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Ein verallgemeinerter Gauss-Kuzmin-Opertor. (A generalized Gauss-Kuzmin operator). (English) Zbl 0709.11040
If \(\alpha_ n\) denotes the nth remainder in the regular continued fraction expansion of \(\alpha =\alpha_ 0\in (0,1)\), and \(m_ n(x)\) is the Lebesgue measure of the set \(\{\alpha:\alpha_ n\leq x\}\), then \(m_ n(x)\to \log_ 2(1+x)\) as \(n\to \infty\), as was stated by Gauss and proved by Kuzmin. The proof entails consideration of the operator G on \(C_ 0[0,1]\), for which \(Gm'_ n(x)=m'_{n+1}(x)\). The present author considers a family of operators \(G_ a\) on \(C_ 0[0,1]\), where \(a\geq 1\) and \(G_ 1=G\). By considering related operators, analogous to those defined by E. A. Wirsing [Acta Arith. 24, 507-528 (1974; Zbl 0283.10032)], the author proves that 1 is the largest eigenvalue of \(G_ a\) and that the corresponding eigenspace is generated by the function \(a/(a+x)\). Typographical errors abound, and the estimate of the constant \(c(p_ 0)\) in Theorem 2 is incorrect, since it depends on the assumption that if \(\int^{1}_{0}p(x)dx=1\), then \(\int^{1}_{0}G_ ap(x)dx=1\), which is false if \(a>1\).

11K50 Metric theory of continued fractions
47A75 Eigenvalue problems for linear operators