Some Properties of the Cauchy Function

Let K(t,s) be the Cauchy function of the linear equation (g sup n) = Summation of ((g sub k)(t)(y sup k)). (1) It is shown that if g sub k is summable in any interval, there are found n points s sub i such that the functions (y sub i)(t) = K(t, s sub i) (i = 1,...,n) are linearly independent. If g sub k is sufficiently smooth (e.g. the equation adjoint to (1) has continuous coefficients), the points s sub i can be chosen arbitrarily within the non-oscillation interval, i.e. an interval in which any non-trivial solution of Eq. (1) has no more than n-1 zeros. The existence of the conditions of smoothness is not clarified. Criteria are given for the sign-preservation of K(t,s). (Author) Trans. of Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika (USSR) n4(41) p9-11 1964, by L. Holtschlag.