The response of a homogeneous time-invariant system to an exponential input is a similar exponential output. I noticed this when teaching Matemática Aplicada (Applied Mathematics) years ago, as a consequence of the approach that I use to introduce Fourier analysis to the engineering students. Basically, I try to show that exponentials are eigenfunctions of linear time-invariant systems, and then look for orthonormal sets of exponentials. I found several arguments that suggest that linear time-invariant systems have exponential eigenfunctions, and that require just the axiomatic definition of linearity (no integral or convolution representation is involved; indeed, there are linear time-invariant systems that have no convolution representations). The simplest argument that I know of uses homogeneity and time-invariance only. Thus, time-invariant systems have exponentials eigenfunctions.

In an interesting article (Homogeneous Time-Invariant Systems, IEEE Signal Processing Letters, vol. 6, n. 4, p. 76-77, Apr. 1999), Vaidyanathan established the invariance of exponentials for homogeneous time-invariant systems, noticed that the concepts of "impulse response" and "frequency response" are of little use for their analysis, and asked about more general classes of systems with exponential eigenfunctions.

This paper argues that the concepts of impulse and frequency response can be useless even for certain linear, time-invariant systems. It discusses the role of time-invariance, commuting linear systems, and conditions under which they have common eigenfunctions. Then it exhibits a class of nonlinear, nonhomogeneous, time-varying systems that still have exponential eigenfunctions. This class contains homogeneous time-invariant systems, FIR filters and generalized feed-forward filters as special cases. This clearly shows that the exponential eigenfunction property does not imply linearity, homogeneity, or time-invariance. This class has been enlarged in a subsequent paper (P. M. Q. Aguiar and F. M. Garcia, Systems with Exponential Eigenfunctions and Exponential-Input / Constant-Output Operators, IEEE Signal Processing Letters, vol. 7, n. 8, p. 219-220, Aug. 2000).

The set of fixed points of a nonexpansive operator is either empty or closed and convex. Under rather general conditions, this shows that the minimum norm solution of an operator equation of the form x=Tx exists and is unique, provided that T is nonexpansive. This holds in any strictly convex Banach space, a class of spaces that includes Hilbert spaces as particular cases. This has consequences in signal and image reconstruction, as well as in other engineering applications.

This is a tutorial on the basic fixed point theorems, which are often found in engineering applications. It discusses Banach's fixed point theorem, also known as the contraction mapping theorem, and Brouwer's theorem, which asserts that every continuous mapping of a closed finite dimensional ball into itself has a fixed point. The proof of Brouwer's theorem depends on a combinatorial lemma (Sperner's lemma). Another important fixed point theorem is Schauder's: any compact convex nonempty subset of a Banach space has the fixed point property.