands for the set of bounded operators from $X$ to itself): $$ A \...
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$T:L^p \to L^p$ , $A$ a dense subset of $L^p$ , $T|_A $ is bounded and has an extension $T'$ . Can we show that $T=T'$ on $L^p$?

$(a)$ Suppose $T:L^p \to L^p$ is a linear operator(might not be bounded) , $A$ a dense subset of $L^p$ , $T|_A $ is a bounded operator , then $T|_A$ can be extend to all of $L^p$ which we call it $T'$ ...
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Interpolation of polynome - is degree needed?

I'm doing my project from numerical methods in math and I'm supposed to create the following program: Inputs: $n \in [0,\infty)$ a sequence $\Gamma_{i=0}^{n} (u_i) \in \mathbb{R}^{n+1}$ ...
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91 views

Complex Interpolation and Intersection

Does it hold $$[X \cap Y, X \cap Z]_\theta = X \cap [Y,Z]_\theta$$ where $X,Y,Z$ are suitable spaces and $[\cdot,\cdot]_\theta $ denotes the complex interpolation functor of order $\theta \in [0,1]$. ...
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Analytic Interpolations

I understand that for any sequence of reals there is an analytic interpolation, and for any positive sequence there is an analytic interpolation that is positive. I am wondering if there is a ...
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Radial Sobolev embedding

I'm trying to prove the following inequality $$\left\||x|f\right\|_{L^\infty(\mathbb{R}^3)}\lesssim \left\|f\right\|_{H^1(\mathbb{R}^3)},$$ for every radial function $f\in H^1(\mathbb{R}^3)$, where $\...
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146 views

Hardy-Littlewood-Sobolev inequality using generalised Young inequality in Lorenz spaces

I want to prove that $$\left| \left| \frac{1}{|x|^a} \ast f \right| \right|_q \lesssim ||f||_p$$ with $1 < p < q < \infty$ and $a= n \left(1 + \frac{1}{q}- \frac{1}{p} \right)$ using a ...
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85 views

Riesz-Thorin theorem in finite-dimensional vector space

In all formulation of Riesz-Thorin complex interpolation theorem I saw (e.g. here), it always involves function spaces like $L^p$. I would like to know if this theorem can be applied to linear ...
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1answer
147 views

Collocation points and trapezoidal rule

I want to construct the classical trapezoidal rule by using a collocation method. I am in the time interval $[t_n, t_n + \tau]$. I know that I have the following conditions for the polynomial $p$ $$...
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159 views

3D interpolation of a signed distance function

I'm trying to find an interpolation for a continuous real function $f:\Bbb [0,1]^3\rightarrow \Bbb R$ from the values it takes at the eight points $\{0,1\}^3$ (the vertices of a unit cube); I have the ...
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1answer
36 views

Error estimation of a Taylor polynomial

Suppose $y : \mathbb{R} \rightarrow \mathbb{R}$ is a signal admitting a Taylor series expansion around zero (Maclaurin series) \begin{equation} y(t) = \sum_{n=0}^\infty \frac{y^{(n)}(0)}{n!}t^n \end{...
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Intuition for Polynomial Interpolation Error

Math StackExchange, Wikipedia and Proof Wiki all offer roughly similar proofs of the error formula for polynomial interpolation of a $n+1$ differentiable function $f(x)$ with a $n$-degree polynomial $...
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1answer
56 views

Interpolation type theorem for multiplication operator on measure space. Is it trivial?

In my notes I found the following result without proof or reference: Let $(\mathcal{M},\mu)$ be a Borel measure space, with $\mu$ positive and $\sigma$-finite. Let $g,h : \mathcal{M} \mapsto \mathbb{...
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60 views

Meromorphic interpolation of number-theoretic functions

A meromorphic interpolation of the harmonic numbers is given by \begin{align} H_n &= \sum_{k=1}^n \frac{1}{k} \\ &= \sum_{k=0}^{n-1} \frac{1}{k+1} \\ &= \sum_{k=0}^{n-1} \int_0^1 x^k \,\...
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90 views

Comparision of real and complex operator norms in Riesz-Thorin theorem

This is an exercise from Dirk Werner's "Funktionalanalysis" (Aufgabe II.5.4): Let $(\Omega,\Sigma, \mu)$ be some $\sigma$-finite measure space. Denote by $L^p_{\mathbb{R}}(\mu)$ and $L^p_{\mathbb{C}}(\...
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27 views

How to interpolate under specific conditions using FFT?

I am solving a very specific problem where I need to apply FFT to interpolate. I am given two sets of distinct points $A = (a_{1}, a_{2}, ... , a_{n})$ and $B = (b_{1}, b_{2}, ..., b_{n})$. Using $B$, ...