Projection to the Set of Shift Orthogonal Functions

TL;DR

This paper introduces a fast, parallelizable algorithm for projecting functions onto the set of shift orthogonal functions, utilizing a new basis and achieving computational complexity of O(M log M).

Contribution

The paper develops a novel class of Shift Orthogonal Basis Functions and an efficient projection algorithm with theoretical analysis.

Findings

Algorithm achieves O(M log M) complexity.

Parallelizable projection method.

Introduction of Shift Orthogonal Basis Functions.

Abstract

This paper presents a fast algorithm for projecting a given function to the set of shift orthogonal functions (i.e. set containing functions with unit $L^2$ norm that are orthogonal to their prescribed shifts). The algorithm can be parallelized easily and its computational complexity is bounded by $O(M\log(M))$, where $M$ is the number of coefficients used for storing the input. To derive the algorithm, a particular class of basis called Shift Orthogonal Basis Functions are introduced and some theory regarding them is developed.