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References
Cf.Banach, Théorie des opérations linéaires, Warszawa 1932, p. 33–34.
Cf.Banach, op. cit. p. 32.
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.
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.
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.
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.
Cf. for exampleH. Bohr, Fastperiodische Funktionen, Ergebn, der Math. Berlin 1932, p. 34–38.
Deutsche Mathematik,3 (1938), 390–402.
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.
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\(\mathop {\lim }\limits_{x \to \infty } \sum\limits_0^{ + \infty } {K\left( {x,t} \right)f\left( t \right)dt = s} \) sums everyalmost periodic function f(t) of a real argument −∞<t<+∞ to its mean value (II) exactly if the condition of «asymtotic orthogonality»\(\mathop {\lim }\limits_{x \to \infty } \sum\limits_0^{ + \infty } {K\left( {x,t} \right)\mathop {\cos }\limits_{\sin } \lambda tdt - o\left( {\lambda real \ne o} \right) } \) is fulfilled. This is certainly the case, if the kernelK(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 setE<(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.
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.
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 elementsa ′μv converging to zero for μ→∞ sums all bounded sequences exactly when lim\(\sum\limits_\nu {\left| {a'_{\mu \nu } } \right|} = o\) holds.
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.
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.
F. Hausdorff, Summationsmethoden und Momentenfolgen, Math. Zeitschrift9 (1921), 74–109.
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Some preliminary results have been published in Zapiski Univ. Leningrad, Math. Ser.12, 30–41 (1941).
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Lorentz, G.G. A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948). https://doi.org/10.1007/BF02393648
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DOI: https://doi.org/10.1007/BF02393648