Convergence in variation and rate of approximation for nonlinear integral operators of convolution type
Let \(K_w:\mathbb R\times \mathbb R\to \mathbb R\), \(w\geq 0\), where \(K_w(t,u)=L_w(t)H_w(u)\), \(H_w:\mathbb R\to\mathbb R\) have property \(H_w(0)=0\) and \(L_w:\mathbb R\to\mathbb R\) is a \(2\pi\)-periodic function. We also assume 1) \(| H_w(u)-H_w(v)| \leq K| u-v| \) for all \(u,v\in\mathbb R\); 2) \(L_w\in L^1_{2\pi}\), \(\| L_w\| _1\) are bounded and \(\lim_{w\to +\infty}A_w:=\)\(\lim_{w\to +\infty}\int^{\pi}_{-\pi}L_w(t)\,dt=1\); 3) For every \(\delta\in (0,\pi)\) there holds \(\lim_{w\to +\infty}B(\delta,w):=\) \(\int_{\delta\leq | t| \leq\pi}| L_w(t)| \,dt=0\); 4) \(V_J(H_w(u)-u)\to 0\) as \(w\to +\infty\) for every bounded interval \(J\subset \mathbb R\). Here \(V_I(f)\) is the variation of \(f\) on \(I\). Let introduce \((T_wf)(s)=\int^{\pi}_{-\pi}K_w(t, f(s-t))\,dt\), \(w>0\), \(s\in\mathbb R\), for \(f\in BV_{2\pi}\). Theorem 1. If \(f\in AC_{2\pi}\) and \(K_w\), \(w>0\), satisfy 1)-4), then \(\lim_{w\to +\infty}V_{[0,2\pi]}(T_wf-f)=0\). Let \(\xi:\mathbb R^+_0\to \mathbb R^+_0\) be even and continuous at \(u=0\), \(\xi(0)=0\) and \(\xi(u)>0\) for \(u>0\). We say that \(L_w\) is a \(\xi\)-singular kernel if (a) \((A_w-1)=O(\xi(w^{-1}))\) as \(w\to +\infty\); (b) \(B(\delta,w)=O(\xi(w^{-1}))\), \(w\to+\infty\), for every \(\delta\in (0,\pi)\). Further \(\tau(t)\) has same properties as \(\xi(t)\) and \(V\operatorname{Lip}(H,\tau)=\{f\in AC_{2\pi}: V_{[0,2\pi]}(\Delta_t(H_w(f))\leq K\tau(t)\}\), where \(K\) is independent of \(w\). Theorem 2. Let \(K_w\) satisfy 1)-4), \(L_w\) being \(\xi\)-singular, \(f\in V\operatorname{Lip}_{2\pi}(H,\tau)\), \(V_j(H_w(u)-u)=O(\xi(w^{-1}))\) for \(w\to+\infty\) and bounded \(J\subset\mathbb R\). If there exists \(\delta>0\) such that \[ \int_{0\leq| t| \leq\delta}| L_w(t)| \tau(t)\,dt=O(\xi(w^{-1})), \] then \(V_{[0,2\pi]}(T_wf-f)=O(\xi(w^{-1}))\). Similar multidimensional results are proved.
scientific article; zbMATH DE number 5117884