By Reinhard Diestel
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Extra resources for A Fourier Analysis And Its Applications
6). In plain language, the derivative of |t| is plus one for positive t and minus one for negative t, just as we know from elementary calculus; at the origin, the value of the derivative is undecided. We proceed to the second derivative: f (t) = 2δ(t) − 0 = 2δ(t). This formula reﬂects the fact that f has derivative zero everywhere outside the origin; whereas at the origin, the delta term indicates that f has a positive jump of two units. This is characteristic of the derivative of a function with jumps.
9. The preceding example can be generalized in the following way: Let ψ : R → R be some function satisfying ψ(s) ≥ 0 and R ψ(s) ds = 1. Putting Kn (s) = nψ(ns), we have a positive summation kernel. The examples should help the reader to understand what is going on: a positive summation kernel creates a weighted mean value of the function f , with the weight being successively concentrated towards the point s = 0. If f is continuous at that point, the limit will yield precisely the value of f at s = 0.
6 *Some simple distributions 29 for all s such that the integral is convergent (see Chapter 3). The Laplace transform of δ cannot be deﬁned in this way. We can, however, modify the deﬁnition so as to include the origin. It is indeed customary to write ∞ f (s) = ∞ f (t)e−st dt = lim k 0− 0 f (t)e−st dt. k With this deﬁnition one ﬁnds that δ(s) = 1 for all s. Similarly, δa (s) = e−as , if a > 0. The Heaviside function, or unit step function, H is deﬁned by H(t) = 0 for t < 0, 1 for t > 0. The value of H(0) is mostly left undeﬁned, because it is normally of no importance.