# Methods of summability and uniform distribution mod 1

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## Abstract

Methods of summability and uniform distribution mod 1 COMPOSITIO MATHEMATICA JOHANNCIGLER Methods of summability and uniform distributionmod 1 Compositio Mathematica, tome 16 (1964), p. 44-51. <http://www.numdam.org/item?id=CM_1964__16__44_0> © Foundation Compositio Mathematica, 1964, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 44 Methods of summability and uniform distribution mod 1 * by Johann Cigler Let {xn} be a sequence of real numbers. If it has a distribution function z(x), i.e. if exists for each continuous function f(x) with period 1, then its asymptotic behavior is characterized to a certain extent by z(x). If no distribution function exists, one may ask for other ways of characterizing the asymptotic behavior of the sequence {xn}. One possibility consists in replacing the arithmetic mean in the definition of uniform distribution by other summability methods. This is what we propose to do in the sequel. The first researches in this direction have been taken up by M. Tsuji [12]. He considered weighted means (M, 03BBn). As is well- known, one says that a sequence s" is summable (M, Ân) to a limit s, if limN~~ (In:;;N 03BBnsn/03A3n~N 03BBn) = s. With respect to questions of uniform distribution one is thus led to the following definition: A sequence {xn} has the (M, A.)-distribution function z(x) mod 1, if for every continuous function f(x) with period 1 the sequence f(xn) is summable (M, 03BBn) to the value fô f(x)dz(x). If z(x) = x, then one calls the sequence {xn} (M, 03BBn)-un

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