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A contribution to the theory of divergent sequences. (English) Zbl 0031.29501

40C05Matrix methods in summability
[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 $$mathop {lim }limits_{x to infty } sumlimits_0 + infty } {Kleft( {x,t} right)fleft( t right)dt = s} $$ sums everyalmost periodic function f(t) of a real argument <t<+to its mean value (II) exactly if the condition of “asymtotic orthogonality” $$mathop {lim }limits_{x to infty } sumlimits_0 + infty } {Kleft( {x,t} right)mathop {cos }limits_{sin } lambda tdt - oleft( {lambda real ne o} right) } $$ is fulfilled. This is certainly the case, if the kernelK(x, t) is equally distributed in the sense that $$mathop {lim }limits_{x to infty } intlimits_E {Kleft( {x,t} right)dt = delta left( E right)} $$ 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.
[11]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.
[12]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 $$sumlimits_nu {left| {a’_{mu nu } } right|} = o$$ holds.
[13]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.
[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 02723425 · doi:10.1007/BF01378337