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On $φ$-convergence and $φ$-density

In: Mathematica Slovaca, vol. 55, no. 3
Eugen Kováč
Detaily:
Rok, strany: 2005, 329 - 351
O článku:
We study $φ$-convergence (a special type of summability method introduced in [SCHOENBERG, I. J.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375]), $φ$-density of subsets of integers (which is equivalent to the $φ$-convergence of the set's indicator function) and $\frak Iφ$-convergence (convergence according to the ideal of all sets with $φ$-density zero in the sense as defined in [KOSTYRKO, P. — ŠALÁT, T.—WILCIŃSKY, W.: $I$-convergence, Real Anal. Exchange 26 (2000-01), 669–686]). We analyze the relation of $φ$-density and other types od densities, in particular asymptotic, logarithmic, and uniform density. We prove the following properties: \roster \item"$\bullet$" $φ$-density can attain only values $0$ and $1$ (whenever it exists). \item"$\bullet$" If $φ$-density exists for a set, then asymptotic and logarithmic densities also exist and attain the same value. \item"$\bullet$" There is a set with $φ$-density zero which does not have uniform density. \item"$\bullet$" There is a sequence which is $φ$-convergent, but the sequence of its absolute values is not. \item"$\bullet$" $\frak Iφ$-convergence is strictly weaker than $φ$-convergence. \endroster
Ako citovať:
ISO 690:
Kováč, E. 2005. On $φ$-convergence and $φ$-density. In Mathematica Slovaca, vol. 55, no.3, pp. 329-351. 0139-9918.

APA:
Kováč, E. (2005). On $φ$-convergence and $φ$-density. Mathematica Slovaca, 55(3), 329-351. 0139-9918.