On multi-dimensional systems; properties of their transfer functions

Abstract

A range of interesting electromagnetic systems like antennas, extraordinary transmission and absorbers have been shown to have certain bandwidth limitations given by a family of sum-rules. Common for the systems are that they are passive, linear and time-translational invariant. In this paper we shortly review the extension from one-dimensional passive systems to multi-dimensional systems, aiming towards constraining the system properties. The most well-known case of system constraints follows from multi-dimensional passivity, where a Schwartz-kernel representation theorem maps Borel-measures with a growth condition to the (complexified) Fourier transform of the transfer function. A weaker form of system constraints follow from generalizations of Kramers-Kronig relations. One such approach is a generalized Cauchy-Bochner representations, under Sobolev space limitations on the transform pair. This approach is closely connected to that the support of the transfer function is within an acute cone. Another approach to system transfer constraints is the multi-dimensional Hilbert-transform, often with square-integrable function requirements. It is observed that the Cauchy-Bochner representation and the multi-dimensional Hilbert transform yield different representations in higher dimensions although they give the same in one dimension. We end the paper with a few explicit examples of functions that satisfy the constraints.