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.
119
questions
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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 ...
0
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0answers
16 views
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 ...
4
votes
1answer
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|_{...
1
vote
0answers
64 views
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\|...
1
vote
0answers
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 $\...
2
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0answers
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 $\...
3
votes
1answer
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 ...
2
votes
1answer
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
$$
A + B = \{a+b : a ...
0
votes
1answer
52 views
Injection of the intersection of Banach spaces on the real interpolation space $(A_0,A_1)_{\theta,\infty}$.
Let $A_0$, $A_1$ be two Banach spaces, both injected continuously in a Hausdorff topological vector space $\mathcal{A}$. Then we can consider the normed spaces $A_0 \cap A_1$ and $A_0 + A_1$ with the ...
2
votes
1answer
24 views
Is there a name for a set of linear functionals $f_i$ that is "sufficiently rich" to uniquely identify a polynomial from the values $f_i p$
I have found this statement in some old lecture notes on interpolation in my lab.
Let $\mathcal{P}_{n}(I)$ be the vector space of polynomials over some open interval $I\subset\mathbb{R}$.
Suppose some ...
0
votes
1answer
30 views
Conditioning of linear systems: Data interpolation
In a given problem, we known that the function
$$ y = ae^x + be^{2x} + ce^{3x}$$
should interpolate a set of data points given by $(x_1, y_1), · · · ,(x_n, y_n)$. However, data is noisy and we have ...
2
votes
0answers
30 views
K- or J-interpolation of uniformly convex Banach spaces
Let $X_0, X_1$ be two real Banach spaces and let $X = (X_0, X_1)_{s, q}$ be the Banach space obtained by the K- or J-method of interpolation. Here, $0 < s < 1$, $1 < q < \infty$.
If one of ...
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0answers
18 views
Showing that a sequence does not have an analytic extension of exponential type
Problem:
Let $H(\delta)$ denote the right half plane given by $Re(z)>-\delta$ for some $0<\delta<1$Suppose I have a sequence, which I will write as a function $\tilde f:\mathbb{N}\to\mathbb{C}...
1
vote
0answers
44 views
Reference for interpolation theory
In the book "Interpolation theory, function spaces, differential operators" by Hans Triebel, I tried to understand the result Theorem~1(a) of section~2.4.2. In particular, I am interested in ...
1
vote
1answer
63 views
Sum of norms over elements is not equal to norm over the whole $\Omega$
In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$:
$$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
0
votes
1answer
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} + ...
4
votes
0answers
105 views
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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0answers
36 views
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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0answers
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 ...
2
votes
1answer
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 ...
2
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2answers
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.
...
0
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0answers
8 views
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 ...
3
votes
1answer
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 ...
1
vote
2answers
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 ...
2
votes
1answer
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}
$$
Let $\phi$ be a non-negative smooth function supported in $\{x\in\mathbb{...
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votes
0answers
59 views
Modified toom-cook algorithm
For these transformation matrices generated using modified toom-cook algorithm , why are they different compared to the following transformation matrices in the picture below ?
We can solve the ...
1
vote
1answer
23 views
toom-cook algorithm matrix G
For this toom-cook algorithm at https://arxiv.org/pdf/1803.10986v1.pdf#page=6 , how do I get the value 4/2 in the matrix G ?
1
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0answers
43 views
What algorithms exist for Polynomial Interpolation of arbitrary degree in two variables.
Is there a way to take a set of points $((x_0, x_1), y)$ of arbitrary length and interpolated them into a polynomial of pre-defined degree.
So we would have a function $y = f(x_0, x_1) = (a_0) + (a_1 ...
2
votes
0answers
221 views
Interchange Of Infimum and Integral
The following question is in regard to a line from the textbook `Interpolation Spaces' by Bergh and Lofstrom. More precisely, it is in regard to the last line on page 110, in the proof of Theorem 5.2....
3
votes
0answers
41 views
Complex interpolation between $L^2$ and $H_0^1$
Let $\Omega \subset \mathbb{R}^n$ be any smooth bounded domain and consider the complex interpolation space $X_\theta=[L^2(\Omega), H_0^1(\Omega)]_\theta$, $0<\theta<1$. I want to show the ...
2
votes
0answers
63 views
Besov or Triebel-Lizorkin spaces versus Lorentz spaces
At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$
$$
\...
2
votes
0answers
23 views
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
$$
\|f\|_A\...
1
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0answers
99 views
Prove the following interpolation inequality
Let $f \in L^1(\mathbf R^N \times \mathbf R^N) \cap L^p(\mathbf R^N \times \mathbf R^N)$. Let $\rho(x) = \int_{\mathbf R^N} f(x,v) dv$. Prove that
\begin{equation}
\|\rho\|_{L^q(\mathbf R^N)} \le C \|...
1
vote
0answers
33 views
Estimate the maximum interpolation error $\tan(\sin(x^3))$
I have proposed to solve this problem but I am already giving up because I cannot find a way out. If someone has come across it and solved it, I appreciate how to face the solution.
I want to ...
2
votes
0answers
24 views
Is there something like an interpolation theorem for Fredholm operators?
Given three Banach spaces $X,Y,Z$ we say that $Z$ is an interpolation space between $X$ and $Y$ if the following holds (Where $B(X)$ stands for the set of bounded operators from $X$ to itself):
$$ A \...
0
votes
0answers
33 views
$T:L^p \to L^p$ , $A$ 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 $T'$ ...
1
vote
0answers
38 views
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 $\Gamma_{i=0}^{n} (u_i) \in \mathbb{R}^{n+1}$ ...
0
votes
1answer
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 $\theta \in [0,1]$. ...
1
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0answers
12 views
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 ...
1
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0answers
90 views
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 $\...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
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 $p$
$$...
2
votes
0answers
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 ...
0
votes
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{...
1
vote
0answers
80 views
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 $...
1
vote
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{...
1
vote
1answer
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 \,\...
3
votes
0answers
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}}(\...
1
vote
0answers
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 $B$, ...
