On the convergence of $ω$@-primitives

Rok, strany: 2003, 687 - 702

We continue the series of research on oscillations of functions [DUSZYŃSKI, Z.—GRANDE, Z.—PONOMAREV, S. P.: On the $ω$@-primitive, Math. Slovaca 51 (2001), 469–476], [EWERT, J.—PONOMAREV, S. P.: Oscillation and $ω$@-primitives, Real Anal. Exchange 26 (2000/2001), 687–702]. In the first part we consider first-countable topological spaces $X$ satisfying some neighborhood conditions (weak enough to imply the metrizability of $X$) and show that given a sequence of upper semicontinuous functions $f_n: X\to [0, ∞)$ converging to an upper semicontinuous function $f:X\to [0,∞)$, there exist functions $F_n, F :X\to [0,∞)$, $n\in \Bbb N$, such that $ω(F_n,·)=f_n$, $n\in\Bbb N$, $ω (F,·)=f$ and $F_n\to F$ in the same sense as $f_n\to f$. By $ω (g,x)$ we denote the oscillation of $g:X\to \Bbb R$ at $x$. Quite different technique had to be employed in the second part of the paper where the analogous result is proved for $X=\Bbb Rⁿ$ equipped with the usual density topology $τ_d$.

ISO 690:

Ewert, J., Ponomarev, S. 2003. On the convergence of $ω$@-primitives. In Mathematica Slovaca, vol. 53, no.1, pp. 687-702. 0139-9918.

APA:

Ewert, J., Ponomarev, S. (2003). On the convergence of $ω$@-primitives. Mathematica Slovaca, 53(1), 687-702. 0139-9918.