Oscillation of certain sets of orthogonal functions
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In the classic memoirs of Sturm and Liouville, two classes of theorems are found concerning sets of orthogonal functions. The first deal with the number of sign-changes in [phi]3, and the second with the number of sign-changes in a polynomial c2[phi]2(x) + ... c3[phi]3(x). The functions of the orthogonal sets with which they deal are solutions of a differential equation containing a single parameter. The question arises as to whether these two classes of theorems are consequences of the mere orthogonality of the function sets, or whether the differential equation is a necessary condition for them. We shall find that orthogonality alone is not a sufficient condition for the oscillation theorems in question; but that with the addition of the hypothesis of the non-vanishing of a certain set of determinants it becomes so. The first chapter will be given to this subject. The later ones will be occupied with the verification for certain special sets of orthogonal functions of the oscillation theorems of Sturm.