Saturation problem for Fejér sum of Neumann-Bessel series (Q2713057)

Let \(\{J_n\}\), \(\{O_n\}\) be sequences of Bessel functions and Neumann polynomials respectively, \(\Gamma\) the unit circle, \(\|\cdot\|\) the supremum norm. For \(f\in L(\Gamma)\), NEWLINE\[NEWLINEA_n(f)= {\varepsilon_n \over\pi i} \oint_\Gamma f O_n d\zeta,\quad B_n(f)= {\varepsilon_n \over\pi i} \oint_\Gamma f J_n d\zeta,\quad \varepsilon_0= {1\over 2},\quad \varepsilon_n=1 (n\geq 1).NEWLINE\]NEWLINE By \(\sigma_n^{(N,B)} (f)\) denote the Fejér sum of \(N-B\) series: \(\sum^\infty_0 (A_nJ_n+ B_nO_n)\). Let NEWLINE\[NEWLINE\overline f(z)= {1\over\pi i}\text{ (P.V.) }\oint_\Gamma {f(\zeta) \over\zeta-z} d\zeta, \quad z\in\Gamma.NEWLINE\]NEWLINE Theorem 1. If \(f\in C(\Gamma)\) and \(\|\sigma_n^{(N,B)}(f)-f\|_\Gamma= o({1\over n})\), then \(f(z)=C J_0(z)+DO_0(z)\), \(C,D\)-constants. Theorem 2. Let \(f\in C(\Gamma)\). If \(\overline f\in\text{lip} 1\), \(z\in\Gamma\), then \(\|\sigma_n^{(N,B)}(f)- f\|_\Gamma= O({1\over n})\). Under the condition: \(|{1\over \pi i}\int_{\Gamma-L_\varepsilon (z)}{f(\zeta) \over\zeta-z}d\zeta |\leq M\), \(z\in\Gamma\), \(0\leq \varepsilon\leq 1\). (Where \(M\) is a constant, \(L_\varepsilon (z)\) is the arc: \(\zeta=e^{is} (\theta-\varepsilon\leq s\leq\theta+ \varepsilon)\), \(z=e^{i\theta}\) the inverse of Theorem 2 is valid.