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

In: Tatra Mountains Mathematical Publications, vol. 34, no. 3
Katarína Lendelová
Detaily:
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.

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