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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.

7
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1answer
202 views

Borderline case of interpolation of Banach spaces

Let $B \subset A$ be Banach spaces with a continuous embedding. Is the inequality $$ \|b\|_B \leq C \sup_{t > 0} \inf_{\tilde{b} \in B} \{ \|b - \tilde{b}\|_B + t \|\tilde{b}\|_A \} \quad \forall ...
0
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1answer
40 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}{...
0
votes
1answer
27 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 ...
0
votes
1answer
108 views

Polynomial interpolation of Bessel function

I used polynomial interpolation of degree 5 to approximate one 0 of the Bessel function $J_0$. I used the following data: With the data above, I computed a polynomial using the images of the Bessel ...
0
votes
1answer
26 views

What is the general descriptor of this class of numerical interpolation schemes?

I have been working on developing some n-dimensional interpolation code for a project of mine, and have developed something that works well. But I am wondering how it would be categorized according to ...
0
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1answer
32 views

Boundedness in $C^{0,\sigma}(I,H^{l-2\sigma}(\mathbb{R}^n))$ by interpolation

I am reading the book "Partial Differenatial Equations III: Nonlinear Equations" by M. Taylor (Google Books) and I am stuck on a claim he makes at page 379. I will reformulate it here. Suppose we ...
6
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0answers
186 views

Relating primal and dual characterization of an (interpolation) norm on $\ell_1+\ell_2$

For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ a'+a''=a\...
3
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0answers
23 views

How to find the minimun of an interpolation polynomial by the Brent's Method?

If we determinate the second-degree interpolation polynomial that pass through the points $(a,f(a)),(b,f(b))$ and $(c,f(c))$ as: $$P_2(x)=f(a)+f[a,b](x-a) +f[a,b,c](x-a)(x-b)$$ Where the values of $...
2
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0answers
39 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 $T$ ...
2
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0answers
23 views

Counterexample in real interpolation

Can we find a linear operator $T$ that maps $L^1$ to $L^{2,\infty}$ and $L^2$ to $L^{2,\infty}$ such that $T$ does not map any $L^p$, $1<p<2$ to $L^{2,\infty}$? Can we find a linear operator $T$ ...
2
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0answers
119 views

Proving the Marcinkiewicz Interpolation Theorem

I am trying to prove a form of the Marcinkiewicz Interpolation Theorem. $T$ mapping a measurable function to a measurable function is sublinear if $$|T(f_1 + f_2)(x)| \leq |T(f_1)(x)| + |T(f_2)(x)|$...
2
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0answers
82 views

What is the smallest polynomial with almost linear condition?

What is the smallest degree univariate polynomial with $f(0)=0$, $f(1)\in(0,0.5)$ and $f(x)={x-1}+\epsilon$ where $\epsilon\in(-0.5,0.5)$ at every $x\in\{2,\dots,a-1,a\}$? Can we have $O((\log a)^c)$ ...
2
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0answers
105 views

Weighted vector norms and interpolation

Let $\mu$ be a probability vector of $\mathbb{R}^n.$ Then we can define the weighted $\ell^p$ $(1 \leq p < \infty)$ norms by $$ \|x\|_{p,\mu} = \left(\sum_{i=1}^n |x_i|^p \mu_i\right)^{1/p},$$ ...
2
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0answers
34 views

Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
1
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0answers
19 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 ...
1
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0answers
25 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}^...
1
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0answers
32 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}\...
1
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0answers
59 views

Choose support for polynomial interpolation

I have the following table, where $t$ is defined in minutes and $v(t)$ in $m / s$: \begin{array}{|c|c|c|c|} \hline t& 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline v(t) & ...
1
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0answers
81 views

Use 1-degree Chebyshev polynom to approximate $\cos(x)$ and calculate the error

The task is to give for $\cos(x)$ the nodes of the interpolation polynom of degree 1 that approximates the function on $[-\pi,\pi]$ the best as well as the related error. I want to solve this task ...
1
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0answers
367 views

Linear interpolation of over time of standard deviation measurements

I have backscatter (radar image value) measurements of corn fields taken at multiple points along the growing season. I can estimate the expected backscatter value of corn by plotting the mean ...
1
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0answers
13 views

Reference for complex interpolation for the product of two normed space

Let $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ and $(X_i,Y_i)$ are interpolation couple. Then it seems obvious that $$ [X,Y]_{\theta}=[X_1,Y_1]_\theta \times [X_2,Y_2]_\theta $$ for every $\theta\in (...
1
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0answers
68 views

Approximation with stability constant 1

I am looking for an approximation method for a typical 1-dimensional approximation problem, that has stability constant 1. The setting We have an interval $[a,b]$ an unknown function $f$ on $[a,b]$ ...
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0answers
41 views

$(X,Y)$ interpolation couple, then $(X^\prime,Y^\prime)$ interpolation couple

Let $(X,Y)$ be an interpolation couple, i.e. there exists a Hausdorff topological vector space $U$ s.t. both $X$ and $Y$ embed continuously into $U$. We define $X\cap Y$ and $X+Y$ in $U$ and turn them ...
1
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0answers
155 views

How to prove that a quadratic function is equal to quadratic spline?

I need to prove that a quadratic spline $S$ and a quadratic function $f$ are equal. We are given points $([X_1,f(X_1)],.....,[X_n,f(X_n)])$ and the boundary condition that the first derivatives of $...
1
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0answers
800 views

Tricubic Interpolation

I am currently writing a plugin for 3D analysis software and I am working with a data grid where certain values are stored at XYZ coordinates, and I need to find an estimated value of a point that ...
1
vote
0answers
42 views

Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
1
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0answers
60 views

Property of intersections of Bochner spaces

My question: Assume I have a function $ u \in H^2(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega))$. Now I want to bound the gradient of $u$. Can I deduce that $u \in H^1(0,T;H^1(\Omega))$ and under which ...
0
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0answers
27 views

Monotonic linear interpolation of monotonic data on a scattered grid

Let $x_1,\dots,x_n\in[0,1]^d$ and $y_1,\dots,y_n\in[0,1]$ satisfy: $$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$ I seek to find a continuous function $f:[0,1]^d\...
0
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0answers
15 views

Hello. How to prove theorem about relationship between the interpolation operator norm and the constant of Lebesgue:

A norm of the linear operator $P_n$ can be defined by $||P_n|| = max_{f \in [a,b]} \frac{||P_n f||}{||f||} $ where on the right-hand side one takes any convenient norm for functions. Taking the $...
0
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0answers
60 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 ...
0
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0answers
45 views

Marcinkiewicz Interpolation for exponents $p_0, q_0, p_1, q_1 \in (0, \infty]$.

I would like to try and extend the Marcinkiewicz Theorem in Folland (Theorem 6.28) using a similar argument in the text. The theorem I am after is as follows: Let $p_0, p_1, q_0, q_1 \in (0, \infty]$...
0
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0answers
45 views

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 ...
0
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0answers
66 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 ...
0
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0answers
90 views

Linear Interpolation Error estimation

Suppose we have applied trilinear interpolation technique over grid points in the space $A\times B \times C \times D={(x,y,z)|x\in A, y\in B,z \in C}$} . The interpolant, which is piecewise function, ...
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0answers
11 views

how to estimate maximum of Lebesgue function of arbitrary nodes?

Denote $S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0}<x_{n1}<\cdots<x_{nn}\leq b\right\}$ with $-\infty<a<b<+\infty$ and $n\geq 1$. For any $x_n\in S_n$, define the ...
0
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0answers
23 views

Romberg's algorithm for integration

what does the row and the column in Romberg's algorithm for integration indicate? I assume that the column represents the integration value with smaller intervals considered for calculating the ...
0
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0answers
39 views

Interpolation polynomial for $f=|x|$.

Why the interpolation polynomials for $f=|x|,x\in[-1,1]$ will oscillate near the endpoint of $[-1,1]$ as $n$ increases? I know that one explanation of the Runge's phenomenon is that the interpolation ...
0
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0answers
47 views

Question above proof of Marcinkiewicz Interpolation Theorem

I am reading through the proof in Tao's lecture notes and I have a question about justifying a "without loss of generality" statement. Statement: Let $T$ be a sublinear operator taking functions on $(...
0
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0answers
40 views

What is the Lagrange Interpolation polynomial of $1/{(x-1)}$?

So what is the value of $$L[x_0,x_1,\ldots,x_n; \; 1/{(x-1)}]=\text{?}$$ I tried writing a Mathematica program to compute it, but I couldn't figure it out...
0
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0answers
342 views

How can Newton's Divided Difference Interpolation accuracy be improved?

Are there any modifications to Newton's divided difference interpolation that can be made in order to improve the accuracy of the value given, specifically for exponential functions such as $y=e^{-x}-...
0
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0answers
120 views

How to determine the nodes of a interpolating polynomial?

Determine the nodes of the first degree interpolating polynomial which approximate $f(x) = cos(x)$ the best on $[-\pi,\pi]$ in the uniform norm. What is the interpolation error? I think I need a ...
0
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0answers
59 views

Interpolation of a subspace of codimension one

I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we ...