Appl Math Optim 55:203–218 (2007)
2007 Springer Science+Business Media, Inc.
Semi-Discrete Ingham-Type Inequalities
and Paola Loreti
D´epartement de Math´ematique, Universit´e Louis Pasteur,
7 Rue Ren´e Descartes, 67084 Strasbourg Cedex, France
Dipartimento di Metodi e Modelli,
Matematici per le Scienze Applicate,
Universit`a degli Studi di Roma “La Sapienza”,
Via A. Scarpa, 16, 00161 Roma, Italy
Abstract. One of the general methods in linear control theory is based on harmonic
and non-harmonic Fourier series. The key of this approach is the establishment of
various suitable adaptations and generalizations of the classical Parseval equality.
A new and systematic approach was begun in our papers – in collaboration
with Baiocchi. Many recent results of this kind, obtained through various Ingham-
type theorems, were exposed recently in . Although this work concentrated on
continuous models, in connection with numerical simulations a natural question
is whether these results also admit useful discrete versions. The purpose of this
paper is to establish discrete versions of various Ingham-type theorems by using our
approach. They imply the earlier continuous results by a simple limit process.
Key Words. Observability, Fourier series, Vibrating strings.
AMS Classiﬁcation. 42C99, 93B07, 35L05, 74K05, 74K10.
In a classical paper devoted to the study of Dirichlet series, Ingham  proved an estimate
for non-harmonic Fourier series, which extended Parseval’s equality to cases where the
exponents do not form an arithmetical sequence. His theorem was an improvement of
earlier results of Paley and Wiener, and was subsequently generalized in many different
directions. Ingham’s original theorem turned out to be quite useful in control theory,