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

40C05 Matrix methods for summability
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[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 · doi:10.1007/BF01378337
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