A note on trigonometric matrices
Garret J. Etgen · Proceedings of the American Mathematical Society · 1966
More recently, the author [3 ] established that the pair { S(x), C(x) has many of the properties of the sine and cosine functions. In particular, S(x) and C(x) satisfy identities on X which are the matrix analogues of the elementary trigonometric identities and, in the event Q(x) is positive definite on X, the pair { S(x), C(x) } has oscillatory behavior which is analogous to the oscillatory behavior of {sin fxq(t)dt, cos f0q(t)dt}, where q(x) is a positive, continuous function on X. In this paper we establish that there exists a positive number p such that if Q(x) is positive definite on X, a is any nonnegative number and b, b > a, has the property fJ tr Q(x)dx _ p, then at least one of S(x) and C(x) has a singularity on a ? x 0 on X. An application of this result is discussed in ?3.