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Positive trigonometric polynomials for strong stability of difference equations
We follow a polynomial approach to analyse strong stability of linear
difference equations with rationally independent delays. Upon application of
the Hermite stability criterion on the discrete-time homogeneous characteristic
polynomial, assessing strong stability amounts to deciding positive
definiteness of a multivariate trigonometric polynomial matrix. This latter
problem is addressed with a converging hierarchy of linear matrix inequalities
(LMIs). Numerical experiments indicate that certificates of strong stability
can be obtained at a reasonable computational cost for state dimension and
number of delays not exceeding 4 or 5
Generalized hyperbolic functions, circulant matrices and functional equations
There is a contrast between the two sets of functional equations f_0(x+y) =
f_0(x)f_0(y) + f_1(x)f_1(y), f_1(x+y) = f_1(x)f_0(y) + f_0(x)f_1(y), and
f_0(x-y) = f_0(x)f_0(y) - f_1(x)f_1(y), f_1(x-y) = f_1(x)f_0(y) - f_0(x)f_1(y)
satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y).
J. Schwaiger and, later, W. F\"org-Rob and J. Schwaiger considered the
extension of these ideas to the case where f is sum of n components. Here we
shorten and simplify the statements and proofs of some of these results by a
more systematic use of matrix notation.Comment: 18 pages; corrected and updated versio
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