Questions tagged [interpolation-theory]

Question 1

I want to show that the fractional Sobolev space $H^\frac{1}{2}$ is equal to the interpolation space $(L^2,H^1)_{\frac{1}{2},2;K}$. In particular I would like to use the K-method as described in the ...

Question 2

I'm currently studying interpolation spaces, but I'm struggling to understand some of the intuition behind real interpolation. There are two equivalent methods to define the real interpolation functor,...

Question 3

I am studying Theorem X.36' (Nelson's Commutator Theorem) on Reed-Simon vol II, here is a short resume: Let $N \geq I$ a self-adjoint operator on a complex Hilbert Space $H$. Suppose we have a ...

Question 4

I am interested in the specific embedding constant of the Lions-Magenes lemma. In particular, let $V \subset H \subset V'$ be a Gelfand triple, where the embeddings are compact. Furthermore, we define ...

Question 5

I am stuck with this problem since a week, if someone could help me that would be amazing. Considering, a smooth 2d map on a closed domain let's take the unit square: $$\begin{cases}\phi : [0,1] \...

Question 6

Setting Consider the following setting: Let $d \in \mathbb{N}$. Lorentz space: $L^{p,q} = L^{p,q}(\mathbb{R}^d; \mathbb{R})$, $1<p<\infty$, $1\leq q \leq \infty$ (I assume familiarity with ...

Question 7

Let $M$ be a closed manifold, and $\Omega\subseteq M$ an open subset. For $s\geq 0$ let $H^s_\bar{\Omega}(M)$ denote the subset of the Sobolev space $H^s(M)$ containing all functions with (...

Question 8

There are two types of boundedness for operators that are commonly used as endpoints in interpolation theory: Weak-type $(1,1)$ boundedness; Hardy space $H^1$ boundedness. Both of these serve as ...

Question 9

Below, every function space is on the domain $\mathbb{R}^N$ for some $N \in \mathbb{N}$, $W^{m,p}$ denotes the classical Sobolev spaces (i.e., having $L^p$ weak derivatives), and $B^{s,p}_q$ denotes ...

Question 10

Suppose we are given a sequence $a_{0}, a_{1}, a_{2}, \ldots$ of real numbers and we define $F:\mathbb{N}\times[0, \infty)\rightarrow\mathbb{R}$ by $$ F(k, \alpha) = \sum_{n=0}^{\infty} \frac{\alpha^{\...

Question 11

Consider a convex and smooth (or at least $C^2$) function $f$ such that: $f(x)>0$ for $x\in (a,b)$ $f'(x)>0$ for $x\in (a,b)$ $f''(x)>0$ for $x\in (a,b)$ Now, we are given an interval $[...

Question 12

It is in the proof of the Riesz interpolation theorem in Chapter 2 section 2, page 56. I formulate it as follows. If $f\in L^p(X)$, write $f^U= f \chi_E$ and $f^L=f-f^U$, where $E=\{x\in X: |f(x)|\geq ...

Question 13

Consider two non-constant real polynomials $f(x)$ and $g(x)$: $$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N $$ $$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M $$ where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ ...

Question 14

Let consider $N$ points in Euclidean space of dimension $n$, $\boldsymbol{r}_i \in \mathbb{R}^n$. Then, consider a polynomial $P(X_1,\dots,X_n)$ of $n$ variables vanishing at those $N$ points: i.e. $P(...

Question 15

I have the question "Find the "natural" cubic spline, the one satisfying $s''(x_{0})=s''(x_{n})=0$ that passes through $$ (x_{i},y_{i})= (0,0), (1,2),(2,1) \text{ and } (3,0 ) $$ <\...

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Questions tagged [interpolation-theory]

Fractional Sobolev space $H^\frac{1}{2}$ is equal to the interpolation space $(L^2,H^1)_{\frac{1}{2},2;K}$

Real interpolation: Why do the J- and K-methods scale everything by 1/t?

On Nelson's Commutator Theorem (Theorem X.36/X.36') in Reed-Simon II: can we show the theorem without realying on interpolation theory?

Constant of Lions-Magenes lemma

L2 norm scaling for a smooth 2d map with N vanishing points

Does the fractional Leibniz rule hold in Lorentz spaces?

Interpolation for Sobolev spaces with support in closed subspace

The boundedness of operators

Besov spaces as the real interpolation spaces of classical Sobolev spaces: $p=1$ or $p=\infty$ case?

Does Newton's forward difference formula / Newton's interpolation formula yield well-defined functions (in one variable) if it is convergent?

Fix first and second derivatives at two points: is convexity possible?

One confusion in Riesz interpolation in Functional analysis by Stein-Shakarchi

Strategies to build functions with given expansions at two different points

What is the minimal degree $d$ of a polynomial in $n$ variables vanishing at $N$ points

Natural Cubic Spline interloping 4 points

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