Shift operator

This article is about shift operators in mathematics. For operators in computer programming languages, see Bit shift. For the shift operator of group schemes, see Verschiebung operator.

In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a).[1] In time series analysis, the shift operator is called the lag operator.

Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution.[2] Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator.

  1. ^ Weisstein, Eric W. "Shift Operator". MathWorld.
  2. ^ Marchenko, 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. ISBN 978-3-540-23235-3. MR 2182783.