Projective view at Optimization Problem for Multiband Filter

TL;DR

This paper revises the optimization problem for multiband filters by introducing a projective formulation, addressing contradictions in previous criteria, and providing a new constructive optimality condition.

Contribution

It proposes a new projective formulation for the multiband filter optimization problem and offers a constructive optimality criterion that resolves previous contradictions.

Findings

Identifies contradictions in existing optimality criteria.

Introduces a projective invariant formulation.

Provides a constructive criterion for optimality.

Abstract

The best uniform rational approximation of the \emph{sign} function on two intervals separated by zero was explicitly solved by E.I. Zolotar\"ev in 1877. This optimization problem is the initial step in the staircase of the so called approximation problems for multiband filters which are of great importance for electrical engineering. We show that known in the literature optimality criterion for this problem may be contradictory since it does not take into account the projective invariance of the problem. We propose a new consistently projective formulation of this problem and give a constructive optimality criterion for it.