Facebook Instagram Twitter RSS Feed Back to top

$L2$ space and $g$-calculus with applications

In: Tatra Mountains Mathematical Publications, vol. 34, no. 3
Katarína Lendelová
Rok, strany: 2006, 243 - 254
O článku:
J. Desatní ková [$Lp$ spaces and $g$-calculus, student scientific work, FPV UMB, Banská Bystrica, 2004] studied $L1$, $L2$ spaces with respect to the Pap's $g$-calculus (see [E. Pap: {sl Null-Additive Set Functions}, Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava, 1995]). She defined $(Loplus1, doplus1)$, $(Loplus2, doplus2)$ spaces and proved that they are complete $oplus$-pseudometric spaces, where $doplus1(f,h)=int olimitsoplus {|fominus h|oplus dP}$ and $doplus2(f,h)=ig(int olimitsoplus {(fominus h)2oplus dP}ig)((1) / (2))oplus$. The aim of this paper is to show a relation between $L2$ space and $Loplus2$ space. We define a mapping $(·,·)oplus:Loplus2 × Loplus2 ightarrow Bbb{R}$ satisfying the following conditions: oster item"(i)" $(f,f)oplus geqslant 0$ for each $fin Loplus2$, smallskip item"(ii)" $(f,h)oplus=(h,f)oplus$ for each $f,hin Loplus2$, smallskip item"(iii)" $(łambda odot f,h)oplus=łambda odot(f,h)oplus$ for each $f,hin Loplus2$ and $łambda in Bbb{R}$, smallskip item"(iv)" $(f,h1 oplus h2)oplus=(f,h1)oplus oplus (f,h2)oplus$ for each $f,h1, h2in Loplus2$. endroster We prove the Cauchy inequality for $Loplus2$ space and the Chebyshev inequality for pseudo-probability $P$ defined by L. M. Nedovi'c and T. Grbi'c: [The pseudo probability, Journal of Electrical Engineering 53 (2002), 27–31]. We show the applications of $Loplus2$ space like a pseudo-dispersion $σ2oplus$ and pseudo-mean value $Eoplus$ in a pseudo-probability.
Ako citovať:
ISO 690:
Lendelová, K. 2006. $L2$ space and $g$-calculus with applications. In Tatra Mountains Mathematical Publications, vol. 34, no.3, pp. 243-254. 1210-3195.

Lendelová, K. (2006). $L2$ space and $g$-calculus with applications. Tatra Mountains Mathematical Publications, 34(3), 243-254. 1210-3195.