$L^{1}$-convergence of Fourier series with complex quasimonotone coefficients

Abstract:

A sequence of Fourier coefficients $\left \{ {\hat f(n)} \right \}$ of a complex function in ${L^1}(T)$ is said to be complex quasimonotone if there exists ${\theta _0}$ such that \[ \Delta \hat f(n) + \frac {\alpha } {n}\hat f(n) \in \left \{ {z|\left | {\arg z} \right | \leqslant {\theta _0} < \frac {\pi } {2}} \right \}\] for some $\alpha \geqslant 0$ and for all $n$. It is proved that Fourier series with asymptotically even and complex quasimonotone coefficients, satisfying \[ \overline {\lim \limits _{n \to \infty } } \;{n^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\Delta \hat f(j)} \right |^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\hat f(j)} \right |^{1/p}} = o(1),\; \lambda \to 1 + 0,\tfrac {1} {p} + \tfrac {1} {q} = 1, \] converges in ${L^1}(T)$-norm if and only if $\hat f(n)\lg \left | n \right | = o(1)$, $n \to \infty$. A recent result of Č V. Stanojević [3] is a special case of the corollary of the main theorem.