Orthonormal Basis Functions for Continuous-time Systems: Completeness and L P {convergence

In this paper, model sets for continuous{time linear time invariant systems that are spanned by xed pole orthonormal bases are investigated. These bases gen-eralise the well known Laguerre and Kautz bases. It is shown that the obtained model sets are complete in all of the Hardy spaces H p ((); 1 p < 1 and the right half plane algebra A(() provided that a mild condition on the choice of basis poles is satissed. As a further extension, the paper shows how orthonormal model sets, that are norm dense in H p ((), 1 p < 1 and which have a prescribed asymptotic order may be constructed. Finally, it is established that the Fourier series formed by orthonormal basis functions converge in all spaces H p ((), 1 < p < 1. The results in this paper have application in system identiication, model reduction and control system synthesis. 1 Notation C the eld of complex numbers. R the eld of real numbers. the open right half plane fs 2 C : Refsg > 0g. the closed right half plane fs 2 C : Refsg 0g. D the open unit disk fz 2 C : jzj < 1g. T the unit circle fz 2 C : jzj = 1g. H p (() the Hardy spaces of functions f(s) analytic on and such that kfk p p = (1=2) sup x>0 R 1 ?1 jf(x + jy)j p dy < 1; 0 < p < 1 and kfk 1 = sup s2 jf(s)j < 1. A(() the right half plane algebra ff : f 2 H 1 (() and continuous on g. A(D) the disk algebra ff : f analytic on D and continuous on Dg. spA the linear span of A. a the complex conjugate of a. O(jsj ?m) The notation f(s) = O(jsj ?m) as jsj ! 1 means that lim sup jsj!1 jsj m jf(s)j < 1 2 Introduction A fundamental idea in various areas of applied mathematics , control theory, signal processing and system analysis is that of decomposing (perhaps innnite dimensional) descriptions of linear time invariant dynamics in terms of an orthonormal basis. This approach is of greatest utility when accurate system descriptions are achieved with only a small number of basis functions. In recognition of this, there has been much work over on the construction, analysis and application of rational orthonormal bases suitable for providing linear …