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.

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1D Interpolating subdivision for lifting schemes

I am looking into wavelet lifting methods first introduced by Swelden, and explained in this paper: Build your own wavelets at home. In this paper (in chapter 2 specifically), they discuss ...
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Compare error bound to theoretical error bound

$P_3(x) = 3x^3 +3$ is an interpolating Lagrange polynomial for $\widetilde{P(x)} = x^4-2x^3-x^2+2x$ generated from the data points $$(-1, 0), (0, 3), (1, 6), (2, 27)$$ $\widetilde{P(x)}$ is itself a ...
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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 ...
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$T:L^p \to L^p$ , $A$ a dense subset of $L^p$ , $T|_A$ is bounded and has an extension $T'$ . Can we show that $T=T'$ on $L^p$?

$(a)$ Suppose $T:L^p \to L^p$ is a linear operator(might not be bounded) , $A$ a dense subset of $L^p$ , $T|_A$ is a bounded operator , then $T|_A$ can be extend to all of $L^p$ which we call it $T'$ ...
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Interpolation of polynome - is degree needed?

I'm doing my project from numerical methods in math and I'm supposed to create the following program: Inputs: $n \in [0,\infty)$ a sequence $\Gamma_{i=0}^{n} (u_i) \in \mathbb{R}^{n+1}$ ...
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Complex Interpolation and Intersection

Does it hold $$[X \cap Y, X \cap Z]_\theta = X \cap [Y,Z]_\theta$$ where $X,Y,Z$ are suitable spaces and $[\cdot,\cdot]_\theta$ denotes the complex interpolation functor of order $\theta \in [0,1]$. ...
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Analytic Interpolations

I understand that for any sequence of reals there is an analytic interpolation, and for any positive sequence there is an analytic interpolation that is positive. I am wondering if there is a ...
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In my notes I found the following result without proof or reference: Let $(\mathcal{M},\mu)$ be a Borel measure space, with $\mu$ positive and $\sigma$-finite. Let g,h : \mathcal{M} \mapsto \mathbb{... 1answer 60 views Meromorphic interpolation of number-theoretic functions A meromorphic interpolation of the harmonic numbers is given by \begin{align} H_n &= \sum_{k=1}^n \frac{1}{k} \\ &= \sum_{k=0}^{n-1} \frac{1}{k+1} \\ &= \sum_{k=0}^{n-1} \int_0^1 x^k \,\... 0answers 90 views Comparision of real and complex operator norms in Riesz-Thorin theorem This is an exercise from Dirk Werner's "Funktionalanalysis" (Aufgabe II.5.4): Let(\Omega,\Sigma, \mu)$be some$\sigma$-finite measure space. Denote by$L^p_{\mathbb{R}}(\mu)$and$L^p_{\mathbb{C}}(\...
I am solving a very specific problem where I need to apply FFT to interpolate. I am given two sets of distinct points $A = (a_{1}, a_{2}, ... , a_{n})$ and $B = (b_{1}, b_{2}, ..., b_{n})$. Using $B$, ...