# A contribution to the theory of divergent sequences

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## Keywords

Natural Number Periodic Function Matrix Method Bounded Sequence Convergent Sequence
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## References

- 2.Cf.Banach, Théorie des opérations linéaires, Warszawa 1932, p. 33–34.Google Scholar
- 1.Cf.Banach, op. cit. p. 32.Google Scholar
- 1.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.Google Scholar
- 1.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.Google Scholar - 1.For a complex sequence
*z*_{n}=*x*_{n}+*iy*_{n}we define Lim*z*_{n}by the aid of (8) or put Lim*z*_{n}= Lim*x*_{n}+*i*Lim*y*_{n}.Google Scholar - 1.A similar definition, where
*x*_{n}is defined for all −∞<*n*<+∞ is given byA. Walther, Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koeffizienten, Hamburger Abh, 6 (1928), p. 217–234.Google Scholar - 2.Cf. for exampleH. Bohr, Fastperiodische Funktionen, Ergebn, der Math. Berlin 1932, p. 34–38.Google Scholar
- 1.Deutsche Mathematik,
*3*(1938), 390–402.Google Scholar - 1.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.Google Scholar
- 1.We shall not treat the similar problem of the regularity of a method with respect to all
*almost periodic*sequences. A regular method for functions*f(t)*which has the form\(\mathop {\lim }\limits_{x \to \infty } \sum\limits_0^{ + \infty } {K\left( {x,t} \right)f\left( t \right)dt = s} \) sums every*almost periodic function f(t)*of a real argument −∞<*t*<+∞ to its mean value (II) exactly if the condition of «asymtotic orthogonality»\(\mathop {\lim }\limits_{x \to \infty } \sum\limits_0^{ + \infty } {K\left( {x,t} \right)\mathop {\cos }\limits_{\sin } \lambda tdt - o\left( {\lambda real \ne o} \right) } \) is fulfilled. This is certainly the case, if the kernel*K(x, t)*is equally distributed in the sense that\(\mathop {\lim }\limits_{x \to \infty } \int\limits_E {K\left( {x,t} \right)dt = \delta \left( E \right)} \) holds for every measurable set*E*<(0, +∞), for which the density in the interval (0, +∞), viz.\(\delta \left( E \right) = \mathop {\lim }\limits_{n \to \infty } \frac{I}{n}\) meas {*E*·(o,*n*)} has a sense.Google Scholar - 1.It may be remarked here that the methods of class\(\mathfrak{A}\) 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\(\mathfrak{A}\). Cf. alsoR. P. Agnew, Bull. Americ. Math. Soc.*52*, 128–132 (1946), where a special case of our theorem 8 is proved.Google Scholar - 1.I. Schur, Journ. für reine und angew. Math.
*151*(1921), 79–111 Theorem III. According to this theorem a regular method\(A' = \left\| {a'_{\mu \nu } } \right\|\) with elements*a*_{μv}^{′}converging to zero for μ→∞ sums all bounded sequences exactly when lim\(\sum\limits_\nu {\left| {a'_{\mu \nu } } \right|} = o\) holds.Google Scholar - 1.We assume that Ω(
*n*) does not alter more than by 1 in every interval of the length 1, a natural condition as a density ω(*n*) always has this property.Google Scholar - 1.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.Google Scholar
- 2.F. Hausdorff, Summationsmethoden und Momentenfolgen, Math. Zeitschrift
*9*(1921), 74–109.CrossRefMathSciNetGoogle Scholar

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© Almqvist & Wiksells Boktryckeri 1948