Operator using Fourier transform I have the following question:
Let $\epsilon >0$. We define the operator $M:= (1-\epsilon \Delta)^{-\frac 12}$, where  $\Delta$ refers to the seconde derivative. 
Let $u \in H^2(\mathbb{T})$ (the Sobolev space of periodic functions
having $H^2$ regularity). We ask to prove the existence of  $v \in L^2(\mathbb{T})$ such that  $$Mu = u + \epsilon v$$ et $\|v\|_{L^2}= \mathcal{O}(1)$   
In the case where the domain of definition is $\mathbb{R}$, we can give  $Mu$ the following sense: 
$$Mu := \mathcal{F}^{-1}\big((1-\epsilon \xi^2)^{-\frac 12}\hat{u}\big).$$ But in our case, could we define $Mu$ in the sense of Fourier series? Namely,
$$Mu := \mathcal{F}^{-1}\big((1-\epsilon n^2)^{-\frac 12}\hat{u}(n)\big).$$
Thank you.
 A: Spectral Theory for selfadjoint operators gives you a way to view functions of a selfadjoint operator $A$ as multiplication operators on the spectrum. For example,
$$
           F(-\Delta)f = \mathcal{F}^{-1}\{F(-s^2)\mathcal{F}\{f\}\}.
$$
Specifically,
$$
        (I-\epsilon\Delta)^{-1}f=\mathcal{F}^{-1}\{(1+\epsilon s^2)^{-1}\mathcal{F}\{f\}\}.
$$
The reason for $-s^2$ instead of $s$ is that the Fourier transform is built for the operator $\frac{1}{i}\frac{d}{dt}$ so that $s^2$ is associated with $\frac{1}{i^2}\frac{d^2}{dt^2}$.
When you're working on the unit circle (or, equivalently, periodic functions on $[-\pi,\pi]$,) the spectrum is discrete. When you treat the spectrum $\sigma(\frac{1}{i}\frac{d}{d\theta})=\mathbb{Z}$ as a discrete space with counting measure, then $\frac{1}{i}\frac{d}{d\theta}$ is transformed to multiplication by $s$ on $\ell^2(\mathbb{Z})$.
The continuous Fourier transform confuses things because the underlying space and the spectral space are the same, and the spectral density measure is uniform. The discrete case better represents what is going on. The space is continuous, but the spectrum is discrete. The spectral density measure remains uniform, but that's to be expected for differentiation.  Other operators may require non-uniform spectral density measures.
