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A contribution to the theory of divergent sequences. (English) Zbl 0031.29501
Reviewer: G. Köthe (Mainz)

40C05Matrix methods in summability
Full Text: DOI
[1] Cf.Banach, Théorie des opérations linéaires, Warszawa 1932, p. 33--34.
[2] Cf.Banach, op. cit. p. 32.
[3] We here make use of the known construction which leads to the proof of the theorem about the continuation of a linear functional, cf.Banach op. cit. p. 27.
[4] It is no use to try to generalize by aid of (8) the notion of almost convergence also for unbounded sequences. In fact it easily follows from (8) that the sequence {x n} is bounded.
[5] For a complex sequencez n=x n+iy n we define Limz n by the aid of (8) or put Limz n= Limx n+i Limy n.
[6] A similar definition, wherex n is defined for all <n<+is given byA. Walther, Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koeffizienten, Hamburger Abh, 6 (1928), p. 217--234.
[7] Cf. for exampleH. Bohr, Fastperiodische Funktionen, Ergebn, der Math. Berlin 1932, p. 34--38.
[8] Deutsche Mathematik,3 (1938), 390--402.
[9] The author gave similar Tauberian theorems for the methods of Cesàro and Abel in a paper “Tauberian theorems and Tauberian conditions{” which is to appear in the Transactions Americ. Math. Soc.
[10] We shall not treat the similar problem of the regularity of a method with respect to allalmost periodic sequences. A regular method for functionsf(t) which has the form $$$\backslash$mathop {$\backslash$lim }$\backslash$limits_{x $\backslash$to $\backslash$infty } $\backslash$sum$\backslash$limits_0{ + $\backslash$infty } {K$\backslash$left( {x,t} $\backslash$right)f$\backslash$left( t $\backslash$right)dt = s} $$ sums everyalmost periodic function f(t) of a real argument <t<+to its mean value (II) exactly if the condition of “asymtotic orthogonality{” $$$\backslash$mathop {$\backslash$lim }$\backslash$limits_{x $\backslash$to $\backslash$infty } $\backslash$sum$\backslash$limits_0{ + $\backslash$infty } {K$\backslash$left( {x,t} $\backslash$right)$\backslash$mathop {$\backslash$cos }$\backslash$limits_{$\backslash$sin } $\backslash$lambda tdt - o$\backslash$left( {$\backslash$lambda real $\backslash$ne o} $\backslash$right) } $$ is fulfilled. This is certainly the case, if the kernelK(x, t) is equally distributed in the sense that $$$\backslash$mathop {$\backslash$lim }$\backslash$limits_{x $\backslash$to $\backslash$infty } $\backslash$int$\backslash$limits_E {K$\backslash$left( {x,t} $\backslash$right)dt = $\backslash$delta $\backslash$left( E $\backslash$right)} $$ holds for every measurable setE<(0, +, for which the density in the interval (0, +, viz. $$$\backslash$delta $\backslash$left( E $\backslash$right) = $\backslash$mathop {$\backslash$lim }$\backslash$limits_{n $\backslash$to $\backslash$infty } $\backslash$frac{I}{n}$$ meas {E{$\cdot$}(o,n)} has a sense.
[11] It may be remarked here that the methods of class $$$\backslash$mathfrak{A}$$ have been investigated byD. Menchoff, Bull. Acad. Sci. URSS, Moscou, ser. math.,1937, 203--229. D. Menchoff proves an interesting theorem about the summability of orthogonal series by methods of class $$$\backslash$mathfrak{A}$$ . Cf. alsoR. P. Agnew, Bull. Americ. Math. Soc.52, 128--132 (1946), where a special case of our theorem 8 is proved.
[12] I. Schur, Journ. für reine und angew. Math.151 (1921), 79--111 Theorem III. According to this theorem a regular method $$A’ = $\backslash$left$\backslash$| {a’_{$\backslash$mu $\backslash$nu } } $\backslash$right$\backslash$|$$ with elementsa {$\mu$}v ’ converging to zero for {$\mu$} sums all bounded sequences exactly when lim $$$\backslash$sum$\backslash$limits_$\backslash$nu {$\backslash$left| {a’_{$\backslash$mu $\backslash$nu } } $\backslash$right|} = o$$ holds.
[13] We assume that {$\Omega$}(n) does not alter more than by 1 in every interval of the length 1, a natural condition as a density {$\omega$}(n) always has this property.
[14] Theorem 12 evidently follows also from the known theorem about the condensation of singularities. Cf.Kaczmarz undSteinhaus, Theorie der Orthogonalreihen, Warschau, 1935 p. 24. The proof given above is nothing but a “geometrical{” proof of this theorem.
[15] F. Hausdorff, Summationsmethoden und Momentenfolgen, Math. Zeitschrift9 (1921), 74--109. · Zbl 48.2005.01 · doi:10.1007/BF01378337