Functions of generalized Wiener classes \(\operatorname {BV}(p(n)\uparrow\infty,\varphi)\) and their Fourier coefficients (Q2702992)

The \(2\pi\)-periodic function \(f\) belongs to generalized Wiener class \(\text{BV}(p(n)\uparrow\infty,\varphi)\) defined by non-decreasing sequences \(p(n)\uparrow\infty\) and \(\varphi(n)\uparrow\infty\), if NEWLINE\[NEWLINE V(f,p,\varphi)=\sup_n\sup_{T(n)}\left[\sum_{k=1}^{m}\left|f(t_{k})- f(t_{k-1})\right|^{p(n)}\right]^{1/p(n)}<\infty, NEWLINE\]NEWLINE where \(T(n)=\{0=t_{0}<t_{1}<\dotsb<t_{m}=2\pi\), \(|t_{k}-t_{k-1}|\geq 2\pi[\varphi(n)]^{-1}\}\). The main result (Theorem 1) is: if the function \(f(x)\sim\frac 12a_0+\sum_{n=1}^\infty (a_{n}\cos nx+b_n\sin nx)\) belongs to \(\text{BV}(p(n)\uparrow\infty,\varphi)\), then \(\max\{|a_{n}|,|b_{n}|\}\leq 20\pi^{-1}V(f,p,\varphi)n^{-1/p(\tau(n))}\), where \(\tau(r)=\min\{m:m\in N,\varphi(m)\geq r\}, r\geq 1\), and this estimate is precise (Theorem 2).