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.

Interpolation theorems are a class of results which can be broadly characterized as follows: given information about an operator defined on two different "endpoint" cases, one can deduce information about the behavior of the operator in "intermediate" cases. Notable results in this vein include the Riesz-Thorin theorem and the Marcinkiewicz interpolation theorem.

Interpolation theorems serve an important role in the fields of analysis and partial differential equations. In these contexts one might study particular interpolation results that are appropriate for the given context. One may also study interpolation more abstractly, in the form of interpolation spaces and functors.

Interpolation theory is sometimes referred to as "interpolation of operators" to distinguish it from "interpolation of functions," in which one approximates a function given partial information about its values: .

128 questions
24
votes
4 answers

What is the goal of harmonic analysis?

I am taking a basic course in harmonic analysis right now. Going in, I thought it was about generalizing Fourier transform / series: finding an alternative representation of some function where something works out nicer than it did before. Now,…
fubal
  • 381
  • 1
  • 12
9
votes
2 answers

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 through all of those. Is there an analogous construct…
7
votes
1 answer

Borderline case of interpolation of Banach spaces

Let $B \subset A$ be Banach spaces with a continuous embedding. Is the inequality $$ \|b\|_B \leq C \sup_{t > 0} \inf_{\tilde{b} \in B} \{ \|b - \tilde{b}\|_B + t \|\tilde{b}\|_A \} \quad \forall b \in B $$ valid for some $C \geq 1$? What if the…
user66081
  • 3,887
  • 15
  • 28
6
votes
0 answers

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 $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ a'+a''=a\}. \tag{1} $$ Now, one can show (Lemma 1, [1])…
Clement C.
  • 63,949
  • 7
  • 61
  • 141
5
votes
3 answers

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 that for each $x\in[a,b]$ exist $\xi_x\in[a,b]$ such…
5
votes
1 answer

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 the differences between them (sublinearity, weak…
4
votes
1 answer

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 $\left. v\right|_{U}\in H^s(U)$ for all $v\in H^s(V)$ when $s\geq 0$…
4
votes
0 answers

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): First, note that $$ ||T||_{p_\vartheta\to…
Pelota
  • 933
  • 1
  • 12
4
votes
1 answer

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 Laplace…
4
votes
2 answers

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 Newton and Lagrange are well known, but I wanted…
Max
  • 1,043
  • 2
  • 7
  • 20
4
votes
0 answers

Example of Holmested formula for real interpolation spaces

Can anybody give me an example of Holmested formula for real interpolation spaces.
King Khan
  • 1,014
  • 6
  • 11
3
votes
1 answer

Analytic extension of $(-1)^n$ satisfying growth condition

Can the function $(-1)^n$, $n=1,2,...$ be extended to an analytic function $f(z)$ defined on the right half complex plane satisfying the growth condition $$|f(x+iy)|\le C e^{Px+A|y|},$$ with $A<\pi$ and $C,P\in\mathbb{R}$? I know that such an…
zbrads2
  • 830
  • 4
  • 10
3
votes
1 answer

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 norms $$ ||a||_{A_0 \cap A_1} = \max\{…
3
votes
1 answer

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 exists $C > 0$ such that for any $f \in W^{s, 2}$…
Three.OneFour
  • 1,547
  • 9
  • 13
1
2 3
8 9