Approximation with best order by Toeplitz-irregular methods of summability of Fourier series (Q1905336)

Let \(f\) be a continuous \(2\pi\)-periodic function with the Fourier coefficients \(a_k\), \(b_k\) and \(\varepsilon_n=\varepsilon_n(f)\) denote the sequence of best uniform approximations of \(f\) by trigonometric polynomials of order \(n\). It is proved that if there is \(C>0\) independent of \(n\) such that \[ {1\over a_0}+\sum^n_{k=1} \min\Biggl\{{1\over |a_k|},{1\over |b_k|}\Biggr\}\leq {C\over \varepsilon_{n-1}},\quad n=1,2,\dots, \] then \(f\) can be approximated with the best order \(O(\varepsilon_n)\) only by regular summability methods. If this condition fails, then \(f(x)\) can be approximated with the best order \(O(\varepsilon_n)\) both by regular and irregular methods. This result is also valid for approximation in the space \(L^p\), \(1\leq p<\infty\).