Deux résultats sur les suites limite-périodiques. (Two results on limit-periodic sequences)
First the author proves the following result. If f is an integrable function in the sense of Riemann and a(n) a Besicovič-limit-periodic sequence, then the convolution limit \[ \lim_{N\to \infty}(1/N)\sum_{n\leq N}f(n/N) a(n)\quad = \] \[ =\quad (\lim_{N\to \infty}(1/N)\sum_{n\leq N}f(n/N)) (\lim_{N\to \infty}(1/N)\sum_{n\leq N}a(n)) = \int^{1}_{0}f(t)dt \cdot \int_{G}A dm \] exists (for the definition of G, dm and \(A\in {\mathcal L}(G,dm)\) see the author's book ''Intégration et théorie des nombres'' (Paris 1986), generalizing a result of \textit{C. Radoux} [Ann. Soc. Sci. Brux., Sér. I 91, 13-18 (1977; Zbl 0347.41022)], where a(n) was Euler's function. For the proof, the function a is approximated by certain partial sums of its Fourier series. Secondly, the author answers a problem, posed by the reviewer, in a survey lecture at the Banach Center 1982 [see Banach Cent. Publ. 17, 463- 498 (1985; Zbl 0598.10049)] on Hankel determinants.
scientific article; zbMATH DE number 4008507