zbMATH — the first resource for mathematics

Topological results of sequences \(\{n_k x\}^\infty_{k=1}\) and their applications in the theory of trigonometric series. (English) Zbl 0924.40006
All results of this paper are based on Theorem 1.1 which describes from the topological point of view the behaviour of fractional parts of numbers \(n_kx\) \((k=1,2,\dots)\), where \(x\in\mathbb{R}\) and \(\{n_k\}^\infty_{k=1}\) is a given sequence of positive integers. In the second part of the paper we give a new proof of the categorical analogue of the well-known theorem of Cantor and Lebesgue.
40J05 Summability in abstract structures
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Full Text: EuDML
[1] N.K. Bari: Trigonometric series. Gos. Izd. Fiz.-Mat. Lit., Moskva, 1961.
[2] B. K. Bel’nov: Some Theorems on distribution of fractional parts. Sibirsk. Mat. Ž. 18 (1977), 512-521. · Zbl 0355.10028
[3] Z. Bukovská: Thin sets in trigonometrical series and quasinormal convergence. Math. Slov. 40 (1990), 53-62. · Zbl 0733.43003
[4] G.H. Hardy, J.E. Littlewood: Some problems on diophantine approximation. Acta Math. 37 (1913), 155-239. · JFM 45.0305.03
[5] S. Kahane: Ideaux de compacts et applications a l’analyse harmonique. Thèse de doctorat, Paris, 1990.
[6] L. Kuipers - H. Niederreiter: Uniform Distribution of Sequences. John Wiley, New York-London-Sydney-Toronto, 1974. · Zbl 0281.10001
[7] C. Kuratowski: Topologie I. PWN, Warszawa, 1958. · Zbl 0078.14603
[8] Th. E. Mott: Some generalizations of the Cantor-Lebesgue theorem. Math. Ann. 152 (1963), 95-119. · Zbl 0119.28902
[9] W. Orlicz: Une generalisation d’un thèoreme de Cantor-Lebesgue. Ann. Soc. Pol. Math. 21 (1948), 38-45. · Zbl 0032.19902
[10] I.P. Natanson: Theory of functions of real variable. Nauka, Moskva, 1974.
[11] W.H. Young: A note on trigonometrical series. Messenger of Math. 38 (1909), 44-48.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.