# 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.

102 questions
Filter by
Sorted by
Tagged with
7 views

### 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 ...
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 ...
11 views

### 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 ?
51 views

### 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 ...
37 views

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....
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 ...
10 views

### 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 ...
45 views

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 \|...
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 ...
20 views

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 ...
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 ...
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 ...
60 views

128 views