Approximation of elements of a generalized Orlicz sequence space by nonlinear, singular kernels (Q1108518)

Let X a real vector space and \(\rho\) (x) be an s-convex modular in X. Let \(X_{\rho}=\{x\in X;\lim_{a\to 0_+}\phi (ax)=0\}\) be the modular space generated by \(\rho\). Then \(\| x\|_{\rho}\inf \{u>0;\rho (x/u^{1/s})\leq 1\}\) is an s-homogeneous F-norm in \(X_{\rho}\) and \(\| x_ n-x\|_{\rho}\to^{n}0\) is equivalent to the condition \(\rho (a(x_ n-x))\to^{n}0\) for every \(a>0\) and is denoted by \(x_ n\to x\). Define \(x_ n\to^{\rho}x\) in \(X_{\rho}\) by the condition \(\rho (a(x_ n-x))\to^{n}0\) for some \(a>0\) depending on \((x_ n).\) If W is a set of indices and M is a filter of subsets of W, then norm- convergence \(x_ n\to^{M}x\) and \(\rho\)-convergence \(x_ w\to^{\rho,M}K\) are defined analogously. Define \(K_{w,j}: {\mathbb{R}}_+\to {\mathbb{R}}_+\) \(j=0,1,..\). for each \(w\in W\), with \(K_{w,j}(0)=0\) for each w and j, and, for every sequence \(x=(t_ j)_ 0^{\infty}\) define the operators: \((T_ wx)_ i=\sum^{i}_{g=0}K_{w,i-j}(| t_ j|)\) and \(T_ wx=((T_ wx)_ i)^{\infty}_{i=0}\). The kernel \(K=(K_{w,j})^{\infty}_{j=0}\), \(w\in W\) is called singular if \((1/c)K_{w,0}(c)\to^{M}1\) for every \(c>0\). Let X be the space of all real sequences and \(\rho (x)=\sum^{\infty}_{i=1}\phi_ i(| t_ i|)\) where \((\phi_ i)_ 0^{\infty}\) is a sequence of nonnegative s-convex functions. The respective modular space \(\ell^{\phi}=X_{\rho}\) is called the generalized Orlicz sequence space. The sequence \((\phi_ i)_ 0^{\infty}\) is called \(\tau_+\)- bounded if there exist \(k_ 1\) and \(k_ 2\geq 1\) and a double-sequence \((\epsilon_{i,j}),\epsilon_{i,j}\geq 0,\epsilon_ i=0,\epsilon_ j=\sum^{\infty}_{i=0}\epsilon_{i,j}\to 0\) as \(j\to \infty \sup_{j\geq 0}\epsilon_ j<\infty\), such that \(\phi_{i+j}(u)\leq k_ 1\phi_ i(k_ 2U)+\epsilon_{i,j}\) for \(U\geq 0\); \(i,j=0,1,2,..\). If \((\phi_ i)\) is \(\rho_+\)-bounded and s-convex \(0<s\leq 1\) and K is a singular kernel such that \(\rho (bx_ w^ j)\to^{M}0\) for \(j=0,1,2,..\). and for all \(b>0\), then \(T_ wx\to^{\rho,M}x\) for each \(x\in \ell^{\phi}\).