Array Flattening Trenasformations via Superoperators on Difference, Unit and Shift Operators

Abstract

In this paper, we introduce a new approach for array flattening. We first establish a relationship between the general array element and a specifically chosen reference element by using unit, shift, forward and backward difference operators. Then, we define an array function by multiplying the forward and backward difference operators with a single scalar variable t. This variable is a perturbating agent since the produced array is equivalent to the original array when t is 1. We define the discrete counterparts of the Taylor series and Taylor formula with remainder in the form of expansion in t powers. We also define array flatness as this array function’s flatness with respect to t. Then the use of an affine transformation via a superoperator (first degree polynomial in the array function on the function operator and the variable operator) to create a new array function makes the flattening possible. This is done by appropriately choosing the super operator’s coefficient functions. The Taylor polynomial’s reciprocal is taken as the first degree term coefficient. The flattened array is obtained by setting t=1 after the flattening in t becomes complete.