Problems of Control and Information Theory 8. (Budapest, 1979)
1979 / 1. szám - Greblicki, W.: Nonparametric system identification by orthogonal series
70 GKEBLICKI: NONPABAMETBIC SYSTEM IDENTIFICATION BY OBTHOGONAL SEBIES We assume that N(n) A Ф'п (x) = / /2(я) 95,(ж) 1=0 is a model. Theorem 2. If conditions (4), (5), (6) and (7) are fulfilled, then lim E f (Ф0(x) — Ф'п (x))2 f(x)dx = 0. (9) n-+°° The proof is similar to that of Theorem 1. It is easy to see that for any model Щф) = ЩФ0) + j (Ф„(х) — Ф(х))2{(х) dx. Therefore by (9) we get Corollary. Under all the assumptions of Theorem 2 lim ЕВ{Ф'„) = Я(Ф0). П-*- ~ From a practical viewpoint, (5) is the only essential assumption in Theorems 1 and 2. It confines us to input signals having a density bounded from zero. It is clear that it can be satisfied only if the space % has a finite measure. If, however Ф0(х) f(x) is square integrable, (5) can be neglected. In this case Theorem 3. lfT0(x)f(x) £L2, and (4), (6), (7) hold, then lim E j" ((Ф0(х) — Ф'п (x))2f2(x) dx — 0. The theorem can be established in the same way as Theorem 1.