The error in discrete \(\Phi\)-approximation
Let \(\phi:\left[ 0,\infty\right) \to\left[ 0,\infty\right) \) be a convex function (\(\phi\left( 0\right) =0,\phi\left( x\right) >0\) for \(x>0\)) and \(\left( \Omega,\mathcal{A},\mu\right)\) a space with a finite measure \(\mu\), where \(\Omega={\mathbf N}\) or a finite part of \(\mathbf{N}\) and \(\mathcal{A}=2^{\Omega}\). Let \(L_{\phi}\left( \Omega,\mathcal{A},\mu\right) =\left\{ f:f \mathcal{A}-\text{measurable such that }\exists \alpha>0\int\phi\left( \alpha\left|f\right|\right) d\mu<\infty\right\}\) and \(L_{\phi}^{\infty}(\Omega,\mathcal{A},\mu) =\{ f:f \mathcal{A}-\text{ measurable and }\int_{\Omega}\phi\left( \alpha\left|f\right|\right) d\mu<\infty\text{ for all }\alpha>0\}.\) If \(C\) is a lattice included in \(L_{\phi}\left( \Omega,\mathcal{A},\mu\right) \), nonvoid and \(\phi\)-closed, an element \(g\in C\) is called a best \(\phi\)-approximation element for \(f\in L_{\phi}\) if \(\int_{\Omega}\phi\left( \left|f-g\right|\right) d\mu=\inf\limits_{h\in C}\int_{\Omega}\phi\left( \left|f-h\right|\right) d\mu\). The set of all best \(\phi\)-approximation elements is denoted by \(M_{\phi,C}\left( f\right).\) Let \(W\) be the lattice of all non-decreasing sequences in \(L_{\phi}\) and let \(f\in D_{\phi}\left( W\right) =\{ f\in L_{\phi}:\sum_{k=1}^{\infty }\phi\left(\left|f_{k}-g_{k}\right|\right) \alpha_{k}<\infty\text{ for some }g\in W\} \), where \(f:\mathbf{N}\to\mathbf{R}\), \(f_{k}=f\left( k\right) \), \(\alpha_{k}=\mu\left( \left\{ \alpha_{k}\right\} \right) \), \(k\in\mathbf{N}.\) Let \(f^{\ast}=\inf\left\{ g:g\in M_{\phi,W}\left( f\right) \right\} \), \[ E^{\ast}=\inf\limits_{h\in W}\sum\limits_{k=1}^{\infty}\phi\left( \left|f_{k}-h_{k}\right|\right) \alpha_{k}=\sum\limits_{k=1}^{\infty}\phi\left( \left|f_{k}-f_{k}^{\ast}\right|\right) \alpha_{k} \] and \[ E_{n}=\inf\limits_{h\in M_{n}}\sum\limits_{k=1}^{n}\phi\left(\left|f_{k}-h_{k}\right|\right) \alpha_{k}\text{,} \] where \(M_{n}=\left\{ h\in\mathbb{R}^{n}:h_{k}\leq h_{k+1},k=1,2,...\right\} .\) The author shows that if \(\gamma=\left( \gamma_{k}\right) _{k\geq 1}\in M_{\phi,W}\left( f\right) \) then \(f_{k}^{n}\downarrow\gamma_{k}\) for every \(k\in\mathbf{N}\). Also, \(E_{n}\leq E_{n+1}\), \(E_{n}\leq E^{\ast}\) and \(E^{\ast}=\lim\limits_{h\to\infty}E_{n}\). The author presents an algorithm for the estimation of the error of \(\phi\)-approximation in \(L_{\phi}^{\infty}\) by nondecreasing functions.
scientific article; zbMATH DE number 1755292
scientific article