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Further, with x = 111, y = signt (t^ 0), equation (7) gives h(t) = h(\t\ sign t) = H\t\) + Msignt) = h(\t|), or h(t) = h(\t\) for all t # 0 . Note that, by definition, sign t = 1 if t > 0, sign t= — 1 if f < 0 and sign 0 = 0, but here the sign function is applied only to t # 0. Compared to (8), we get the following. An application to information theory 27 Theorem 3. The general solution h:M*-> R of (7) is given by h(x) = /(log | x |) /or a// rea/ x # 0, (9) w/zere / is aw arbitrary solution of (3).
Therefore the following series is convergent on L(N): /(x) = / + xC + ^ C 2 + " + ^ C ' l + . . 2 (x2*0), (48) nl where / is the identity operator on N. The series (48) defines /(x) = e*c. (49) Then (49) is a solution of (47) for all x, y in R+ as is proved by the usual manipulation of convergent series. However, the general solution of (47) is not of the form (49). An extensive literature has been devoted to the problem of finding regularity conditions for a solution / of (47) so that it reduces to (48).
Conversely, let D be a linear operator mapping S9 a subspace of N9 into N. An interesting question is to decide when there exists a solution / of (50), with /(0) = /, and for which (51) holds with Dn = D(n). If Q = N and if D is bounded, f(x) = exD provides a solution for which / is continuous. There exist deeper results when 3 is dense in N and when D is a closed linear operator. For the existence of a solution / of (50), with /(0) = / and with lim x - 0 /(x)n — n = 0 for all n in N such that (51) holds, a necessary and sufficient condition is the existence of real numbers M and w such that, for every real X > w, (A/ — £>)"J exists and satisfies the inequalities Exercises and further results 1.
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