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

MSC:
 40C05 Matrix methods in summability
Keywords:
Series, summability
Full Text:
References:
 [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 $$\backslashmathop {\backslashlim }\backslashlimits_{x \backslashto \backslashinfty } \backslashsum\backslashlimits_0{ + \backslashinfty } {K\backslashleft( {x,t} \backslashright)f\backslashleft( t \backslashright)dt = s}$$ sums everyalmost periodic function f(t) of a real argument <t<+to its mean value (II) exactly if the condition of “asymtotic orthogonality{” $$\backslashmathop {\backslashlim }\backslashlimits_{x \backslashto \backslashinfty } \backslashsum\backslashlimits_0{ + \backslashinfty } {K\backslashleft( {x,t} \backslashright)\backslashmathop {\backslashcos }\backslashlimits_{\backslashsin } \backslashlambda tdt - o\backslashleft( {\backslashlambda real \backslashne o} \backslashright) }$$ is fulfilled. This is certainly the case, if the kernelK(x, t) is equally distributed in the sense that $$\backslashmathop {\backslashlim }\backslashlimits_{x \backslashto \backslashinfty } \backslashint\backslashlimits_E {K\backslashleft( {x,t} \backslashright)dt = \backslashdelta \backslashleft( E \backslashright)}$$ holds for every measurable setE<(0, +, for which the density in the interval (0, +, viz. $$\backslashdelta \backslashleft( E \backslashright) = \backslashmathop {\backslashlim }\backslashlimits_{n \backslashto \backslashinfty } \backslashfrac{I}{n}$$ meas {E{$\cdot$}(o,n)} has a sense. [11] It may be remarked here that the methods of class $$\backslashmathfrak{A}$$ 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 $$\backslashmathfrak{A}$$ . 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 $$A’ = \backslashleft\backslash| {a’_{\backslashmu \backslashnu } } \backslashright\backslash|$$ with elementsa {$\mu$}v ’ converging to zero for {$\mu$} sums all bounded sequences exactly when lim $$\backslashsum\backslashlimits_\backslashnu {\backslashleft| {a’_{\backslashmu \backslashnu } } \backslashright|} = o$$ 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