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given the operator

$ P_{\Lambda } = (f\in L^{2}(R)^{even}| f(q)=0 , |q| \ge \Lambda) $

what does it mean? the operator $ P_{\Lambda} $ acts over a function $f(q) $ by setting this (even) function to zero whenever $ |q| \ge \Lambda $

how could i evaluate its Eigenvalues and Eigenfunctions ?? thanks

$ P_{\Lambda} T_{n}= \mu _{n} T_{n} $ , can we have a mening to $ P_{\Lambda}^{k} $ ? its power iteration, does this operator have a geommetric meaning ?

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1 Answer 1

up vote 2 down vote accepted

You probably mean $P_\Lambda f(q)=0$. It is multiplication by $\mathbb{1}_{[-\Lambda,\Lambda]}$, the indicator function of $[-\Lambda,\Lambda]$, it restricts functions to that interval, in a sense.

This operator is a projection, $P^2=P$. It is the identity on its image, functions supported on $[-\Lambda,\Lambda]$, so its only eigenvalue is $1$, with eigenvectors all functions in its image.

A last remark is that this is really an operation you will find alot in analysis and physics, cutting some parts of a function (for instance frequencies higher than $\Lambda$, when applied to a Fourier transform $f=\hat g$).

EDIT: A big mistake: $0$ is also an eigenvalue: $P_\Lambda f=0=0\cdot f$ for functions supported outside $[-\Lambda,\Lambda]$ (that is, equal to $0$ in that interval). This holds more generally for projections, for all eigenfunction $f$, either $Pf=f$ or $Pf=0$. Apologies.

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