An explanation of the commuting operator "miracle" in time and band limiting

TL;DR

This paper explores the mathematical properties of time and band limiting operators, expressing them through the confluent Heun operator, and derives explicit formulas for different bandwidth regimes.

Contribution

It introduces a novel connection between time-band limiting operators and the confluent Heun operator, providing explicit formulas for small and large bandwidths.

Findings

Explicit formulas for time and band limiting operators in different bandwidth regimes

Connection established between these operators and the confluent Heun operator

Results recover known Fourier transform properties

Abstract

Time and band limiting operators are expressed as functions of the confluent Heun operator arising in the spheroidal wave equation. Explicit formulas are obtained when the bandwidth parameter is either small or large and results on the complete Fourier transform are recovered.