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

##### MSC:
 40C05 Matrix methods for summability
summability
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##### 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 $$$$\backslash$$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. · JFM 48.2005.01
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