4.5.3 ANALYSIS OF EUCLID S ALGORITHM 347 Gauss s problem, (Free web hosting services)
4.5.3 ANALYSIS OF EUCLID S ALGORITHM 347 Gauss s problem, namely to find the asymptotic behavior of Fn(x) -lg(1 + x), was not really resolved until 1974, when Eduard Wirsing published a beautiful analysis of the situation [Acta Arithmetica 24 (1974), 507-5281. We shall study the simplest aspects of Wirsing s approach here, since his method is an instructive use of linear operators. If G is any function of x defined for 0 5 x 5 1, let SG be the function defined by -(25) SG(x) = kg (G(i) - G( &))- Thus, S is an operator that changes one function into another. In particular, by (23) we have F,+l(x) = SF,(x), hence F, = SnFo. (26) (In this discussion F, stands for a distribution function, not for a Fibonacci number.) Note that S is a linear operator ; i.e., S(cG) = c(SG) for all con- stants c, and S(G1 + Ga) = SG1 + SG2. Now if G has a bounded first derivative, we can differentiate (25) term by term to show that (SG) (x)= k-l (k ; x)2 G( &): (27) - hence SG also has a bounded first derivative. (Term-by-term differentiation of a convergent series is justified when the series of derivatives is uniformly convergent; cf. K. Knopp, Theory and Application of Infinite series (Glasgow: Blackie, 1951), $47.) Let H = SG, and let g(x) = (1 + x)G (x), h(x) = (1 + x)H (x). It follows that h(x)=c -1+x (1+&o) k>l (k+xY - In other words, h = Tg, where T is the linear operator defined by (28)