which may be interpreted operationally through its formal Taylor expansion in t; and whose action on the monomial xn is evident by the binomial theorem, and hence on all series inx, and so all functions f(x) as above. This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.
The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.
In general, as illustrated above, if F is a function on an abelian groupG, and h is an element of G, the shift operator Tg maps F to
Properties of the shift operatorEdit
The shift operator acting on real- or complex-valued functions or sequences is a linear operator which preserves most of the standard norms which appear in functional analysis. Therefore, it is usually a continuous operator with norm one.
Action on Hilbert spacesEdit
The shift operator acting on two-sided sequences is a unitary operator on ℓ2(Z). The shift operator acting on functions of a real variable is a unitary operator on L2(R).
In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:
^ abMarchenko, V. A. (2006). "The generalized shift, transformation operators, and inverse problems". Mathematical events of the twentieth century. Berlin: Springer. pp. 145–162. doi:10.1007/3-540-29462-7_8. MR 2182783.
^Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), ISBN 978-0828400336 .
^M Hamermesh (1989), Group Theory and Its Application to Physical Problems
(Dover Books on Physics), Hamermesh ISBM 978-0486661810 , Ch 8-6, pp 294-5 ,
^p 75 of Georg Scheffers (1891): Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, 1891. ISBN 978-3743343078 online
^ abAczel, J (2006), Lectures on Functional
Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, ISBN 978-0486445236 .
^"A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.