By N. Dinculeanu
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Additional info for Integration on Locally Compact Spaces
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Integration is a smoothing or averaging process. The integral of a jagged function is smoother with more rounded peaks, and for a piecewise-continuous function, integration eliminates the jump discontinuities. As an example, if then D 0 -y(0 = tu(t) and AT2/(0 = i'2«(0These are plotted in Fig. 2, and whereas f(t) is discontinuous at t = 0, 52 Linear systems / o t Fig. 2: Effect of integration applied to «(/) ( 0 Fig. 3: Idealized ( );D^lu(t) (- t ) and more realistic (—) switch performance. AT 7X0 a n ( * a ^ subsequent integrals are continuous.
5, and for every e =£ 0 f£t) is well defined. The limit of the sequence as e -> 0 is a function having the properties (i) and (ii). Other functions could be used to form the sequence. To avoid difficulties that might occur because of the piecewise continuity of the function in Eq. 5 t Fig. 5: The rectangular pulse function fc(t) of Eq. 4) for e — 1, 1/2, and 1/3. 6) which is entirely smooth. This is plotted in Fig. 6 for e = 1/4, 1/20, and 1/100. Still others could suffice and as e -> 0 all specify a limit function that is a particular example of a generalized function.
14. = 2. = 6«(0, with x(0) = x7(0) = (D2 + 22) -h •11. JC'(O) = 12e , with x(0) = - 2 , 6. 10. - 0, r 2)JC(0 = 3^~ r (sin2/)w(/), with x(0) = 2, xr(0) = 1 (Z) + 4)x(/) 4- 3Z)0" x(/) = 2 with x(0) = 0, / 2^- M(/), JC'(O) = x"(0) = 1. with x(0) = - 1 . (Z) - l) x(/) = t{u{t) - u(t - 1)}, with x(0) = 0, x'(0) = 3. CHAPTER 3 Linear systems Having now arrived at the heart of our subject, we show how the mathematical techniques that have been discussed can be used to solve some of the problems associated with linear systems.