By Reinhard Diestel
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Additional info for A Fourier Analysis And Its Applications
6 *Some simple distributions In this section, we introduce, in an informal way, a sort of generalization of the notion of a function. 10. In Sec. 3 (on the wave equation) we saw diﬃculties in the usual requirement that solutions of a diﬀerential equation of order n shall actually have (maybe even continuous) derivatives of order n. Quite natural solutions are disqualiﬁed for reasons that seem more of a “bureaucratic” nature than physically motivated. This indicates that it would be a good thing to widen the notion of diﬀerentiability in one way or another.
We shall soon see that Laplace transforms give us a new technique for solving these equations. We shall also be able to solve more general problems, like integral equations of this kind: t 0 t f (t − x) f (x) dx + 3 f (x) dx + 2t = 0, t > 0. 2) 0 Another consequence of the theorem is worth emphasizing: if a Laplace transform exists for one value of s, then it is also deﬁned for all larger 42 3. Laplace and Z transforms values of s. If we are dealing with several diﬀerent transforms having various domains, we can always be sure that they are all deﬁned at least in one common semi-inﬁnite interval.
Now assume that f is piecewise constant, which means that I (still assumed to be compact) is subdivided into a ﬁnite number of subintervals Ik = (ak−1 , ak ), k = 1, 2, . . , N (a0 = a, aN = b), and that f (u) has a certain constant value ck for u ∈ Ik . This means that we can write N f (u) = ck gk (u), k=1 where gk (u) = 1 on Ik and gk (u) = 0 outside of Ik . We get N b a N b f (u) sin λu du = ck gk (u) sin λu du = a k=1 ak ck sin λu du. ak−1 k=1 This is a sum of ﬁnitely many terms, and by the preceding case each of these terms tends to zero as λ → ∞.
A Fourier Analysis And Its Applications by Reinhard Diestel