Fourier coefficients of piecewise-monotone functions from the class \(\text{Lip}(\alpha,p)\) (Q1275999)

In the paper are considered \(2\pi\)-periodic piecewise monotone functions integrable on the interval \((0,2\pi)\). The set of all such functions is denoted by \(KM\). The main results of the paper are as follows: If \(f\in KM\), \(0<\alpha<p^{-1}\), \(p\geq 1\), then \(f\in\text{Lip}(\alpha,p)\Leftrightarrow | c_n(f)| = O(| n| ^{1/p-1-\alpha})\), \(n\to\infty\). If \(f\in KM\cap\text{Lip}(\alpha,p)\), \(1\geq\alpha>p^{-1}\), \(1<p\leq\infty\), then \(| c_n(f)| = O(| n| ^{-1})\), \(n\to\infty\). If \(1>\alpha>p^{-1}\), \(1<p\leq\infty\), then a function \( f\in KM\cap\text{Lip}(\alpha,p)\) is found such that \(| c_n(f)| \neq o(| n| ^{-1})\), \(n\to\infty\).