# 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.

77 questions
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 ...
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 ...
1answer
202 views

3answers
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### 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 ...
2answers
89 views

0answers
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### 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 $O((\log a)^c)$ ...
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 $$\|x\|_{p,\mu} = \left(\sum_{i=1}^n |x_i|^p \mu_i\right)^{1/p},$$ ...
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 ...
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 ...
1answer
58 views

1answer
532 views