Discrete trigonometric matrix functions.(English)Zbl 0888.39010

The author studies discrete sine $$S(t)$$ and cosine $$C(t)$$ matrix functions, which are defined as the unique solutions $$Y(t)=S(t)$$, $$Z(t)=C(t)$$ of the initial value problem $Y(t+1)= \cos Q(t)Y(t)+ \sin Q(t)Z(t) \tag{*}$
$Z(t+1)= -\sin Q(t)Y(t)+ \cos Q(t)Z(t), \quad Y(a) =0,\;Z(a) =I,$ where $$Q(t)$$ is an $$n\times n$$ Hermitian matrix function on the set $$\{a,a+1, \dots\}$$ and the coefficient matrices in (*) are defined by their Maclaurin series. The author proves several basic identities, among them these similar to sine and cosine for sums or differences of two angles (in the ordinary case). Next, the definition of generalised zeros is introduced and separation properties are considered. Sufficient conditions for oscillation of $$C(t)$$ and $$S(t)$$ are given in terms of $$tr\sin Q(t)$$.
Discrete tangent and cotangent functions are defined in natural way, and some properties of these functions are presented.

MSC:

 39A12 Discrete version of topics in analysis 39A10 Additive difference equations