PREFACE where the kink in the string moves around the circuit with the velocity of sound along the string, and it is this motion of the kink which triggers the bow to adhere and detach from the string. Thus, in a bowed string, we hear the period of the kink, i.e. the length of the "closed geodesic". In an appendix to this Chapter we describe Gelfand's celebrated results on the Plancherel formula for the complex semi-simple Lie groups in terms of the method of stationary phase. Chapter VII. Compound Asymptotics We go back to the type of problems discussed in Chapter II. We have more machinery at our disposal now, so we can formulate the results of Chapter II more systematically.We define objects on manifolds which we call asymptotics. They are functions (densities, half-densities, etc.) which depend on a large parameter T. Any two such objects are identified if they have the same asymptotic growth as r oo. Following Leray we define the Fourier transform of an asymptotic on Rn, and use it to analyze the singularities of the asymptotic. More generally given an asymptotic [y] on a manifold X we define a subset F[y] of T* X which we call its "frequency set" (in analogy with the wave front set of Hormander) and which gives us rather precise information about where high frequency oscillations of [y] are located. Asymptotics have some obvious functorial properties which we discuss in §3. In §4 we develop a symbol calculus for asymptotics. Here these are two parallel theories—one associated to "exact" Lagrangian manifolds (for which {\/2TT)OL is exact) and one associated to "integral" Lagrangian manifolds for which (l/27r)oc E H{(A, Z), i. e. \/(2TT)X has integral cycles. The first of these is associated to asymptotics depending on a continuous parameter and the second to asymptotics associated to a discrete parameter. (These subtleties did not enter in Chapter VI because there we dealt with homogeneous Lagrangian manifold and a vanishes when restricted to a homogeneous Lagrangian manifold.) We then discuss the subprincipal symbol and the transport equation for asymptotics and indicate how integral asymptotics should be used, in conjuction with half forms, to obtain quantization conditions. In particular, our point of view is quite different from that espoused in Chapter II. Even in dimension zero the asymptotic property of an arbitrary asymptotic can be quite complicated. The only general result known is Bernstein's theorem which describes asymptotic properties of integrals of the form j a{z)eira{z) dz for functions a with isolated singularities. This is discussed in §5.
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