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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.

22
votes
4answers
521 views

What is the goal of harmonic analysis?

I am taking a basic course in harmonic analysis right now. Going into there I thought it was about generalizing the idea of the Fourier transformation/fourier series: Finding an alternative ...
7
votes
2answers
398 views

Analytic “Lagrange” interpolation for a countably infinite set of points?

Suppose I have a finite set of points on the real plane, and I want to find the univariate polynomial interpolating all of them. Lagrange interpolation gives me the least-degree polynomial going ...
7
votes
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 ...
6
votes
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\...
5
votes
3answers
110 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 ...
4
votes
2answers
89 views

Inequality of p-norm for Hardy-Littlewood maximum function

I have a question that let $1<p<\infty$, and f $\in L^{p}(\mathbb{R}^{n})$ . I want to prove that $\|Mf\|_{p}\leq2(3^{n}p')^{\frac{1}{p}}\|f\|_{p}$ whereas p' is given by $\frac{1}{p'}+\frac{1}{...
4
votes
2answers
255 views

Who knows this formula for polynomial interpolation?

For my high school math project I studied polynomial interpolation: given a set of points $(x_0,y_0),...,(x_n,y_n)$, find the polynomial of degree $n$ that passes through all points. The solutions by ...
4
votes
1answer
470 views

Real vs Complex Interpolation

The two major classical interpolation theorems in analysis are Riesz-Thorin Theorem (complex method) and Marcinkiewicz Theorem (real method). One can see the statements of the theorems and realize ...
3
votes
2answers
350 views

Lagrange linear, quadratic, and cubic interpolations maximum interpolation error functions comparison

Using Theorem 1, calculate that the maximum interpolation error that is bounded for linear, quadratic, and cubic interpolations. Then compare the found error to the bounds given by Theorem 2. The ...
3
votes
1answer
122 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 ...
3
votes
1answer
59 views

An elementary (interpolation) inequality?

I see the following inequality from a paper $\| f\|_{L^1(\mathbb{R})}^2 \leq C\| xf\|_{L^2(\mathbb{R})} \| f\|_{L^2(\mathbb{R})} $. Although the authors say it's elementary. I try some hours and can'...
3
votes
1answer
528 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
3
votes
1answer
58 views

How to prove $\|P|X|^\theta\|_{p/\theta} \le \|P|X|\|_p^{\theta}$

Let $ 0<p\le \theta<1$. Let $X$ be a self-adjoint bounded linear operator on a Hilbert space $H$ and $P$ is a projection on $H$. Why do we have $$\|P|X|^\theta\|_{p/\theta} \le \|P|X|\|_p^{\...
3
votes
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 $...
3
votes
0answers
44 views

Example of Holmested formula for real interpolation spaces [closed]

Can anybody give me an example of Holmested formula for real interpolation spaces.
2
votes
1answer
196 views

Applying the complex interpolation method to $L^p$ spaces

I am currently reading about the abstract complex interpolation method, which is a generalization of the Riesz-Thorin theorem. I have been trying to see how to apply it to $L^p$ spaces. My question ...
2
votes
1answer
121 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 ...
2
votes
2answers
44 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 ...
2
votes
1answer
110 views

Weak convergence in intersection of Bochner spaces

First off, my question has some similarities to Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces (on MathOverflow). To be specific, we have a Gelfand triple $V \subset H \...
2
votes
1answer
107 views

Trace method is interpolation method

Let $X,Y$ be Banach spaces, $1\leq p\leq \infty$ and $\theta\in (0,1)$. Then we want to define the real interpolation space $(X_0,X_1)_{\theta,p}$. This is possible via the trace method: Define the ...
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
2
votes
1answer
25 views

One parametric family that interpolates continuously between identity and natural logarithm on (0,1]

I am looking for a family of continuous functions $f_p$, $(0,1]\to\mathbb{R}$, and $p\in [1,\infty)$ that fulfill $$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$ for $x\in (0,1]$. I ...
1
vote
1answer
58 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 ,...
1
vote
1answer
39 views

Given $l(f)$ a linear operator, show that it integrates all polynomials of degree $\leq n$ exactly.

Given $l(f)$ a linear operator, if $l(F)$ integrates $1,x,...,x^n$ exactly, show that it integrates all polynomials of degree $\leq n$ exactly. How would I even go about this? I was shown this ...
1
vote
1answer
39 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 ...
1
vote
1answer
30 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 $x\in E$ ...
1
vote
1answer
96 views

What is this Linear Interpolation Method, Brain Teaser!!

Hey geniuses out there! Definitely got a brain teaser here (for me anyway!). I need some help. I'm developing some calculation software that needs to match all calculation results with an existing ...
1
vote
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
vote
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
vote
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
vote
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
vote
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
vote
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
vote
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
vote
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 $[a,b]$ ...
1
vote
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
vote
1answer
51 views

Boundedness of functions in complex interpolation method

In the method of complex interpolation one evaluates traces of suitable holomorphic functions on the strip. I have looked in the book of Lunardi and the one of Bergh/Löfström and in both this "...
1
vote
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
vote
1answer
532 views

Prove the Newton's divided-difference polynomial

Given $(x,y)=(-1,3),(0,1),(1,-1),(-2,-1),(2,3)$. Show that both the Newton's divided-difference polynomial below, interpolate the data. $P(x)=3-2(x+1)+(x+1)x(x-1)$ & $Q(x)=-1+4(x+2)-3(x+2)(x+...
1
vote
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
vote
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
votes
1answer
42 views

Can I decompose the Lagrange interpolating polynomial of the sum of 2 functions into 2 separate Lagrange polynomials? [closed]

For example:L[x0,x1,...,xn;f+g]=?=L[x0,x1,...,xn;f] + L[x0,x1,...,xn;g] where f and g are ordinary functions, not neccessarily polynomials......
0
votes
1answer
483 views

Is the order of the interpolation points in Newton's interpolation polynomial important?

I have been given an assignment to compute the polynomial interpolant of a function, but in the assignment question it says $x_0=1,x_1=0.5, x_2=2$, and I was wondering if I would have to re-label them ...
0
votes
1answer
89 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,...