Editing Shift operator (section)

==Generalization==

[[Jean Delsarte]] introduced the notion of '''generalized shift operator''' (also called '''generalized displacement operator'''); it was further developed by [[Boris Levitan]].<ref name = mar/><ref>{{SpringerEOM|id=g/g043800|first=B.M.|last=Levitan|author-link=Boris Levitan|first2=G.L.|last2=Litvinov|title=Generalized displacement operators}}</ref><ref>{{SpringerEOM|id=A/a011970|first=E.A.|last= Bredikhina|title=Almost-periodic function}}</ref>

A family of operators {{tmath|\{L^x\}_{x \in X} }} acting on a space {{math|Φ}} of functions from a set {{mvar|X}} to {{tmath|\C}} is called a family of generalized shift operators if the following properties hold:
# [[Associative property|Associativity]]: let <math>(R^y f)(x) = (L^x f)(y).</math> Then <math>L^x R^y = R^y L^x.</math>
# There exists {{mvar|e}} in {{mvar|X}} such that {{mvar|L<sup>e</sup>}} is the [[Identity function|identity operator]].
In this case, the set {{mvar|X}} is called a [[hypergroup]].