Chapter 1 Introduction This opening chapter presents Atkinson's three different interpretations of the second-order difference equation (1.0.1) V[p(n)Ay(n)] + q(n)y(n) = Xw(n)y(n) , n = a,..., b, found in Ref. 2, where A is the forward difference operator (A/(n) = f(n -f 1) — /(^)), and V is the backward difference operator (A/(n) = f(n) — f(n — 1)). They are different in flavor, ranging from mechanics, to network theory, to probability theory. However, they are all confined to problems which are finite. Moreover, they are all open to extensions, including the infinite discrete case, which illustrate the closeness between discrete and continuous. 1.1 The Vibrating String Consider a weightless string, stretched between two fixed points, executing small harmonic vibrations and bearing a discrete sequence of particles with masses ao,... , a m -i- Let — be the distance between an and a n +i(n = 0,... , m— cn 2), and let un be the displacement of the particle an at time t. Suppose further that the string extends to length beyond a m _i and beyond ao, and c m - i c_i denote the tension of the string at an by Tr and Tt on the right and the left, respectively (see Figure 1.1). By Newton's second law of dynamics, (1.1.1) a n =T r sin0 r +7/sin0*. Also, since the particle an does not move horizontally, (1.1.2) Tr cos0r = Tt cosOi = To. Received by the editor October 19, 1992. 1

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