Timeline for How to compute $\sin(\frac{d}{dx})f(x)$?
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| Dec 20, 2022 at 7:27 | comment | added | memorial | As regards references, the rigorous treatment of the spectral theorem can be found in most standard introductions to functional analysis (e.g., Rudin)- I am not familiar with less rigorous treatments and have no access to a suitable library at the moment. You might try the classic by Courant and Hilbert on Methods in Mathematical Physics. | |
| Dec 20, 2022 at 7:22 | comment | added | memorial | As regards the second question, if $T$ is a normal operator on a Hilbert space, then you can define $g(T)$ for any measurable function defined on the spectrum of $T$ (which is a subset of the complex plane). In the case of a self-adjoint operator ( as is $i \frac d{dx}$), then it is a subset of the reals. Analyticity is only required for the functional calculus of general (i.e. not normal) operators on Banach spaces. | |
| Dec 19, 2022 at 16:44 | comment | added | Mirar | A question regarding to your answer: " g is a measurable real function on the line". Does this mean that g does not have to be analytic? Say, in discrete domain, if T has a complex eigenvalue and g is not analytic, can we still define g(T)? | |
| Dec 19, 2022 at 16:39 | comment | added | Mirar | +1 Thanks for the time. Please forgive my imprecise mathematical language. I am engineer and just started to explore this area of mathematics. I really appreciate it if you could provide a reference (preferablly application oriented) | |
| Dec 19, 2022 at 16:18 | history | edited | memorial | CC BY-SA 4.0 |
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| Dec 19, 2022 at 16:12 | history | edited | memorial | CC BY-SA 4.0 |
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| Dec 19, 2022 at 15:58 | history | edited | memorial | CC BY-SA 4.0 |
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| Dec 19, 2022 at 15:49 | history | answered | memorial | CC BY-SA 4.0 |