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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Interpolating a function S(a,b) on a path b(a) to find S(b(a))

I feel this should have a simple answer but I am unable to understand, hence I am posting it here. I have a function S(t,k) where k and t are two independent variables. So this function is a surface ...
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1answer
67 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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How do I find Δ^n (1/n), with h being the interval of difference?

I have no clue about solving this. Generally the difference operators are applied on a function. But in this case, it is being applied on a constant. How should I proceed ?
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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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1answer
37 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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44 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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1answer
16 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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35 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 ...
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76 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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36 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 ...
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Failing to demonstrate a minimization for 4-neighbors interpolation

I need to verify a minimization problem in the domain of linear interpolations. In particular, the formula I'm trying to demonstrate is reported in this paper, in the section "Upsampled FMM". Fast ...
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45 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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75 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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24 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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20 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 \...
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30 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 $T'$ ...
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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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1answer
41 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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basis representation of cubic spline interpolant

If i do cubic spline interpolation (periodic), lets say on $n$ pairs $(x,f(x))$, i will get $n-1$ piecewise polynomials $$s_i(x) = a_i(x-x_i)^3+b_i(x-x_i)^2+c_i(x-x_i)+d_i \;\; \text{on} \;\;[x_i,x_{...
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1answer
49 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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27 views

Definition of compatible couple of Banach spaces

Let $X$, $Y$ be Banach spaces. In many books on interpolation theory the following definition is found: The pair $(X,Y)$ is said to form a compatible couple of Banach spaces if there is a ...
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50 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
60 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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112 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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29 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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58 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 $...
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1answer
41 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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1answer
47 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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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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24 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 $B$, ...
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28 views

Interpolation between log and polynomials using Riesz-Thorin

I consider two $L^1$ weighted spaces, with $m_1(x) = e + x, m_0 = \ln(e+x)$. It is known that the Riesz-Thorin interpolation theorem holds for $L^p$-weighted space. I have an operator $T: L^1(m_1) \to ...
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1answer
289 views

Interpolation inequality for Holder continuous functions.

Let $\Omega$ be a bounded open connected set in $\mathbb{R}^n$ with $C^1$ boundary and let $0<\alpha<1$. Then there exists a real number $\sigma_0>0$ and a dimensional constant $C>0$ such ...
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1answer
137 views

What is the geometric meaning of this null-determinant?

While reading about interpolation I came across the following equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton follow by using the ...
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67 views

Question about linear splines.

I have a question about linear splines. A computer package I am using has the option to use linear splines, but there is no user manual I can find. I am having problems with the kind of inputs the ...
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1answer
90 views

How to derive the Newton polynomial with LA

Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in newtonian form: $$N(x)=\sum _{j=0}^k[y_0,\ldots ,...
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1answer
128 views

Formula for coefficients of interpolation polynomial

(Question) What is the (conventional) formula for coefficients of interpolation polynomial? Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,...
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Prove $ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $

Prove: $ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $ for $ 0 \leq x \leq 1 $ Using the fact that $|| (u-u_h)' || \leq \underset{0 \leq y \leq 1}{\max} |u''(y)| $ and ...
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1answer
60 views

Newton-Gregory interpolation with divided differences calculations for new extra interpolation point

So lets suppose that we have the following $x_0=-1, y_0=2$ $x_1= 0, y_1=1$ $x_2=1, y_2=2$ $x_3=3, y_3=10$ and we know that all the above $x_i,y_i$ belong to $p_2(x)=x^2+1$ , and we want to add ...
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59 views

Reference for compact embedding for Holder space on $\mathbb{R}^n$

Suppose $0<\alpha<\beta$, and $\Omega$ is a bounded subset of $\mathbb{R}^n$. Then the Holder space $C^{\beta}(\Omega)$ is compactly embedded into $C^{\alpha}(\Omega)$. But if $\Omega=\mathbb{R}^...
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177 views

What is the basic difference between interpolation & inference?

In mathematics we've studied interpolation as predicting the structure of a function from it's given finitely many values from which consequently we use to construct certain function which nearly ...
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1answer
42 views

For $f(x) = \tan(\pi \cdot x)$, find the Interpolation $Q(x) = b_0 + b_1 x + b_2 \frac{1}{x-\frac{1}{2}}$

Progress so far: In a previous task, I determined a polynomial interpolation using a system of linear equations. The data points to be used were $(0, f(0)), (\frac{1}{6}, f(\frac{1}{6})), (\frac{1}{...
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1answer
36 views

Interpolation- Lagrange polynomial

Let $x_0,x_1,...,x_n$ will be different real numbers. Show, that: $f[x_0,x_1,...,x_n]=\sum_{i=0}^m\frac{f(x_i)}{\Phi '(x)}$ where $\Phi (x)=(x-x_0)(x-x_1)...(x-x_m)$ So, I have some problems.How to ...
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117 views

Interpolation with a new point of $f'$

Background: (Lagrange Interpolation) Let $f\in C^{n+1}([a,b])$ and $x_0,...,x_n\in[a,b]$. If they are different there is a unique $p_n\in\mathcal{P}_n$ such that $p_n(x_i)=f(x_i)$. Also, we have ...
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2answers
73 views

Approximation using Lagrange Interpolation

I am aware of the formula of this method. However, is it true that the method produces more accurate polynomial when the $x$ points are closer to each other? if so or not, why? Moreover, If I am ...
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1answer
55 views

Construction of real interpolation Banach space

Let $(A,\| \cdot \|_A)$, $(B,\| \cdot \|_B)$ and $(E,\| \cdot \|_E)$ real Banach spaces such that $A\subseteq E$ and $B\subseteq E$ with continuous injections. Let $0 < \theta < 1$. For $x\in E$ ...
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107 views

Understanding multilinear interpolation

Do I understand the interpolation of bounded multilinear operators on Banach spaces correct ? For simplicity, let us Consider bilinear operators on $L^p$ spaces. Suppose the bilinear operator $T$ ...
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42 views

Interpolation theory

Consider the interpolation space $Z=(X,Y)_{\theta,p}$, in the case $Y\subseteq X$ do we have that the following norm: $x\longrightarrow\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\...