By Reinhard Diestel

ISBN-10: 0387950141

ISBN-13: 9780387950143

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**Extra resources for A Fourier Analysis And Its Applications**

**Example text**

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. 11. Ever since the days of Newton, physicists have been dealing with situations where some physical entity assumes a very large magnitude during a very short period of time; often this is idealized so that the value is inﬁnite at one point in time.

Find f (s). 2 Operations The Laplace transformation obeys some simple rules of computation and also some less simple rules. The simplest ones are collected in the following table. Everywhere we assume that s takes suﬃciently large values, as discussed at the end of the preceding section. 1. L[αf + βg](s) = αf (s) + βg(s), if α and β are constants. 2. L[eat f (t)](s) = f (s − a), if a is a constant (damping rule). 3. If we deﬁne f (t) = 0 for t < 0 and if a > 0, then L[f (t − a)](s) = e−as f (s) (delaying rule).

It would be convenient to have some simple set of conditions on a function f that ensure that the Laplace transform is absolutely convergent for some value of s. Such a set of conditions is given in the following deﬁnition. 1 Let k be a positive number. Assume that f has the following properties: (i) f is continuous on [0, ∞[ except possibly for a ﬁnite number of jump discontinuities in every ﬁnite subinterval; (ii) there is a positive number M such that |f (t)| ≤ M ekt for all t ≥ 0. Then we say that f belongs to the class Ek .

### A Fourier Analysis And Its Applications by Reinhard Diestel

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