Timeline for Spectral multiplier and Littlewood-Paley projection
Current License: CC BY-SA 3.0
5 events
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| May 23, 2017 at 15:56 | comment | added | Christian Remling | For example, you could work with a spectral representation (= unitary transformation so that $L_a$ becomes multiplication by the variable in the new Hilbert space $\bigoplus L^2(\mathbb R, \rho_n)$), and then $\psi_N(\sqrt{L_a})$ is multiplication by $\psi_N(\sqrt{t})$ there. I don't think this can be worked out very explicitly in your example (though I might be wrong about this). If you want to read up on the general background, then you can use any book that discuss the spectral theorem in Hilbert spaces. | |
| May 23, 2017 at 5:11 | comment | added | XYZ | @CR: Thanks. Do I need to find spectral representation to define $\psi_N(\sqrt{\mathcal L}_a)f$ precisely? ( Any reference? where I could find this..., I think, paper I am reading authors have not mentioned explicit formula for this...) PS: I am wondering without explicit formula(representation), how one could get the further insights.. | |
| May 22, 2017 at 16:26 | comment | added | Christian Remling | For any self-adjoint operator $L$ and any measurable function $f$ on the spectrum, $f(L)$ is defined, via the spectral theorem, but to make it more explicit, you'd have to know more about the spectral representation (which for the Laplacian can be implemented using the FT, but it won't be as easy for $L_a$). However, these topics seem too basic for this site; math.stackexchange.com is the right place to ask (more concrete questions tend to work better, though). | |
| May 22, 2017 at 16:22 | review | First posts | |||
| May 22, 2017 at 16:24 | |||||
| May 22, 2017 at 16:11 | history | asked | XYZ | CC BY-SA 3.0 |