Celestial Mechanics. Part I by Shlomo Sternberg

By Shlomo Sternberg

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2) = OL(U *ψι) + ß(u *φ 2 ) f° r scalars a,/?. To sum up: convolution with fixed u G 2' is a linear translation invariant and continuous mapping of CQ into C00. Conversely, we have the following theorem. 16. Let U be a linear translation invariant, and continuous mapping of CQ into C00. Then there exists a unique u e ^ ' such that Ucp = M * φ for all φ G Qf. PROOF. Let φ(χ) = φ(-χ). Then the mapping φ -> (£/φ)(0) is linear on CJ and, by hypothesis, it is continuous. Hence it defines a unique distribution u such that

Let {un} be a sequence in S'. If un -» u weakly then un-> u strongly and u e S'. 25 (see, for example, Friedman [Fl]). 28. A distribution u belongs to S' iff for some m, / we can write u = D\\ + x2)m,2v, where v is a bounded continuous function on the real line (the derivative is, of course, a weak derivative). Moreover, for some other m, the same representation holds where v can now be chosen as a continuous L 2 function. 28 yields the next theorem. 29. If {ua} in S' vary in some bounded set then, for some fixed m, /, u = D\l 4- x2)ml2va, where {νΛ} are uniformly bounded continuous functions.

The classical Laplace transform is defined for certain restricted classes of functions and the inversion is, similarly, limited. In the distributional setting, however, the Laplace transform achieves a certain structural completeness that ensures a greater flexibility in applications. In the next chapter the full scope of the distributional Laplace transform will be used in an essential way. Most of the basic ideas of this chapter are due to Schwartz [Sc2] although there are many differences in detail in the present version.

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