In using the shift operator, which is really just an expression of a Maclaurin (or Taylor) series expansion:


Or, as a generalization:


How would I generalize the shift operator to deal with a condition of periodicity such as:


I think of some kind of condition such as:


However, it's an operator, so that really doesn't make sense here. Intuition says it'll have to be an imaginary exponential? Or am I looking at this wrong and this is actually a differential equation putting restrictions on our (probably function) $f(\overrightarrow{x})$.

Note: I'm working on a fairly general (pseudo) Riemannian manifold, so the derivative will be the covariant one in general. Hence, the shift operator will be in general non-commutative.

  • 1
    $\begingroup$ You can view this as eigenvalue problem. Periodic functions will be eigenfunctions of your operator, with eigenvalue of $1$. $\endgroup$ – Ruslan Dec 19 '16 at 9:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.